arXiv subjects

Argyrios Deligkas

Publications and source records attributed to Argyrios Deligkas.

60 records · Page 4Linked to original sources

arXiv2016

On the Complexity of Weighted Greedy Matchings

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GreedyMatching. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GreedyMatching is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GreedyMatching is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider two natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GreedyMatching on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is $ρ$, then for every $ε>0$ the randomized MRG algorithm of [Aronson et al., Rand. Struct. Alg. 1995] gives a $(ρ-ε)$-approximation for the maximum cardinality matching. We conjecture that a tight bound for $ρ$ is $\frac{2}{3}$; we prove our conjecture true for two subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for $ρ$) is a long-standing open problem [Poloczek and Szegedy; FOCS 2012]. This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.

cs.DM↗

arXiv2016

Lipschitz Continuity and Approximate Equilibria

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in $L_1$, $L_2^2$, and $L_\infty$ biased games, and then use these algorithms to provide strongly polynomial algorithms that find $2/3$, $5/7$, and $2/3$ approximations for these norms, respectively.

cs.GT↗

arXiv2016

An Empirical Study on Computing Equilibria in Polymatrix Games

The Nash equilibrium is an important benchmark for behaviour in systems of strategic autonomous agents. Polymatrix games are a succinct and expressive representation of multiplayer games that model pairwise interactions between players. The empirical performance of algorithms to solve these games has received little attention, despite their wide-ranging applications. In this paper we carry out a comprehensive empirical study of two prominent algorithms for computing a sample equilibrium in these games, Lemke's algorithm that computes an exact equilibrium, and a gradient descent method that computes an approximate equilibrium. Our study covers games arising from a number of interesting applications. We find that Lemke's algorithm can compute exact equilibria in relatively large games in a reasonable amount of time. If we are willing to accept (high-quality) approximate equilibria, then we can deal with much larger games using the descent method. We also report on which games are most challenging for each of the algorithms.

cs.GT↗

arXiv2015

Distributed Methods for Computing Approximate Equilibria

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing \emph{approximate well-supported Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring $O(n \log n)$ payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing \emph{approximate Nash equilibria}: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium.

cs.GT↗

arXiv2014

Computing Approximate Nash Equilibria in Polymatrix Games

In an $ε$-Nash equilibrium, a player can gain at most $ε$ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in $[0,1]$, the best-known$ε$ achievable in polynomial time is 0.3393. In general, for $n$-player games an $ε$-Nash equilibrium can be computed in polynomial time for an $ε$ that is an increasing function of $n$ but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $ε$ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general $n$-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant $ε$ such that computing an $ε$-Nash equilibrium of a polymatrix game is \PPAD-hard. Our main result is that a $(0.5+δ)$-Nash equilibrium of an $n$-player polymatrix game can be computed in time polynomial in the input size and $\frac{1}δ$. Inspired by the algorithm of Tsaknakis and Spirakis, our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $(0.5+δ)$-Nash equilibrium in a two-player Bayesian game.

cs.GT↗

arXiv2013

Revenue Maximization via Hiding Item Attributes

We study probabilistic single-item second-price auctions where the item is characterized by a set of attributes. The auctioneer knows the actual instantiation of all the attributes, but he may choose to reveal only a subset of these attributes to the bidders. Our model is an abstraction of the following Ad auction scenario. The website (auctioneer) knows the demographic information of its impressions, and this information is in terms of a list of attributes (e.g., age, gender, country of location). The website may hide certain attributes from its advertisers (bidders) in order to create thicker market, which may lead to higher revenue. We study how to hide attributes in an optimal way. We show that it is NP-hard to solve for the optimal attribute hiding scheme. We then derive a polynomial-time solvable upper bound on the optimal revenue. Finally, we propose two heuristic-based attribute hiding schemes. Experiments show that revenue achieved by these schemes is close to the upper bound.

cs.GT↗