141 research outputs found

    On the fine-grained complexity of rainbow coloring

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    The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in kk colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k2k\ge 2, there is no algorithm for Rainbow k-Coloring running in time 2o(n3/2)2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set SS of pairs of vertices and we ask if there is a coloring in which all the pairs in SS are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by S|S|. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer qq and we ask if there is a coloring in which at least qq anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by qq and has a kernel of size O(q)O(q) for every k2k\ge 2 (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]

    Deciding first-order properties of nowhere dense graphs

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    Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

    Parameterized Complexity of Maximum Edge Colorable Subgraph

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    A graph HH is {\em pp-edge colorable} if there is a coloring ψ:E(H){1,2,,p}\psi: E(H) \rightarrow \{1,2,\dots,p\}, such that for distinct uv,vwE(H)uv, vw \in E(H), we have ψ(uv)ψ(vw)\psi(uv) \neq \psi(vw). The {\sc Maximum Edge-Colorable Subgraph} problem takes as input a graph GG and integers ll and pp, and the objective is to find a subgraph HH of GG and a pp-edge-coloring of HH, such that E(H)l|E(H)| \geq l. We study the above problem from the viewpoint of Parameterized Complexity. We obtain \FPT\ algorithms when parameterized by: (1)(1) the vertex cover number of GG, by using {\sc Integer Linear Programming}, and (2)(2) ll, a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p+kp+k, where kk is one of the following: (1)(1) the solution size, ll, (2)(2) the vertex cover number of GG, and (3)(3) l - {\mm}(G), where {\mm}(G) is the size of a maximum matching in GG; we show that the (decision version of the) problem admits a kernel with O(kp)\mathcal{O}(k \cdot p) vertices. Furthermore, we show that there is no kernel of size O(k1ϵf(p))\mathcal{O}(k^{1-\epsilon} \cdot f(p)), for any ϵ>0\epsilon > 0 and computable function ff, unless \NP \subseteq \CONPpoly

    The Minimum Shared Edges Problem on Grid-like Graphs

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    We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route pp paths from a start vertex to a target vertex in a given graph while using at most kk edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number pp of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number pp of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter kk, pp, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs
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