269 research outputs found

    On-line partitioning of width w posets into w^O(log log w) chains

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    An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width ww into wO(loglogw)w^{O(\log{\log{w}})} chains. This improves over previously best known algorithms using wO(logw)w^{O(\log{w})} chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in wO(w)nw^{O(\sqrt{w})}n time, where ww is the width and nn is the size of a presented poset.Comment: 16 pages, 10 figure

    An easy subexponential bound for online chain partitioning

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    Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width ww into w14lgww^{14\lg w} chains. We improve this to w6.5lgw+7w^{6.5 \lg w + 7} with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

    Two New Bounds on the Random-Edge Simplex Algorithm

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    We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2^d on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.Comment: 10 page

    Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance

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    We consider a Metropolis-Hastings method with proposal kernel N(x,hG1(x))\mathcal{N}(x,hG^{-1}(x)), where xx is the current state. After discussing specific cases from the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of G1(x)G^{-1}(x) can change the ergodicity properties compared to the Random Walk Metropolis case N(x,hΣ)\mathcal{N}(x,h\Sigma), either for the better or worse. In higher dimensions we use a specific example to show that judicious choice of G1(x)G^{-1}(x) can produce a chain which will converge at a geometric rate to its limiting distribution when probability concentrates on an ever narrower ridge as x|x| grows, something which is not true for the Random Walk Metropolis.Comment: 15 pages + appendices, 4 figure

    First-Fit is Linear on Posets Excluding Two Long Incomparable Chains

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    A poset is (r + s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo

    The on-line width of various classes of posets.

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    An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\\u27edi proved that any on-line algorithm could be forced to use (w+12)\binom{w+1}{2} chains to partition a poset of width ww. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size dd. We prove two results for this problem. First, we prove that any on-line algorithm can be forced to use (2o(1))(w+12)(2-o(1))\binom{w+1}{2} chains to partition a 22-dimensional poset of width ww. Second, we prove that any on-line algorithm can be forced to use (21d1o(1))(w+12)(2-\frac{1}{d-1}-o(1))\binom{w+1}{2} chains to partition a poset of width ww presented via a realizer of size dd. Chrobak and \\u27Slusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most 3w23w-2 chains, where ww is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most 2w12w-1 chains, over 3030 years later, it remains unknown whether a more optimal algorithm exists. We improve upon previously known bounds and show that any on-line algorithm can be forced to use 32w\lceil\frac{3}{2}w\rceil chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for w=3w=3. We also consider entirely new variants and present the results for those

    On Exact Algorithms for Permutation CSP

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    In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables VV and a set of constraints C, in which constraints are tuples of elements of V. The goal is to find a total ordering of the variables, π :V[1,...,V]\pi\ : V \rightarrow [1,...,|V|], which satisfies as many constraints as possible. A constraint (v1,v2,...,vk)(v_1,v_2,...,v_k) is satisfied by an ordering π\pi when π(v1)<π(v2)<...<π(vk)\pi(v_1)<\pi(v_2)<...<\pi(v_k). An instance has arity kk if all the constraints involve at most kk elements. This problem expresses a variety of permutation problems including {\sc Feedback Arc Set} and {\sc Betweenness} problems. A naive algorithm, listing all the n!n! permutations, requires 2O(nlogn)2^{O(n\log{n})} time. Interestingly, {\sc Permutation CSP} for arity 2 or 3 can be solved by Held-Karp type algorithms in time O(2n)O^*(2^n), but no algorithm is known for arity at least 4 with running time significantly better than 2O(nlogn)2^{O(n\log{n})}. In this paper we resolve the gap by showing that {\sc Arity 4 Permutation CSP} cannot be solved in time 2o(nlogn)2^{o(n\log{n})} unless ETH fails
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