On Coloring Resilient Graphs

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 $r$-resiliently $k$-colorable graphs, which are those $k$-colorable graphs that remain $k$-colorable even after the addition of any $r$ 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 $k$-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

Approximate Graph Coloring by Semidefinite Programming

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter $\alpha\in\mathbb{N}$. Nevertheless, here it is proved that the probability of requiring a value of $\alpha>k$ to obtain a solution for a random graph decreases exponentially: $P(\alpha>k) \leq 2^{-(k+1)}$, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances

Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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

Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

In this paper, we study the $k$-forest problem in the model of resource augmentation. In the $k$-forest problem, given an edge-weighted graph $G(V,E)$, a parameter $k$, and a set of $m$ demand pairs $\subseteq V \times V$, the objective is to construct a minimum-cost subgraph that connects at least $k$ demands. The problem is hard to approximate---the best-known approximation ratio is $O(\min\{\sqrt{n}, \sqrt{k}\})$. Furthermore, $k$-forest is as hard to approximate as the notoriously-hard densest $k$-subgraph problem. While the $k$-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the $k$-forest problem can be viewed as to remove at most $m-k$ demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the $k$-forest problem that, for every $\epsilon>0$, removes at most $m-k$ demands and has cost no more than $O(1/\epsilon^{2})$ times the cost of an optimal algorithm that removes at most $(1-\epsilon)(m-k)$ demands