Abstract

We study the following basic problem called Bi-Covering. Given a graph $G(V,E)$, find two (not necessarily disjoint) sets $A\subseteq V$ and $B\subseteq V$ such that $A\cup B = V$ and such that every edge $e$ belongs to either the graph induced by $A$ or the graph induced by $B$. The goal is to minimize $\max\{|A|,|B|\}$. This is the most simple case of the Channel Allocation problem [R. Gandhi et al., Networks, 47 (2006), pp. 225--236]. A solution that outputs $V,\emptyset$ gives ratio at most 2. We show that under a similar strong Unique Games Conjecture by Bansal and Khot [Optimal long code test with one free bit, in Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS'09, IEEE, 2009, pp. 453--462] there is no $2-\epsilon$ ratio algorithm for the problem, for any constant $\epsilon>0$. Given a bipartite graph, Max-Bi-Clique is a problem of finding the largest $k\times k$ complete bipartite subgraph. For the Max-Bi-Clique problem, a constant factor hardness was known un...

Highlights

  • We study the Bi-Covering problem - Given a graph G(V, E), find two sets A, B ⊆ V such that A ∪ B = V and that every edge e ∈ E belongs to either the graph induced by A or to the graph induced by B

  • We show that Bi-Covering problem is hard to approximate within any factor strictly less than 2 assuming a strong Unique Games Conjecture (UGC) similar to the one in [5]

  • Assuming a strong Unique Games Conjecture (Conjecture 12), given a graph G(V, E), it is NP-hard to distinguish between following two cases: 1. G has Bi-Covering of size at most (1/2 + )|V |. 2

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Summary

Introduction

The Channel Allocation Problem can be described as follows: there is a universe of topics, a fixed number of channels and a set of requests where each request is a subset of topics. The optimization version of Channel Allocation Problem asks for a way to satisfy all the request by minimizing the maximum number of topics sent through a channel. If we fix the number of channels to k = 2 the optimization problem exactly corresponds to the Bi-Covering problem. The optimization problem asks for two subsets A and B of V minimizing max{|A|, |B|} such that A ∪ B = V and every edge is totally contained in a graph induced by either A or B

Our Results
Algorithmic Results
Inapproximability of Bi-Covering
Preliminaries
Dictatorship Test
Completeness
Soundness
Actual Reduction
Labeling
Full Text
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