Tim Roughgarden

STOC Conference 2024 Invited Paper

The Computer in the Sky (Keynote)

• Tim Roughgarden

ICML Conference 2023 Conference Paper

Formalizing Preferences Over Runtime Distributions

• Devon R. Graham
• Kevin Leyton-Brown
• Tim Roughgarden

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e. g. , SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm’s expected utility from runtime samples.

NeurIPS Conference 2023 Conference Paper

No-Regret Learning with Unbounded Losses: The Case of Logarithmic Pooling

• Eric Neyman
• Tim Roughgarden

For each of $T$ time steps, $m$ experts report probability distributions over $n$ outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as *logarithmic pooling* -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i. e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains $O(\sqrt{T} \log T)$ expected regret as compared with the best weights in hindsight.

NeurIPS Conference 2023 Conference Paper

Utilitarian Algorithm Configuration

• Devon Graham
• Kevin Leyton-Brown
• Tim Roughgarden

We present the first nontrivial procedure for configuring heuristic algorithms to maximize the utility provided to their end users while also offering theoretical guarantees about performance. Existing procedures seek configurations that minimize expected runtime. However, very recent theoretical work argues that expected runtime minimization fails to capture algorithm designers' preferences. Here we show that the utilitarian objective also confers significant algorithmic benefits. Intuitively, this is because mean runtime is dominated by extremely long runs even when they are incredibly rare; indeed, even when an algorithm never gives rise to such long runs, configuration procedures that provably minimize mean runtime must perform a huge number of experiments to demonstrate this fact. In contrast, utility is bounded and monotonically decreasing in runtime, allowing for meaningful empirical bounds on a configuration's performance. This paper builds on this idea to describe effective and theoretically sound configuration procedures. We prove upper bounds on the runtime of these procedures that are similar to theoretical lower bounds, while also demonstrating their performance empirically.

AAAI Conference 2022 Conference Paper

Strictly Proper Contract Functions Can Be Arbitrage-Free

• Eric Neyman
• Tim Roughgarden

We consider mechanisms for truthfully eliciting probabilistic predictions from a group of experts. The standard approach — using a proper scoring rule to separately reward each expert — is not robust to collusion: experts may collude to misreport their beliefs in a way that guarantees them a larger total reward no matter the eventual outcome. It is a long-standing open question whether there is a truthful elicitation mechanism that makes any such collusion (also called arbitrage) impossible. We resolve this question positively, exhibiting a class of strictly proper arbitrage-free contract functions. These contract functions have two parts: one ensures that the total reward of a coalition of experts depends only on the average of their reports; the other ensures that changing this average report hurts the experts under at least one outcome.

FOCS Conference 2021 Conference Paper

Smoothed Analysis with Adaptive Adversaries

• Nika Haghtalab
• Tim Roughgarden
• Abhishek Shetty

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by a multiplicative factor from the uniform distribution; nature then samples an input from this distribution. This interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm's current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i. e. , an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: Online learning, Online discrepancy and Dispersion in online optimization. We show that in these setting, we can get bounds that match bounds we can get for non-adaptive adversaries.

JMLR Journal 2020 Journal Article

Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

• Rad Niazadeh
• Tim Roughgarden
• Joshua R. Wang

In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular function is also coordinate-wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasi-linear time. Both these results improve upon prior work (Bian et al. 2017; Soma and Yoshida, 2017). Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

TCS Journal 2020 Journal Article

Prior-free multi-unit auctions with ordered bidders

• Sayan Bhattacharya
• Elias Koutsoupias
• Janardhan Kulkarni
• Stefano Leonardi
• Tim Roughgarden
• Xiaoming Xu

