2,012 research outputs found

    Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

    Full text link
    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 vGv \in G stores its distance to the so-called hubs SvVS_v \subseteq V, chosen so that for any u,vVu,v \in V there is wSuSvw \in S_u \cap S_v belonging to some shortest uvuv 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)|E(G)| = O(n), for which we show a lowerbound of n2O(logn)\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(nRS(n)c)O(\frac{n}{RS(n)^{c}}) for some 0<c<10 < c < 1, where RS(n)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 n2(logn)o(1)\frac{n}{2^{(\log n)^{o(1)}}} would require a breakthrough in the study of lower bounds on RS(n)RS(n), which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of 12O(logn)SumIndex(n)\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n), where SumIndex(n)SumIndex(n) is the communication complexity of the Sum-Index problem over ZnZ_n. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Θ(n2(logn)c)\Theta(\frac{n}{2^{(\log n)^c}}) for some 0<c<10<c < 1

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

    Full text link
    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

    The Query-commit Problem

    Full text link
    In the query-commit problem we are given a graph where edges have distinct probabilities of existing. It is possible to query the edges of the graph, and if the queried edge exists then its endpoints are irrevocably matched. The goal is to find a querying strategy which maximizes the expected size of the matching obtained. This stochastic matching setup is motivated by applications in kidney exchanges and online dating. In this paper we address the query-commit problem from both theoretical and experimental perspectives. First, we show that a simple class of edges can be queried without compromising the optimality of the strategy. This property is then used to obtain in polynomial time an optimal querying strategy when the input graph is sparse. Next we turn our attentions to the kidney exchange application, focusing on instances modeled over real data from existing exchange programs. We prove that, as the number of nodes grows, almost every instance admits a strategy which matches almost all nodes. This result supports the intuition that more exchanges are possible on a larger pool of patient/donors and gives theoretical justification for unifying the existing exchange programs. Finally, we evaluate experimentally different querying strategies over kidney exchange instances. We show that even very simple heuristics perform fairly well, being within 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in the graph. In such a time-sensitive application, this result motivates the use of committing strategies
    corecore