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

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 $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ 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

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

We consider a Metropolis-Hastings method with proposal kernel $\mathcal{N}(x,hG^{-1}(x))$, where $x$ 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 $G^{-1}(x)$ can change the ergodicity properties compared to the Random Walk Metropolis case $\mathcal{N}(x,h\Sigma)$, either for the better or worse. In higher dimensions we use a specific example to show that judicious choice of $G^{-1}(x)$ can produce a chain which will converge at a geometric rate to its limiting distribution when probability concentrates on an ever narrower ridge as $|x|$ 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

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.

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 $\binom{w+1}{2}$ chains to partition a poset of width $w$. 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 $d$. We prove two results for this problem. First, we prove that any on-line algorithm can be forced to use $(2-o(1))\binom{w+1}{2}$ chains to partition a $2$-dimensional poset of width $w$. Second, we prove that any on-line algorithm can be forced to use $(2-\frac{1}{d-1}-o(1))\binom{w+1}{2}$ chains to partition a poset of width $w$ presented via a realizer of size $d$. 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 $3w-2$ chains, where $w$ 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 $2w-1$ chains, over $30$ 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 $\lceil\frac{3}{2}w\rceil$ chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for $w=3$. We also consider entirely new variants and present the results for those

On Exact Algorithms for Permutation CSP

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