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Register a new userVertex adjacencies in the set covering polyhedron
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We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvatal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.
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Let $mathcal{P}$ be an $mathcal{H}$-polytope in $mathbb{R}^d$ with vertex set $V$. The vertex centroid is defined as the average of the vertices in $V$. We prove that computing the vertex centroid of an $mathcal{H}$-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for $mathcal{H}$-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an $epsilon$ distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an $mathcal{H}$-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to emph{any} ``sufficiently non-trivial (for example constant) distance, can be used to construct a fully polynomial approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of $d^{{1/2}-delta}$ for any fixed constant $delta>0$.
Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=sum_{v in S} w(v).$ A non-empty subset $S subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $mathrm{s}(G,w)$ and connected weighted safe number $mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $mathrm{s}(G,w) le mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $mathrm{s}(G,w)=mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.
We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.
Let $mathcal{S}_n$ denote the set of permutations of ${1,2,dots,n}$. The function $f(n,s)$ is defined to be the minimum size of a subset $Ssubseteq mathcal{S}_n$ with the property that for any $rhoin mathcal{S}_n$ there exists some $sigmain S$ such that the Hamming distance between $rho$ and $sigma$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kezdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kezdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!< f(n,s)< 3s!(n-s)log n$ for $1leq sleq n-2$ and that [f(n,s)>leftlfloor frac{2+sqrt{2s-2}}{2}rightrfloor frac{n}{2}] if $sgeq 3$.
We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $Rcup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $Rcup B$, like ${1,3}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $Rcup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $Rcup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $ble rle 3b$, then at least $frac{8}{9}(r+b-8)$ points of $Rcup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.
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