Vera Traub

SODA Conference 2024 Conference Paper

Single-Source Unsplittable Flows in Planar Graphs

• Vera Traub
• Laura Vargas Koch
• Rico Zenklusen

FOCS Conference 2024 Conference Paper

The Bidirected Cut Relaxation for Steiner Tree has Integrality Gap Smaller Than 2

• Jaroslaw Byrka
• Fabrizio Grandoni 0001
• Vera Traub

The Steiner tree problem is one of the most prominent problems in network design. Given an edge-weighted undirected graph and a subset of the vertices, called terminals, the task is to compute a minimum-weight tree containing all terminals (and possibly further vertices). The best-known approximation algorithms for Steiner tree involve enumeration of a (polynomial but) very large number of candidate components and are therefore slow in practice. A promising ingredient for the design of fast and accurate approximation algorithms for Steiner tree is the bidirected cut relaxation (BCR): bidirect all edges, choose an arbitrary terminal as a root, and enforce that each cut containing some terminal but not the root has one unit of fractional edges leaving it. BCR is known to be integral in the spanning tree case [Edmonds'67], i. e. , when all the vertices are terminals. For general instances, however, it was not even known whether the integrality gap of BCR is better than the integrality gap of the natural undirected relaxation, which is exactly 2. We resolve this question by proving an upper bound of 1. 9988 on the integrality gap of BCR.

STOC Conference 2023 Conference Paper

A (1. 5+ε)-Approximation Algorithm for Weighted Connectivity Augmentation

• Vera Traub
• Rico Zenklusen

Connectivity augmentation problems are among the most elementary questions in Network Design. Many of these problems admit natural 2-approximation algorithms, often through various classic techniques, whereas it remains open whether approximation factors below 2 can be achieved. One of the most basic examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In WCAP, one is given an undirected graph together with a set of additional weighted candidate edges, and the task is to find a cheapest set of candidate edges whose addition to the graph increases its edge-connectivity. We present a (1.5+ε)-approximation algorithm for WCAP, showing for the first time that factors below 2 are achievable. On a high level, we design a well-chosen local search algorithm, inspired by recent advances for Weighted Tree Augmentation. To measure progress, we consider a directed weakening of WCAP and show that it has highly structured planar solutions. Interpreting a solution of the original problem as one of this directed weakening allows us to describe local exchange steps in a clean and algorithmically amenable way. Leveraging these insights, we show that we can efficiently search for good exchange steps within a component class of link sets that is closely related to bounded treewidth subgraphs of circle graphs. Moreover, we prove that an optimum solution can be decomposed into smaller components, at least one of which leads to a good local search step as long as we did not yet achieve the claimed approximation guarantee.

STOC Conference 2022 Conference Paper

Breaching the 2-approximation barrier for the forest augmentation problem

• Fabrizio Grandoni 0001
• Afrouz Jabal Ameli
• Vera Traub

The basic goal of survivable network design is to build cheap networks that guarantee the connectivity of certain pairs of nodes despite the failure of a few edges or nodes. A celebrated result by Jain [Combinatorica'01] provides a 2-approximation for a wide class of these problems. However nothing better is known even for very basic special cases, raising the natural question whether any improved approximation factor is possible at all. In this paper we address one of the most basic problems in this family for which 2 is still the best-known approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. we can assume to be a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2-edge-connected. Several better-than-2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP), see e.g. [Grandoni, Kalaitzis, Zenklusen - STOC'18; Cecchetto, Traub, Zenklusen - STOC'21] and references therein. Recently this was achieved also for the weighted version of TAP [Traub, Zenklusen - FOCS'21], and for the k-connectivity generalization of TAP [Byrka, Grandoni, Jabal-Ameli - STOC'20; Cecchetto, Traub, Zenklusen - STOC'21]. These results heavily exploit the fact that the input graph is connected, a condition that does not hold in FAP. In this paper we breach the 2-approximation barrier for FAP. Our result is based on two main ingredients. First, we describe a reduction to the Path Augmentation Problem (PAP), the special case of FAP where the input graph is a collection of disjoint paths. Our reduction is not approximation preserving, however it is sufficiently accurate to improve on a factor 2 approximation. Second, we present a better-than-2 approximation algorithm for PAP, an open problem on its own. Here we exploit a novel notion of implicit credits which might turn out to be helpful in future related work.

