42,086 research outputs found

    Connected and internal graph searching

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    This paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.Postprint (published version

    Contraction Obstructions for Connected Graph Searching

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    We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for k=2k=2, where kk is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite

    ALPS: A Linear Program Solver

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    ALPS is a computer program which can be used to solve general linear program (optimization) problems. ALPS was designed for those who have minimal linear programming (LP) knowledge and features a menu-driven scheme to guide the user through the process of creating and solving LP formulations. Once created, the problems can be edited and stored in standard DOS ASCII files to provide portability to various word processors or even other linear programming packages. Unlike many math-oriented LP solvers, ALPS contains an LP parser that reads through the LP formulation and reports several types of errors to the user. ALPS provides a large amount of solution data which is often useful in problem solving. In addition to pure linear programs, ALPS can solve for integer, mixed integer, and binary type problems. Pure linear programs are solved with the revised simplex method. Integer or mixed integer programs are solved initially with the revised simplex, and the completed using the branch-and-bound technique. Binary programs are solved with the method of implicit enumeration. This manual describes how to use ALPS to create, edit, and solve linear programming problems. Instructions for installing ALPS on a PC compatible computer are included in the appendices along with a general introduction to linear programming. A programmers guide is also included for assistance in modifying and maintaining the program

    Structurally Parameterized d-Scattered Set

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    In dd-Scattered Set we are given an (edge-weighted) graph and are asked to select at least kk vertices, so that the distance between any pair is at least dd, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any d≥2d\ge2, an O∗(dtw)O^*(d^{\textrm{tw}})-time algorithm, where tw\textrm{tw} is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - dd-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if kk is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A 2O(td2)2^{O(\textrm{td}^2)}-time algorithm parameterized by tree-depth (td\textrm{td}), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter ϵ>0\epsilon > 0, runs in time O∗((tw/ϵ)O(tw))O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})}) and returns a d/(1+ϵ)d/(1+\epsilon)-scattered set of size kk, if a dd-scattered set of the same size exists
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