989 research outputs found

    On Coloring Resilient Graphs

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    We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study rr-resiliently kk-colorable graphs, which are those kk-colorable graphs that remain kk-colorable even after the addition of any rr new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for kk-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

    Improved Inapproximability Results for Maximum k-Colorable Subgraph

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    We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1-1/k of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than 32/33 is NP-hard. Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of edges of a k-colorable graph.Comment: 16 pages, 2 figure

    Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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    A hypergraph is said to be χ\chi-colorable if its vertices can be colored with χ\chi colors so that no hyperedge is monochromatic. 22-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 22-colorable kk-uniform hypergraph, it is NP-hard to find a 22-coloring miscoloring fewer than a fraction 2k+12^{-k+1} of hyperedges (which is achieved by a random 22-coloring), and the best algorithms to color the hypergraph properly require n11/k\approx n^{1-1/k} colors, approaching the trivial bound of nn as kk increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 22-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 22-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy k\ell \ll \sqrt{k}, we give an algorithm to color the it with nO(2/k)\approx n^{O(\ell^2/k)} colors. However, for the maximization version, we prove NP-hardness of finding a 22-coloring miscoloring a smaller than 2O(k)2^{-O(k)} (resp. kO(k)k^{-O(k)}) fraction of the hyperedges when =O(logk)\ell = O(\log k) (resp. =2\ell=2). Assuming the UGC, we improve the latter hardness factor to 2O(k)2^{-O(k)} for almost discrepancy-11 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k)(k-\ell)-coloring such that each hyperedge is polychromatic with all these colors, we give a 22-coloring algorithm that miscolors at most kΩ(k)k^{-\Omega(k)} of the hyperedges when k\ell \ll \sqrt{k}, and complement this with a matching UG hardness result showing that when =k\ell =\sqrt{k}, it is hard to even beat the 2k+12^{-k+1} bound achieved by a random coloring.Comment: Approx 201

    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

    Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

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    A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs. Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover

    Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

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    We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

    Improved Hardness of Approximating Chromatic Number

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    We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2^{K^{1/3}} colors. This improves the previous result of K versus K^{O(log K)} in Khot [14]
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