Improved Inapproximability Results for Maximum k-Colorable Subgraph

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

Partitioned Sampling of Public Opinions Based on Their Social Dynamics

Public opinion polling is usually done by random sampling from the entire population, treating individual opinions as independent. In the real world, individuals' opinions are often correlated, e.g., among friends in a social network. In this paper, we explore the idea of partitioned sampling, which partitions individuals with high opinion similarities into groups and then samples every group separately to obtain an accurate estimate of the population opinion. We rigorously formulate the above idea as an optimization problem. We then show that the simple partitions which contain only one sample in each group are always better, and reduce finding the optimal simple partition to a well-studied Min-r-Partition problem. We adapt an approximation algorithm and a heuristic algorithm to solve the optimization problem. Moreover, to obtain opinion similarity efficiently, we adapt a well-known opinion evolution model to characterize social interactions, and provide an exact computation of opinion similarities based on the model. We use both synthetic and real-world datasets to demonstrate that the partitioned sampling method results in significant improvement in sampling quality and it is robust when some opinion similarities are inaccurate or even missing

On the Hardness of Approximating MAX k-CUT and its Dual

. We study the Max k-Cut problem and its dual, the Min k-Partition problem. In the Min k-Partition problem, given a graph G = (V; E) and positive edge weights, we want to find an edge set of minimum weight whose removal makes G k-colorable. For the Max k-Cut problem we show that, if P 6= NP, no polynomial time approximation algorithm can achieve a relative error better than 1=34k. It is well known that a relative error of 1=k is obtained by a naive randomized heuristic. For the Min k-Partition problem, we show that for k ? 2 and for every ffl ? 0, there exists a constant ff such that the problem cannot be approximated within ffjV j 2\Gammaffl , even for dense graphs. Both problems are directly related to the frequency allocation problem for cellular (mobile) telephones, an application of industrial relevance. 1 Introduction In order to motivate the problems studied in this paper, consider the frequency allocation problem for cellular telephones. We are given a fixed set of k freque..