891 research outputs found

    (Non-)existence of Polynomial Kernels for the Test Cover Problem

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    The input of the Test Cover problem consists of a set VV of vertices, and a collection E={E1,...,Em}{\cal E}=\{E_1,..., E_m\} of distinct subsets of VV, called tests. A test EqE_q separates a pair vi,vjv_i,v_j of vertices if ∣{vi,vj}∩Eq∣=1.|\{v_i,v_j\}\cap E_q|=1. A subcollection TβŠ†E{\cal T}\subseteq {\cal E} is a test cover if each pair vi,vjv_i,v_j of distinct vertices is separated by a test in T{\cal T}. The objective is to find a test cover of minimum cardinality, if one exists. This problem is NP-hard. We consider two parameterizations the Test Cover problem with parameter kk: (a) decide whether there is a test cover with at most kk tests, (b) decide whether there is a test cover with at most ∣Vβˆ£βˆ’k|V|-k tests. Both parameterizations are known to be fixed-parameter tractable. We prove that none have a polynomial size kernel unless NPβŠ†coNP/polyNP\subseteq coNP/poly. Our proofs use the cross-composition method recently introduced by Bodlaender et al. (2011) and parametric duality introduced by Chen et al. (2005). The result for the parameterization (a) was an open problem (private communications with Henning Fernau and Jiong Guo, Jan.-Feb. 2012). We also show that the parameterization (a) admits a polynomial size kernel if the size of each test is upper-bounded by a constant

    Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

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    For digraphs GG and HH, a homomorphism of GG to HH is a mapping $f:\ V(G)\dom V(H)suchthat such that uv\in A(G)implies implies f(u)f(v)\in A(H).If,moreover,eachvertex. If, moreover, each vertex u \in V(G)isassociatedwithcosts is associated with costs c_i(u), i \in V(H),thenthecostofahomomorphism, then the cost of a homomorphism fis is \sum_{u\in V(G)}c_{f(u)}(u).Foreachfixeddigraph. For each fixed digraph H,theminimumcosthomomorphismproblemfor, the minimum cost homomorphism problem for H,denotedMinHOM(, denoted MinHOM(H),canbeformulatedasfollows:Givenaninputdigraph), can be formulated as follows: Given an input digraph G,togetherwithcosts, together with costs c_i(u),, u\in V(G),, i\in V(H),decidewhetherthereexistsahomomorphismof, decide whether there exists a homomorphism of Gto to H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes
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