Abstract
We study the problem of 2Catalog Segmentation which is one of the several variants of segmentation problems, introduced by Kleinberg et al., that naturally arise in data mining applications. Formally, given a bipartite graph $G = (U, V, E)$ and parameter $r$, the goal is to output two subsets $V_1, V_2 subseteq V$, each of size $r$, to maximize, $sum_{u \in U} max {E(u, V_1), E(u, V_2)},$ where $E(u, V_i)$ is the set of edges between $u$ and the vertices in $V_i$ for $i = 1, 2$. There is a simple 2approximation for this problem, and stronger approximation factors are known for the special case when $r = V/2$. On the other hand, it is known to be NPhard, and Feige showed a constant factor hardness based on an assumption of average case hardness of random 3SAT.
In this paper we show that there is no PTAS for $2$Catalog Segmentation assuming that NP does not have subexponential time probabilistic algorithms, i.e. NP $\not\subseteq \cap_{\eps > 0}$ BPTIME($2^{n^\eps}$). In order to prove our result we strengthen the analysis of the QuasiRandom PCP of Khot, which we transform into an instance of $2$Catalog Segmentation. Our improved analysis of the QuasiRandom PCP proves stronger properties of the PCP which might be useful in other applications.
BibTeX  Entry
@InProceedings{saket:LIPIcs:2010:2885,
author = {Rishi Saket},
title = {{QuasiRandom PCP and Hardness of 2Catalog Segmentation}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)},
pages = {447458},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897231},
ISSN = {18688969},
year = {2010},
volume = {8},
editor = {Kamal Lodaya and Meena Mahajan},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2885},
URN = {urn:nbn:de:0030drops28858},
doi = {10.4230/LIPIcs.FSTTCS.2010.447},
annote = {Keywords: Hardness of Approximation, PCPs, Catalog Segmentation}
}
Keywords: 

Hardness of Approximation, PCPs, Catalog Segmentation 
Collection: 

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010) 
Issue Date: 

2010 
Date of publication: 

14.12.2010 