SODA Conference 2022 Conference Paper

Local Search for Weighted Tree Augmentation and Steiner Tree

• Vera Traub
• Rico Zenklusen

We present a technique that allows for improving on some relative greedy procedures by well-chosen (non-oblivious) local search algorithms. Relative greedy procedures are a particular type of greedy algorithm that start with a simple, though weak, solution, and iteratively replace parts of this starting solution by stronger components. Some well-known applications of relative greedy algorithms include approximation algorithms for Steiner Tree and, more recently, for connectivity augmentation problems. The main application of our technique leads to a (1. 5 + ∊ )-approximation for Weighted Tree Augmentation, improving on a recent relative greedy based method with approximation factor 1 + ln 2 + ∊ ≈ 1. 69. Furthermore, we show how our local search technique can be applied to Steiner Tree, leading to an alternative way to obtain the currently best known approximation factor of ln 4 + ∊. Contrary to prior methods, our approach is purely combinatorial without the need to solve an LP. Nevertheless, the solution value can still be bounded in terms of the well-known hypergraphic LP, leading to an alternative, and arguably simpler, technique to bound its integrality gap by ln 4.

FOCS Conference 2021 Conference Paper

A Better-Than-2 Approximation for Weighted Tree Augmentation

• Vera Traub
• Rico Zenklusen

We present an approximation algorithm for Weighted Tree Augmentation with approximation factor 1 + $\ln 2+\varepsilon < 1. 7$. This is the first algorithm beating the longstanding factor of 2, which can be achieved through many standard techniques. −

STOC Conference 2021 Conference Paper

Bridging the gap between tree and connectivity augmentation: unified and stronger approaches

• Federica Cecchetto
• Vera Traub
• Rico Zenklusen

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a k -edge-connected graph G =( V , E ) and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to G makes the graph ( k +1)-edge-connected. If k is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP)—i.e., G is a spanning tree—for which significant progress has been achieved recently, leading to approximation factors below 1.5 (the currently best factor is 1.458). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below 2 for CAP by presenting a 1.91-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below 1.5, based on a new analysis technique. Through these ingredients, we obtain a -approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned 1.91-approximation for CAP and also the previously best approximation factor of 1.458 for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below 1.5.

STOC Conference 2020 Conference Paper

An improved approximation algorithm for ATSP

• Vera Traub
• Jens Vygen

We revisit the constant-factor approximation algorithm for the asymmetric traveling salesman problem by Svensson, Tarnawski, and Végh [STOC 2018]. We improve on each part of this algorithm. We avoid the reduction to irreducible instances and thus obtain a simpler and much better reduction to vertebrate pairs. We also show that a slight variant of their algorithm for vertebrate pairs has a much smaller approximation ratio. Overall we improve the approximation ratio from 506 to 22+ε for any ε > 0. This also improves the upper bound on the integrality ratio from 319 to 22.

STOC Conference 2020 Conference Paper

Reducing path TSP to TSP

• Vera Traub
• Jens Vygen
• Rico Zenklusen

We present a black-box reduction from the path version of the Traveling Salesman Problem (Path TSP) to the classical tour version (TSP). More precisely, we show that given an α-approximation algorithm for TSP, then, for any ε >0, there is an (α+ε)-approximation algorithm for the more general Path TSP. This reduction implies that the approximability of Path TSP is the same as for TSP, up to an arbitrarily small error. This avoids future discrepancies between the best known approximation factors achievable for these two problems, as they have existed until very recently. A well-studied special case of TSP, Graph TSP, asks for tours in unit-weight graphs. Our reduction shows that any α-approximation algorithm for Graph TSP implies an (α+ε)-approximation algorithm for its path version. By applying our reduction to the 1.4-approximation algorithm for Graph TSP by Sebő and Vygen, we obtain a polynomial-time (1.4+ε)-approximation algorithm for Graph Path TSP, improving on a recent 1.497-approximation algorithm of Traub and Vygen. We obtain our results through a variety of new techniques, including a novel way to set up a recursive dynamic program to guess significant parts of an optimal solution. At the core of our dynamic program we deal with instances of a new generalization of (Path) TSP which combines parity constraints with certain connectivity requirements. This problem, which we call Φ-TSP, has a constant-factor approximation algorithm and can be reduced to TSP in certain cases when the dynamic program would not make sufficient progress.

SODA Conference 2018 Conference Paper

Approaching for the s - t -path TSP

• Vera Traub
• Jens Vygen

FOCS Conference 2018 Conference Paper

Beating the Integrality Ratio for s-t-Tours in Graphs

• Vera Traub
• Jens Vygen

Among various variants of the traveling salesman problem, the s-t-path graph TSP has the special feature that we know the exact integrality ratio, 3/2, and an approximation algorithm matching this ratio. In this paper, we go below this threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP with approximation ratio 1. 497. Our algorithm can be viewed as a refinement of the 3/2-approximation algorithm by Sebo and Vygen [16], but we introduce several completely new techniques. These include a new type of ear-decomposition, an enhanced ear induction that reveals a novel connection to matroid union, a stronger lower bound, and a reduction of general instances to instances in which s and t have small distance (which works for general metrics).