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Martin Olsen

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7 papers

2 author rows

AAAI Conference 2025 Conference Paper

Approximating Optimal Labelings for Temporal Connectivity

Daniele Carnevale
Gianlorenzo D'Angelo
Martin Olsen

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-step the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time a and the overall number of labels is minimum. The problem, called Minimum Aged Labeling (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. Problem MAL has previously been proved to be NP-complete on undirected graphs and APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than O(log n) when a >= 2, unless P = NP, and a factor better than 2^[log^(1-ε) n] when a >= 3, unless NP is contained in DTIME(2^(polylog(n))), where n is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower-bounds. In particular, we show that the approximation depends on a relation between a and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called Diameter Constrained Spanning Subgraph (DCSS) and show that our hardness results also apply to DCSS.

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TMLR Journal 2025 Journal Article

On the Hardness of Computing Counterfactual and Semi-factual Explanations in XAI

André Artelt
Martin Olsen
Kevin Tierney

Providing clear explanations to the choices of machine learning models is essential for these models to be deployed in crucial applications. Counterfactual and semi-factual explanations have emerged as two mechanisms for providing users with insights into the outputs of their models. We provide an overview of the computational complexity results in the literature for generating these explanations, finding that in many cases, generating explanations is computationally hard. We strengthen the argument for this considerably by further contributing our own inapproximability results showing that not only are explanations often hard to generate, but under certain assumptions, they are also hard to approximate. We discuss the implications of these complexity results for the XAI community and for policymakers seeking to regulate explanations in AI.

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TCS Journal 2021 Journal Article

Generalised online colouring problems in overlap graphs

Marc Demange
Martin Olsen

In this paper we consider an online version of different colouring problems in overlap graphs, motivated by some stacking problems. The instance is a system of time intervals presented in non-decreasing order of the left endpoint. We consider the usual colouring problem as well as b-bounded colouring (colour class have a maximum capacity b) and the same problems in the complement graph. We also consider the case where at most b intervals of the same colour can intersect. For all these versions we obtain a logarithmic competitive ratio w. r. t. the maximum ratio of interval lengths while the best known ratio for the usual colouring was linear. To our knowledge it is the first time the other variants are considered in online overlap graphs. Moreover, in the offline case, a pre-processing allows us to deduce a logarithmic approximation ratio w. r. t. the maximum number of pairwise disjoint intervals in the system. Our method is based on a partition of the overlap graph into permutation graphs, leading to a competitive-preserving reduction of the problem in overlap graphs to the same problem in permutation graphs. We think that this new partition problem by itself is of interest for future work.

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AAAI Conference 2019 Conference Paper

Coverage Centrality Maximization in Undirected Networks

Gianlorenzo D’Angelo
Martin Olsen
Lorenzo Severini

Centrality metrics are among the main tools in social network analysis. Being central for a user of a network leads to several benefits to the user: central users are highly influential and play key roles within the network. Therefore, the optimization problem of increasing the centrality of a network user recently received considerable attention. Given a network and a target user v, the centrality maximization problem consists in creating k new links incident to v in such a way that the centrality of v is maximized, according to some centrality metric. Most of the algorithms proposed in the literature are based on showing that a given centrality metric is monotone and submodular with respect to link addition. However, this property does not hold for several shortest-path based centrality metrics if the links are undirected. In this paper we study the centrality maximization problem in undirected networks for one of the most important shortestpath based centrality measures, the coverage centrality. We provide several hardness and approximation results. We first show that the problem cannot be approximated within a factor greater than 1 − 1/e, unless P = NP, and, under the stronger gap-ETH hypothesis, the problem cannot be approximated within a factor better than 1/no(1), where n is the number of users. We then propose two greedy approximation algorithms, and show that, by suitably combining them, we can guarantee an approximation factor of Ω(1/ √ n). We experimentally compare the solutions provided by our approximation algorithm with optimal solutions computed by means of an exact IP formulation. We show that our algorithm produces solutions that are very close to the optimum.

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AAAI Conference 2017 Conference Paper

On Covering Codes and Upper Bounds for the Dimension of Simple Games

Martin Olsen

Consider a situation with n agents or players, where some of the players form a coalition with a certain collective objective. Simple games are used to model systems that can decide whether coalitions are successful (winning) or not (losing). A simple game can be viewed as a monotone boolean function. The dimension of a simple game is the smallest positive integer d such that the simple game can be expressed as the intersection of d threshold functions, where each threshold function uses a threshold and n weights. Taylor and Zwicker have shown that d is bounded from above by the number of maximal losing coalitions. We present two new upper bounds both containing the Taylor-Zwicker bound as a special case. The Taylor-Zwicker bound implies an upper bound of n n/2. We improve this upper bound signiﬁcantly by showing constructively that d is bounded from above by the cardinality of any binary covering code with length n and covering radius 1. This result supplements a recent result where Olsen et al. showed how to construct simple games with dimension |C| for any binary constant weight SECDED code C with length n. Our result represents a major step in the attempt to close the dimensionality gap for simple games.

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ECAI Conference 2016 Conference Paper

On the Construction of High-Dimensional Simple Games

Martin Olsen
Sascha Kurz
Xavier Molinero

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i. e. , "yes" and "no", every voting system can be described by a (monotone) Boolean function χ :{0, 1}n→ {0, 1}. However, its naive encoding needs 2nbits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using n weights and one threshold. For heterogeneous agents, one can represent χ as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiringand provided a construction guaranteeing. The magnitude of the worst-case situation was to be determined by Elkind et al. in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number k for a subclass of voting systems. As an application, we give a construction for k≥ 2n− o( n) , i. e. , there is no gain from a representation complexity point of view.

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TCS Journal 2014 Journal Article

On the approximability of the link building problem

Martin Olsen
Anastasios Viglas

We consider the LINK BUILDING problem, which involves maximizing the PageRank value of a given target vertex in a directed graph by adding k new links that point to the target (backlinks). We present a theorem describing how the topology of the graph affects the choice of potential new backlinks. Based on this theorem we show that no fully polynomial-time approximation scheme (FPTAS) exists for LINK BUILDING unless P = NP and we also show that LINK BUILDING is W [ 1 ] -hard. Furthermore, we show that this problem is in the class APX by presenting the polynomial time algorithm r-Greedy, which selects new backlinks in a greedy fashion and results in a new PageRank value for the target vertex that is within a constant factor from the best possible. We also consider the naive algorithm π-Naive, where we choose backlinks from vertices with high PageRank values compared to the out-degree and show that this algorithm performs much worse on certain graphs compared to our constant factor approximation. Finally, we provide a lower bound for the approximation ratio of our r-Greedy algorithm.

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