Prior-free auctions are robust auctions that assume no distribution over bidders' valuations and provide worst-case (input-by-input) approximation guarantees. In contrast to previous work on this topic, we pursue good prior-free auctions with non-identical bidders. Prior-free auctions can approximate meaningful benchmarks for non-identical bidders only when sufficient qualitative information about the bidder asymmetry is publicly known. We consider digital goods auctions where there is a total ordering of the bidders that is known to the seller, where earlier bidders are in some sense thought to have higher valuations. We use the framework of Hartline and Roughgarden (STOC'08) to define an appropriate revenue benchmark: the maximum revenue that can be obtained from a bid vector using prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. This monotone-price benchmark is always as large as the well-known fixed-price benchmark F ( 2 ), so designing prior-free auctions with good approximation guarantees is only harder. By design, an auction that approximates the monotone-price benchmark satisfies a very strong guarantee: it is, in particular, simultaneously near-optimal for essentially every Bayesian environment in which bidders' valuation distributions have nonincreasing monopoly prices, or in which the distribution of each bidder stochastically dominates that of the next. Even when there is no distribution over bidders' valuations, such an auction still provides a quantifiable input-by-input performance guarantee. In this paper, we design a simple O ( 1 ) -competitive prior-free auction for digital goods with ordered bidders. We also extend the monotone-price benchmark and our O ( 1 ) -competitive prior-free auction to multi-unit settings with limited supply.

NeurIPS Conference 2020 Conference Paper

Smoothed Analysis of Online and Differentially Private Learning

• Nika Haghtalab
• Tim Roughgarden
• Abhishek Shetty

Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al. , 2009, Alon et al. , 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations.

SODA Conference 2020 Conference Paper

The Complexity of Contracts

• Paul Dütting
• Tim Roughgarden
• Inbal Talgam-Cohen

SODA Conference 2019 Conference Paper

Communication Complexity of Discrete Fair Division

• Benjamin Plaut
• Tim Roughgarden

SODA Conference 2018 Conference Paper

Almost Envy-Freeness with General Valuations

• Benjamin Plaut
• Tim Roughgarden

The goal of fair division is to distribute resources among competing players in a “fair” way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.

NeurIPS Conference 2018 Conference Paper

Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

• Rad Niazadeh
• Tim Roughgarden
• Joshua Wang

In this paper we study the fundamental problems of maximizing a continuous non monotone submodular function over a hypercube, with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1/2 approximation algorithm for continuous submodular function maximization; this approximation factor of is the best possible for algorithms that use only polynomially many queries. For the special case of DR-submodular maximization, we provide a faster 1/2-approximation algorithm that runs in (almost) linear time. Both of these results improve upon prior work [Bian et al. , 2017, Soma and Yoshida, 2017, Buchbinder et al. , 2012]. Our first algorithm is a single-pass algorithm that uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm is a faster single-pass algorithm that exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.

NeurIPS Conference 2017 Conference Paper

Online Prediction with Selfish Experts

• Tim Roughgarden
• Okke Schrijvers

We consider the problem of binary prediction with expert advice in settings where experts have agency and seek to maximize their credibility. This paper makes three main contributions. First, it defines a model to reason formally about settings with selfish experts, and demonstrates that ``incentive compatible'' (IC) algorithms are closely related to the design of proper scoring rules. Second, we design IC algorithms with good performance guarantees for the absolute loss function. Third, we give a formal separation between the power of online prediction with selfish experts and online prediction with honest experts by proving lower bounds for both IC and non-IC algorithms. In particular, with selfish experts and the absolute loss function, there is no (randomized) algorithm for online prediction---IC or otherwise---with asymptotically vanishing regret.

JAIR Journal 2017 Journal Article

The Price of Anarchy in Auctions

• Tim Roughgarden
• Vasilis Syrgkanis
• Eva Tardos

This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings. This theory complements traditional economic techniques, which generally focus on exact and optimal solutions and are accordingly limited to relatively stylized settings. We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information and deterministic strategies; extension theorems, which extend such guarantees to randomized strategies, no-regret learning outcomes, and incomplete-information settings; and composition theorems, which extend such guarantees from simpler to more complex auctions. Combining these tools yields tight worst-case approximation guarantees for the equilibria of many widely-used auction formats.

