5,364 research outputs found

    Complexity of C_k-Coloring in Hereditary Classes of Graphs

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    For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw

    Approximately coloring graphs without long induced paths

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    It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on tt vertices, for fixed tt. We propose an algorithm that, given a 3-colorable graph without an induced path on tt vertices, computes a coloring with max{5,2t122}\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\} many colors. If the input graph is triangle-free, we only need max{4,t12+1}\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\} many colors. The running time of our algorithm is O((3t2+t2)m+n)O((3^{t-2}+t^2)m+n) if the input graph has nn vertices and mm edges

    Data Reduction for Graph Coloring Problems

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    This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

    A Time Hierarchy Theorem for the LOCAL Model

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    The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as O(1),O(logn),O(logn),2O(logn)O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}, etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical 2122\frac{1}{2}-Coloring}. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the kk-level Hierarchical 2122\frac{1}{2}-Coloring problem is Θ(n1/k)\Theta(n^{1/k}), for kZ+k\in\mathbb{Z}^+. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized no(1)n^{o(1)}-time algorithm solving the LCL can be transformed into a deterministic O(logn)O(\log n)-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges ω(logn)\omega(\log^* n)---o(logn)o(\log n) or ω(logn)\omega(\log n)---no(1)n^{o(1)}. 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in O(TLLL)O(T_{LLL}) time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be Ω(loglogn)\Omega(\log\log n) and O(logn)O(\log n)

    List coloring in the absence of a linear forest.

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    The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H

    Online Multi-Coloring with Advice

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    We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c
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