The number of matchings in random graphs

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical Mechanic

A More Reliable Greedy Heuristic for Maximum Matchings in Sparse Random Graphs

We propose a new greedy algorithm for the maximum cardinality matching problem. We give experimental evidence that this algorithm is likely to find a maximum matching in random graphs with constant expected degree c>0, independent of the value of c. This is contrary to the behavior of commonly used greedy matching heuristics which are known to have some range of c where they probably fail to compute a maximum matching

Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$