STOC Conference 2017 Invited Paper

Why prices need algorithms (invited talk)

• Tim Roughgarden
• Inbal Talgam-Cohen

FOCS Conference 2016 Conference Paper

On the Communication Complexity of Approximate Fixed Points

• Tim Roughgarden
• Omri Weinstein

We study the two-party communication complexity of finding an approximate Brouwer fixed point of a composition of two Lipschitz functions g o f: [0, 1] n → [0, 1] n, where Alice holds f and Bob holds g. We prove an exponential (in n) lower bound on the deterministic communication complexity of this problem. Our technical approach is to adapt the Raz-McKenzie simulation theorem (FOCS 1999) into geometric settings, thereby "smoothly lifting" the deterministic query lower bound for finding an approximate fixed point (Hirsch, Papadimitriou and Vavasis, Complexity 1989) from the oracle model to the two-party model. Our results also suggest an approach to the well-known open problem of proving strong lower bounds on the communication complexity of computing approximate Nash equilibria. Specifically, we show that a slightly "smoother" version of our fixed-point computation lower bound (by an absolute constant factor) would imply that: The deterministic two-party communication complexity of finding an ∈ = Ω(1/log 2 N)-approximate Nash equilibrium in an N × N bimatrix game (where each player knows only his own payoff matrix) is at least N γ for some constant γ > 0. (In contrast, the nondeterministic communication complexity of this problem is only O(log 6 N)). ; The deterministic (Number-In-Hand) multiparty communication complexity of finding an ∈ = Ω(1)-Nash equilibrium in a k-player constant-action game is at least 2 Ω(k/log k) (while the nondeterministic communication complexity is only O(k)).

STOC Conference 2016 Conference Paper

The price of anarchy in large games

• Michal Feldman
• Nicole Immorlica
• Brendan Lucier
• Tim Roughgarden
• Vasilis Syrgkanis

We present an analysis framework for bounding the price of anarchy (POA) in games that have many players, as in many of the games most pertinent to computer science applications. We use this framework to demonstrate that, in many of the models in which the POA has been studied, the POA in large games is much smaller than the worst-case bound. Our framework also differentiates between mechanisms with similar worst-case performance, such as simultaneous uniform-price auctions and greedy combinatorial auctions, thereby providing new insights about which mechanisms are likely to perform well in realistic settings.

IJCAI Conference 2016 Conference Paper

Why Prices Need Algorithms

• Tim Roughgarden
• Inbal Talgam-Cohen

Computational complexity has already had plenty to say about the computation of economic equilibria [Fischer et al. , 2006; Chen et al. , 2009b; 2009a; Daskalakis et al. , 2009; Papadimitriou and Wilkens, 2011]. However, understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. In this note we survey our main results from [Roughgarden and Talgam-Cohen, 2015], which show that the existence of equilibria in markets is inextricably connected to the computational complexity of related optimization problems, such as revenue or welfare maximization. We demonstrate how this relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare maximization.

ICML Conference 2015 Conference Paper

How Hard is Inference for Structured Prediction?

• Amir Globerson
• Tim Roughgarden
• David A. Sontag
• Cafer Yildirim

Structured prediction tasks in machine learning involve the simultaneous prediction of multiple labels. This is often done by maximizing a score function on the space of labels, which decomposes as a sum of pairwise elements, each depending on two specific labels. The goal of this paper is to develop a theoretical explanation of the empirical effectiveness of heuristic inference algorithms for solving such structured prediction problems. We study the minimum-achievable expected Hamming error in such problems, highlighting the case of 2D grid graphs, which are common in machine vision applications. Our main theorems provide tight upper and lower bounds on this error, as well as a polynomial-time algorithm that achieves the bound.

