2,954 research outputs found

    Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

    Full text link
    The study of graph products is a major research topic and typically concerns the term f(GH)f(G*H), e.g., to show that f(GH)=f(G)f(H)f(G*H)=f(G)f(H). In this paper, we study graph products in a non-standard form f(R[GH]f(R[G*H] where RR is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight n1ϵn^{1-\epsilon}-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where nn is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NPRPNP\neq RP (the weakest possible assumption). (2) A tight n1/2ϵn^{1/2-\epsilon} hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where nn denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint kk-cycles for large kk. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

    Improved Inapproximability Results for Maximum k-Colorable Subgraph

    Full text link
    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

    The min-max edge q-coloring problem

    Full text link
    In this paper we introduce and study a new problem named \emph{min-max edge qq-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer qq. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most qq different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge qq-coloring is NP-hard, for any q2q \ge 2. 2. A polynomial time exact algorithm for min-max edge qq-coloring on trees. 3. Exact formulas of the optimal solution for cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial lower bound of the optimal solution with respect to the average degree of the graph. 5. An approximation algorithm for planar graphs.Comment: 16 pages, 5 figure

    Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

    Get PDF
    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

    Improved Hardness of Approximating Chromatic Number

    Full text link
    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]

    A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

    Get PDF
    In the unsplittable flow problem on a path, we are given a capacitated path PP and nn tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge ee of PP, the total demand of selected tasks that use ee does not exceed the capacity of ee. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of O(logn)O(\log n). The approximation ratio of our algorithm is 7+ϵ7+\epsilon for any ϵ>0\epsilon>0. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a (2+ϵ)(2+\epsilon)-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

    Inapproximability of Combinatorial Optimization Problems

    Full text link
    We survey results on the hardness of approximating combinatorial optimization problems

    Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

    Get PDF
    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
    corecore