15,432 research outputs found

    Finding tight Hamilton cycles in random hypergraphs faster

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    In an rr-uniform hypergraph on nn vertices a tight Hamilton cycle consists of nn edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of rr vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random rr-uniform hypergraphs with edge probability at least Clog3n/nC \log^3n/n. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for p=ω(1/n)p=\omega(1/n) for r=3r=3 and p=(e+o(1))/np=(e + o(1))/n for r4r\ge 4 using a second moment argument. Moreover our algorithm is superior to previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities pn1+εp\ge n^{-1+\varepsilon}, while the algorithm of Nenadov and \v{S}kori\'c is a randomised quasipolynomial time algorithm working for edge probabilities pClog8n/np\ge C\log^8n/n.Comment: 17 page

    A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs

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    In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version

    A domination algorithm for {0,1}\{0,1\}-instances of the travelling salesman problem

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    We present an approximation algorithm for {0,1}\{0,1\}-instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio 1n1/291-n^{-1/29}. In other words, given a {0,1}\{0,1\}-edge-weighting of the complete graph KnK_n on nn vertices, our algorithm outputs a Hamilton cycle HH^* of KnK_n with the following property: the proportion of Hamilton cycles of KnK_n whose weight is smaller than that of HH^* is at most n1/29n^{-1/29}. Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio 1/2o(1)1/2-o(1) for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant CC such that n1/29n^{-1/29} cannot be replaced by exp((logn)C)\exp(-(\log n)^C) in the result above.Comment: 29 pages (final version to appear in Random Structures and Algorithms
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