NeurIPS Conference 2015 Conference Paper

On the Pseudo-Dimension of Nearly Optimal Auctions

• Jamie Morgenstern
• Tim Roughgarden

This paper develops a general approach, rooted in statistical learning theory, to learning an approximately revenue-maximizing auction from data. We introduce t-level auctions to interpolate between simple auctions, such as welfare maximization with reserve prices, and optimal auctions, thereby balancing the competing demands of expressivity and simplicity. We prove that such auctions have small representation error, in the sense that for every product distribution F over bidders’ valuations, there exists a t-level auction with small t and expected revenue close to optimal. We show that the set of t-level auctions has modest pseudo-dimension (for polynomial t) and therefore leads to small learning error. One consequence of our results is that, in arbitrary single-parameter settings, one can learn a mechanism with expected revenue arbitrarily close to optimal from a polynomial number of samples.

FOCS Conference 2014 Conference Paper

Barriers to Near-Optimal Equilibria

• Tim Roughgarden

This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worst-case quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games, welfare-maximization in combinatorial auctions with "black-box" bidder valuations, and welfare-maximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worst-case price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players' utility functions. For this reason, our lower bounds are particularly significant for problems of game design -- ranging from the design of simple combinatorial auctions to the computation of tolls for routing networks -- where the goal is to design a game that has only near-optimal equilibria. For example, our results imply that the simultaneous first-price auction format is optimal among all "simple combinatorial auctions" in several settings.

STOC Conference 2014 Conference Paper

Private matchings and allocations

• Justin Hsu
• Zhiyi Huang 0002
• Aaron Roth 0001
• Tim Roughgarden
• Zhiwei Steven Wu

We consider a private variant of the classical allocation problem : given k goods and n agents with individual, private valuation functions over bundles of goods, how can we partition the goods amongst the agents to maximize social welfare? An important special case is when each agent desires at most one good, and specifies her (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph. Private matching and allocation problems have not been considered in the differential privacy literature, and for good reason: they are plainly impossible to solve under differential privacy. Informally, the allocation must match agents to their preferred goods in order to maximize social welfare, but this preference is exactly what agents wish to hide! Therefore, we consider the problem under the relaxed constraint of joint differential privacy : for any agent i , no coalition of agents excluding i should be able to learn about the valuation function of agent i . In this setting, the full allocation is no longer published---instead, each agent is told what good to get. We first show that with a small number of identical copies of each good, it is possible to efficiently and accurately solve the maximum weight matching problem while guaranteeing joint differential privacy. We then consider the more general allocation problem, when bidder valuations satisfy the gross substitutes condition. Finally, we prove that the allocation problem cannot be solved to non-trivial accuracy under joint differential privacy without requiring multiple copies of each type of good.

STOC Conference 2014 Conference Paper

The sample complexity of revenue maximization

• Richard Cole 0001
• Tim Roughgarden

NeurIPS Conference 2013 Conference Paper

Marginals-to-Models Reducibility

• Tim Roughgarden
• Michael Kearns

We consider a number of classical and new computational problems regarding marginal distributions, and inference in models specifying a full joint distribution. We prove general and efficient reductions between a number of these problems, which demonstrate that algorithmic progress in inference automatically yields progress for “pure data” problems. Our main technique involves formulating the problems as linear programs, and proving that the dual separation oracle for the Ellipsoid Method is provided by the target problem. This technique may be of independent interest in probabilistic inference.

