31 research outputs found

    Improved Bounds for the Graham-Pollak Problem for Hypergraphs

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    For a fixed rr, let fr(n)f_r(n) denote the minimum number of complete rr-partite rr-graphs needed to partition the complete rr-graph on nn vertices. The Graham-Pollak theorem asserts that f2(n)=n1f_2(n)=n-1. An easy construction shows that fr(n)(1+o(1))(nr/2)f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}, and we write crc_r for the least number such that fr(n)cr(1+o(1))(nr/2)f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}. It was known that cr<1c_r < 1 for each even r4r \geq 4, but this was not known for any odd value of rr. In this short note, we prove that c295<1c_{295}<1. Our method also shows that cr0c_r \rightarrow 0, answering another open problem

    Clique versus Independent Set

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    Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O(nlogn)O(n^{\log n}), and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant cHc_H for which we find a O(ncH)O(n^{c_H}) CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a O(nck)O(n^{c_k}) CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, cHc_H is of order O(HlogH)O(|H| \log |H|) resulting from Vapnik-Chervonenkis dimension, and on the other side, ckc_k is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by O(nlogn)O(n^{\log n}) instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator

    Graph Theory

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    Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

    Constraint Generation Algorithm for the Minimum Connectivity Inference Problem

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    Given a hypergraph HH, the Minimum Connectivity Inference problem asks for a graph on the same vertex set as HH with the minimum number of edges such that the subgraph induced by every hyperedge of HH is connected. This problem has received a lot of attention these recent years, both from a theoretical and practical perspective, leading to several implemented approximation, greedy and heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear Programming (MILP) formulations have been experimented, all representing connectivity constraints by the means of graph flows. In this work, we investigate the efficiency of a constraint generation algorithm, where we iteratively add cut constraints to a simple ILP until a feasible (and optimal) solution is found. It turns out that our method is faster than the previous best flow-based MILP algorithm on random generated instances, which suggests that a constraint generation approach might be also useful for other optimization problems dealing with connectivity constraints. At last, we present the results of an enumeration algorithm for the problem.Comment: 16 pages, 4 tables, 1 figur
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