 
Summary: Hardness of Fully Dense Problems
Nir Ailon
Noga Alon
July 7, 2005
Abstract
In the past decade, there has been a stream of work in designing approximation schemes for
dense instances of NPHard problems. These include the work of Arora, Karger and Karpinski
from 1995 and that of Frieze and Kannan from 1996. We address the problem of proving
hardness results for (fully) dense problems, which has been neglected despite the fruitful effort
put in upper bounds. In this work we prove hardness results of dense instances of a broad family
of CSP problems, as well as a broad family of ranking problems which we refer to as CSP
Rank. Our techniques involve a construction of a pseudorandom hypergraph coloring, which
generalizes the wellknown Paley graph, recently used by Alon to prove hardness of feedback
arcset in tournaments.
1 Introduction
Dense instances of MAXSNP problems are known to be easier to approximate than the general
case [4, 5, 8, 1215, 17]. In 1995, Arora et al. [8] proved that there exist approximation schemes
for MAXSNP problems for dense instances, and introduced the technique of smooth programs.
Later in 1996, Frieze et al. [14] proved an efficient verion of the regularity lemma and used it as a
general framework for approximation scheme for MAXSNP problems. (Some earlier special cases