STOC Conference 2012 Conference Paper

Prior-free auctions with ordered bidders

• Stefano Leonardi 0001
• Tim Roughgarden

Prior-free auctions are robust auctions that assume no distribution over bidders' valuations and provide worst-case (input-by-input) approximation guarantees. In contrast to previous work on this topic, we pursue good prior-free auctions with non-identical bidders. Prior-free auctions can approximate meaningful benchmarks for non-identical bidders only when "sufficient qualitative information" about the bidder asymmetry is publicly known. We consider digital goods auctions where there is a total ordering of the bidders that is known to the seller, where earlier bidders are in some sense thought to have higher valuations. We use the framework of Hartline and Roughgarden (STOC '08) to define an appropriate revenue benchmark: the maximum revenue that can be obtained from a bid vector using prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. This monotone-price benchmark is always as large as the well-known fixed-price benchmark F (2) , so designing prior-free auctions with good approximation guarantees is only harder. By design, an auction that approximates the monotone-price benchmark satisfies a very strong guarantee: it is, in particular, simultaneously near-optimal for essentially every Bayesian environment in which bidders' valuation distributions have nonincreasing monopoly prices, or in which the distribution of each bidder stochastically dominates that of the next. Of course, even if there is no distribution over bidders' valuations, such an auction still provides a quantifiable input-by-input performance guarantee. In this paper, we design a simple prior-free auction for digital goods with ordered bidders, the Random Price Restriction (RPR) auction. We prove that its expected revenue on every bid profile b is Ω(M(b)/log*n), where M denotes the monotone-price benchmark and log*n denotes the number of times that the log 2 operator can be applied to n before the result drops below a fixed constant.

SODA Conference 2012 Conference Paper

Sketching valuation functions

• Ashwinkumar Badanidiyuru
• Shahar Dobzinski
• Hu Fu 0001
• Robert Kleinberg
• Noam Nisan
• Tim Roughgarden

Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally let f be a function mapping subsets of some ground set [ n ] to the non-negative real numbers. We say that f′ is an α-sketch of f if for every set S, the value f′ ( S ) lies between f ( S )/α and f ( S ), and f′ can be specified by poly( n ) bits. We show that for every subadditive function f there exists an α-sketch where α = n 1/2 · O (polylog( n )). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o ( n 1/2 ) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O ( n 1/3 ) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n 1−ε. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.

IJCAI Conference 2011 Conference Paper

Flexible Tree Matching

• Ranjitha Kumar
• Jerry O. Talton
• Salman Ahmad
• Tim Roughgarden
• Scott R. Klemmer

Tree-matching problems arise in many computational domains. The literature provides several methods for creating correspondences between labeled trees; however, by definition, tree-matching algorithms rigidly preserve ancestry. That is, once two nodes have been placed in correspondence, their descendants must be matched as well. We introduce flexible tree matching, which relaxes this rigid requirement in favor of a tunable formulation in which the role of hierarchy can be controlled. We show that flexible tree matching is strongly NP-complete, give a stochastic approximation algorithm for the problem, and demonstrate how structured prediction techniques can learn the algorithm's parameters from a set of example matchings. Finally, we present results from applying the method to tasks in Web design.

STOC Conference 2011 Conference Paper

From convex optimization to randomized mechanisms: toward optimal combinatorial auctions

• Shaddin Dughmi
• Tim Roughgarden
• Qiqi Yan

We design an expected polynomial time, truthful in expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS) , which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ≠ NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm , rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1/e)-approximation of the optimal welfare.

SODA Conference 2011 Conference Paper

Local Smoothness and the Price of Anarchy in Atomic Splittable Congestion Games

• Tim Roughgarden
• Florian Schoppmann

SODA Conference 2011 Conference Paper

Welfare Guarantees for Combinatorial Auctions with Item Bidding

• Kshipra Bhawalkar
• Tim Roughgarden

We analyze the price of anarchy (POA) in a simple and practical non-truthful combinatorial auction when players have subadditive valuations for goods. We study the mechanism that sells every good in parallel with separate second-price auctions. We first prove that under a standard “no overbidding” assumption, for every subadditive valuation profile, every pure Nash equilibrium has welfare at least 50% of optimal — i. e. , the POA is at most 2. For the incomplete information setting, we prove that the POA with respect to Bayes-Nash equilibria is strictly larger than 2 — an unusual separation from the full-information model — and is at most 2 ln m, where m is the number of goods.

FOCS Conference 2010 Conference Paper

Black-Box Randomized Reductions in Algorithmic Mechanism Design

• Shaddin Dughmi
• Tim Roughgarden

We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality'' between linear perturbations of the objective function of an optimization problem and of its feasible set, and use the “primal'' and “dual'' viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.

