1,997 research outputs found

    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

    Lower Bounds for the Graph Homomorphism Problem

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    The graph homomorphism problem (HOM) asks whether the vertices of a given nn-vertex graph GG can be mapped to the vertices of a given hh-vertex graph HH such that each edge of GG is mapped to an edge of HH. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 22-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound 2Ω(nloghloglogh)2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound 2O(nlogh)2^{{\mathcal O}(n\log{h})} is almost asymptotically tight. We also investigate what properties of graphs GG and HH make it difficult to solve HOM(G,H)(G,H). An easy observation is that an O(hn){\mathcal O}(h^n) upper bound can be improved to O(hvc(G)){\mathcal O}(h^{\operatorname{vc}(G)}) where vc(G)\operatorname{vc}(G) is the minimum size of a vertex cover of GG. The second lower bound hΩ(vc(G))h^{\Omega(\operatorname{vc}(G))} shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph HH, it is known that HOM(G,H)(G,H) can be solved in time (f(Δ(H)))n(f(\Delta(H)))^n and (f(tw(H)))n(f(\operatorname{tw}(H)))^n where Δ(H)\Delta(H) is the maximum degree of HH and tw(H)\operatorname{tw}(H) is the treewidth of HH. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number χ(H)\chi(H) does not exceed tw(H)\operatorname{tw}(H) and Δ(H)+1\Delta(H)+1, it is natural to ask whether similar upper bounds with respect to χ(H)\chi(H) can be obtained. We provide a negative answer to this question by establishing a lower bound (f(χ(H)))n(f(\chi(H)))^n for any function ff. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

    Totally frustrated states in the chromatic theory of gain graphs

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    We generalize proper coloring of gain graphs to totally frustrated states, where each vertex takes a value in a set of `qualities' or `spins' that is permuted by the gain group. (An example is the Potts model.) The number of totally frustrated states satisfies the usual deletion-contraction law but is matroidal only for standard coloring, where the group action is trivial or nearly regular. One can generalize chromatic polynomials by constructing spin sets with repeated transitive components.Comment: 14 pages, 2 figure

    Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

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    Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.Comment: 13 p

    An elementary chromatic reduction for gain graphs and special hyperplane arrangements

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    A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page

    Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs

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    A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z^d, the lists are order ideals in the integer lattice Z^d, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n x d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added references, clarified examples. 35 p

    Algorithms for the minimum sum coloring problem: a review

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    The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex coloring problem which has a number of AI related applications. Due to its theoretical and practical relevance, MSCP attracts increasing attention. The only existing review on the problem dates back to 2004 and mainly covers the history of MSCP and theoretical developments on specific graphs. In recent years, the field has witnessed significant progresses on approximation algorithms and practical solution algorithms. The purpose of this review is to provide a comprehensive inspection of the most recent and representative MSCP algorithms. To be informative, we identify the general framework followed by practical solution algorithms and the key ingredients that make them successful. By classifying the main search strategies and putting forward the critical elements of the reviewed methods, we wish to encourage future development of more powerful methods and motivate new applications
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