Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1-\epsilon)$-approximation for the number of perfect matchings of a $\lambda$-dense bipartite graph, using $O(n^{\frac{1-2\lambda}{8\lambda}+\epsilon^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>\lambda>0$, a $\lambda$-dense bipartite graph has all degrees greater than $(\lambda+\frac{1}{2})n$. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes

Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterise the vertices of these polytopes. We define the combinatorial problem “Another completely labeled Gale string” whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful

Paired-Domination Game Played in Graphs\u3csup\u3e∗\u3c/sup\u3e

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979-991]. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph G by two players, named Dominator and Pairer. They alternately take turns choosing vertices of G such that each vertex chosen by Dominator dominates at least one vertex not dominated by the vertices previously chosen, while each vertex chosen by Pairer is a vertex not previously chosen that is a neighbor of the vertex played by Dominator on his previous move. This process eventually produces a paired-dominating set of vertices of G; that is, a dominating set in G that induces a subgraph that contains a perfect matching. Dominator wishes to minimize the number of vertices chosen, while Pairer wishes to maximize it. The game paired-domination number γgpr(G) of G is the number of vertices chosen when Dominator starts the game and both players play optimally. Let G be a graph on n vertices with minimum degree at least 2. We show that γgpr(G) ≤ 45 n, and this bound is tight. Further we show that if G is (C4, C5)-free, then γgpr(G) ≤ 43 n, where a graph is (C4, C5)-free if it has no induced 4-cycle or 5-cycle. If G is 2-connected and bipartite or if G is 2-connected and the sum of every two adjacent vertices in G is at least 5, then we show that γgpr(G) ≤ 34 n

Shapley Meets Shapley

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page