STOC Conference 2010 Conference Paper

Interactive privacy via the median mechanism

• Aaron Roth 0001
• Tim Roughgarden

We define a new interactive differentially private mechanism --- the median mechanism --- for answering arbitrary predicate queries that arrive online. Given fixed accuracy and privacy constraints, this mechanism can answer exponentially more queries than the previously best known interactive privacy mechanism (the Laplace mechanism, which independently perturbs each query result). With respect to the number of queries, our guarantee is close to the best possible, even for non-interactive privacy mechanisms. Conceptually, the median mechanism is the first privacy mechanism capable of identifying and exploiting correlations among queries in an interactive setting. We also give an efficient implementation of the median mechanism, with running time polynomial in the number of queries, the database size, and the domain size. This efficient implementation guarantees privacy for all input databases, and accurate query results for almost all input distributions. The dependence of the privacy on the number of queries in this mechanism improves over that of the best previously known efficient mechanism by a super-polynomial factor, even in the non-interactive setting.

STOC Conference 2009 Conference Paper

Intrinsic robustness of the price of anarchy

• Tim Roughgarden

The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument . Second, we show that every bound derived via a smoothness argument extends automatically , with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight , in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA.

STOC Conference 2009 Conference Paper

Universally utility-maximizing privacy mechanisms

• Arpita Ghosh
• Tim Roughgarden
• Mukund Sundararajan

A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy , which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism M u that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism M u for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.

SODA Conference 2008 Conference Paper

Designing networks with good equilibria

• Ho-Lin Chen
• Tim Roughgarden
• Gregory Valiant

SODA Conference 2008 Conference Paper

Metric clustering via consistent labeling

• Robert Krauthgamer
• Tim Roughgarden

STOC Conference 2008 Conference Paper

Optimal mechanism design and money burning

• Jason D. Hartline
• Tim Roughgarden

Mechanism design is now a standard tool in computer science for aligning the incentives of self-interested agents with the objectives of a system designer. There is, however, a fundamental disconnect between the traditional application domains of mechanism design (such as auctions) and those arising in computer science (such as networks): while monetary "transfers" (i.e., payments) are essential for most of the known positive results in mechanism design, they are undesirable or even technologically infeasible in many computer systems. Classical impossibility results imply that the reach of mechanisms without transfers is severely limited. Computer systems typically do have the ability to reduce service quality--routing systems can drop or delay traffic, scheduling protocols can delay the release of jobs, and computational payment schemes can require computational payments from users (e.g., in spam-fighting systems). Service degradation is tantamount to requiring that users "burn money", and such "payments" can be used to influence the preferences of the agents at a cost of degrading the social surplus. We develop a framework for the design and analysis of "money-burning mechanisms" to maximize the residual surplus-the total value of the chosen outcome minus the payments required. Our primary contributions are the following. * We define a general template for prior-free optimal mechanism design that explicitly connects Bayesian optimal mechanism design, the dominant paradigm in economics, with worst-case analysis. In particular, we establish a general and principled way to identify appropriate performance benchmarks in prior-free mechanism design. * For general single-parameter agent settings, we characterize the Bayesian optimal money-burning mechanism. * For multi-unit auctions, we design a near-optimal prior-free money-burning mechanism: for every valuation profile, its expected residual surplus is within a constant factor of our benchmark, the residual surplus of the best Bayesian optimal mechanism for this profile. * For multi-unit auctions, we quantify the benefit of general transfers over money-burning: optimal money-burning mechanisms always obtain a logarithmic fraction of the full social surplus, and this bound is tight.

