An improvement on Łuczak's connected matchings method

A connected matching in a graph G is a matching contained in a connected component of G. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007)

Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games

Cooperative games provide a framework for fair and stable profit allocation in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are such solution concepts that characterize stability of cooperation. In this paper, we study the algorithmic issues on the least-core and nucleolus of threshold cardinality matching games (TCMG). A TCMG is defined on a graph $G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$ meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the problems of computing least-core value, finding and verifying least-core payoff are all polynomial time solvable. We also provide a general characterization of the least core for a large class of TCMG. Next, based on Gallai-Edmonds Decomposition in matching theory, we give a concise formulation of the nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When the threshold $T$ is relevant to the input size, we prove that the nucleolus can be obtained in polynomial time in bipartite graphs and graphs with a perfect matching

Gallai-Edmonds decomposition of unicyclic graphs from null space

In this paper, we compute the Gallai-Edmonds decomposition of a unicyclic graph G using linear algebraic tools. More precisely, the Gallai-Edmonds decomposition of G is obtained from the null space associated with adjacency matrices of its subtrees

LP duality in infinite hypergraphs

AbstractIn any graph there exist a fractional cover and a fractional matching satisfying the complementary slackness conditions of linear programming. The proof uses a Gallai-Edmonds decomposition result for infinite graphs. We consider also the same problem for infinite hypergraphs, in particular in the case that the edges of the hypergraph are intervals on the real line. We prove an extension of a theorem of Gallai to the infinite case

Pairing games and markets

Pairing Games or Markets studied here are the non-two-sided NTU generalization of assignment games. We show that the Equilibrium Set is nonempty, that it is the set of stable allocations or the set of semistable allocations, and that it has has several notable structural properties. We also introduce the solution concept of pseudostable allocations and show that they are in the Demand Bargaining Set. We give a dynamic Market Procedure that reaches the Equilibrium Set in a bounded number of steps. We use elementary tools of graph theory and a representation theorem obtained here

Faster Algorithms for Half-Integral T-Path Packing

Let G = (V, E) be an undirected graph, a subset of vertices T be a set of terminals. Then a natural combinatorial problem consists in finding the maximum number of vertex-disjoint paths connecting distinct terminals. For this problem, a clever construction suggested by Gallai reduces it to computing a maximum non-bipartite matching and thus gives an O(mn^1/2 log(n^2/m)/log(n))-time algorithm (hereinafter n := |V|, m := |E|). Now let us consider the fractional relaxation, i.e. allow T-path packings with arbitrary nonnegative real weights. It is known that there always exists a half-integral solution, that is, one only needs to assign weights 0, 1/2, 1 to maximize the total weight of T-paths. It is also known that an optimum half-integral packing can be found in strongly-polynomial time but the actual time bounds are far from being satisfactory. In this paper we present a novel algorithm that solves the half-integral problem within O(mn^1/2 log(n^2/m)/log(n)) time, thus matching the complexities of integral and half-integral versions