FOCS Conference 2008 Conference Paper

Truthful Approximation Schemes for Single-Parameter Agents

• Peerapong Dhangwatnotai
• Shahar Dobzinski
• Shaddin Dughmi
• Tim Roughgarden

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines (Q||C max ), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming P not equal to NP). Our algorithmic techniques are flexible and also yield, among other results, a monotone deterministic quasi-PTAS for Q||C max and a monotone randomized PTAS for max-min scheduling on related machines.

SODA Conference 2006 Conference Paper

Bottleneck links, variable demand, and the tragedy of the commons

• Richard Cole 0001
• Yevgeniy Dodis
• Tim Roughgarden

STOC Conference 2006 Conference Paper

New trade-offs in cost-sharing mechanisms

• Tim Roughgarden
• Mukund Sundararajan

A cost-sharing mechanism is a protocol that collects bids from a set of players, selects a subset of the players to receive a service (incurring a subset-dependent cost), and determines a price to charge each of these players. Three standard requirements for cost-sharing mechanisms are incentive compatibility, which states that players are motivated to bid their true valuation for the service; budget-balance, meaning that the prices charged should recover the cost incurred; and efficiency, which states that the cost incurred and the valuations of the players served should be traded off in an optimal way. These three goals have been known to be mutually incompatible for thirty years. As a result, nearly all work on cost-sharing mechanisms in the economics and theoretical computer science literatures has focused on achieving two of these goals while completely ignoring the third.We show that incentive-compatibility, budget-balance, and approximate efficiency are simultaneously achievable for a wide range of cost functions, where efficiency is measured using the social cost---the sum of the incurred service cost and the excluded valuations. In particular, we prove such guarantees for well-known mechanisms for all submodular cost functions and for the Steiner tree cost function. We also prove a generic, quantifiable trade-off between the objectives of efficiency and budget-balance in groupstrategyproof cost-sharing mechanisms.

SODA Conference 2005 Conference Paper

Computing equilibria in multi-player games

• Christos H. Papadimitriou
• Tim Roughgarden

SODA Conference 2005 Conference Paper

Selfish routing with atomic players

• Tim Roughgarden

SODA Conference 2004 Conference Paper

A stronger bound on Braess's Paradox

• Henry C. Lin
• Tim Roughgarden
• Éva Tardos

SODA Conference 2004 Conference Paper

The maximum latency of selfish routing

• Tim Roughgarden

FOCS Conference 2004 Conference Paper

The Price of Stability for Network Design with Fair Cost Allocation

• Elliot Anshelevich
• Anirban Dasgupta 0001
• Jon M. Kleinberg
• Éva Tardos
• Tom Wexler
• Tim Roughgarden

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

FOCS Conference 2003 Conference Paper

Approximation Via Cost-Sharing: A Simple Approximation Algorithm for the Multicommodity Rent-or-Buy Problem

• Anupam Gupta 0001
• Amit Kumar 0001
• Martin Pál
• Tim Roughgarden

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constant-factor approximation algorithm for this problem was recently given by Kumar et al. ; however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12-approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a "fair" manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse - those ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms.

STOC Conference 2003 Conference Paper

Pricing network edges for heterogeneous selfish users

• Richard Cole 0001
• Yevgeniy Dodis
• Tim Roughgarden

We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimum-latency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency ).It is well known that the outcome of selfish routing (a Nash equilibrium ) does not minimize the total latency. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing , which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated.This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic aspects of edge pricing and the unfortunate possibility that an efficient routing of traffic might only be achieved with exorbitant taxes. Motivated by these shortcomings, we extend this classical work on edge pricing in several different directions and prove the following results.We prove that the edges of a single-commodity network can always be priced so that an optimal routing of traffic arises as a Nash equilibrium, even for very general heterogeneous populations of network users.When there are only finitely many different types of network users and all edge latency functions are convex, we show how to compute such edge prices efficiently.We prove that an easy-to-check mathematical condition on the population of heterogeneous network users is both necessary and sufficient for the existence of edge prices that induce an optimal routing while requiring only moderate taxes.

STOC Conference 2003 Conference Paper

Simpler and better approximation algorithms for network design

• Anupam Gupta 0001
• Amit Kumar 0001
• Tim Roughgarden

We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. We give a randomized 3.55-approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the best-known performance guarantee for the problem. We give a 5.55-approximation algorithm for virtual private network design . Previously, constant-factor approximation algorithms were known only for special cases of this problem. We give a simple constant-factor approximation algorithm for the single-sink buy-at-bulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.

FOCS Conference 2002 Conference Paper

A Constant-Factor Approximation Algorithm for the Multicommodity Rent-or-Buy Problem

• Amit Kumar 0001
• Anupam Gupta 0001
• Tim Roughgarden

We present the first constant factor approximation algorithm for network design with multiple commodities and economies of scale. We consider the rent-or-buy problem, a type of multicommodity buy-at-bulk network design in which there are two ways to install capacity on any given edge. Capacity can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. Recent work on buy-at-bulk network design has concentrated on the special case where all sinks are identical; existing constant factor approximation algorithms for this special case make crucial use of the assumption that all commodities ship flow to the same sink vertex and do not obviously extend to the multicommodity rent-or-buy problem. Prior to our work, the best heuristics for the multicommodity rent-or-buy problem achieved only logarithmic performance guarantees and relied on the machinery of relaxed metrical task systems or of metric embeddings. By contrast, we solve the network design problem directly via a novel primal-dual algorithm.

SODA Conference 2002 Conference Paper

How unfair is optimal routing?

• Tim Roughgarden

STOC Conference 2002 Conference Paper

The price of anarchy is independent of the network topology

• Tim Roughgarden

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a directed network in which each edge possesses a latency function describing the common latency incurred by all traffic on the edge as a function of the edge congestion. Given a rate of traffic between each pair of nodes in the network, we aspire toward an assignment of traffic to paths in which the sum of all travel times (the total latency ) is minimized; however, in many settings network users are free to route their traffic in a selfish manner, without regard to the total latency. We therefore assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths (a Nash equilibrium ) will not minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy , defined as the worst possible ratio between the total latency of a Nash equilibrium and of a minimum-latency routing of the traffic.In this paper, we show that the underlying network topology plays no role in the determination of the price of anarchy. Specifically, we show that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a single-commodity instance on a set of parallel links. In the special case where the class of allowable latency functions includes all of the constant functions, we prove that a network with only two parallel links suffices to achieve the worst-possible ratio. Informally, these results imply that the inefficiency inherent in a flow at Nash equilibrium stems from the inability of selfish users to discern which of two competing routes is superior and not from the topological complexity arising from the diverse intersections of many paths belonging to different commoditie.Our proof techniques also give powerful methods for computing the price of anarchy with respect to an arbitrary class of latency functions. We apply these methods to function classes that have been well studied in the literature (such as degree-bounded polynomials and functions of the form $\ell( x ) = ( u — x ) —1 } that are used to model edges with capacity u ), thereby achieving the first tight analyses of the price of anarchy for significant classes of latency functions outside the class of linear functions.

FOCS Conference 2001 Conference Paper

Designing Networks for Selfish Users is Hard

• Tim Roughgarden

We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n nodes and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = NP). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no (4/3 - /spl epsi/)-approximation algorithm for the problem (for any /spl epsi/ > 0, unless P = NP); the existence of a 4/3-approximation algorithm follows easily from existing work, proving this hardness result sharp.

STOC Conference 2001 Conference Paper

Stackelberg scheduling strategies

• Tim Roughgarden

We study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with load-dependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of “selfishly controlled” and “centrally controlled” jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games , games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a em Stackelberg strategy ) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system. In this paper, we prove that it is NP-hard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible.

FOCS Conference 2000 Conference Paper

How Bad is Selfish Routing?

• Tim Roughgarden
• Éva Tardos

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times-the total latency-is minimized. In many settings, including the Internet and other large-scale communication networks, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. We quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and non-decreasing in the edge congestion.