Some optimal inapproximability results
Johan H˚
astad
Royal Institute of Technology
Sweden
email:[email protected]
September 4, 1998
Abstract
We prove optimal, up to an arbitrary
ǫ >
0, inapproximability results
for MaxE
k
Sat
1
for
k
≥
3, maximizing the number of satisfied linear equa
tions in an overdetermined system of linear equations modulo a prime
p
and Set Splitting. As a consequence of these results we get improved lower
bounds for the efficient approximability of many optimization problems
studied previously. In particular, for MaxE2Sat, MaxCut, MaxDiCut,
and Vertex cover. For MaxE2Sat the obtained lower bound is essentially
22
/
21
≈
1
.
047 while the strongest upper bound is around 1
.
074.
1
Introduction
We know that many natural optimization problems are NPhard. This means
that they are probably hard to solve exactly in the worst case.
In practice,
however, it is many times sufficient to get reasonable good solutions for all (or
even most) instances. In this paper we study the existence of polynomial time
approximation algorithms for some of the basic NPcomplete problems.
We
say that an algorithm is a
C
approximation algorithm if it for each instance
produces an answer that is at most off by a factor
C
from the optimal answer.
The fundamental question is for a given NPcomplete problem, for what value
of
C
can we hope for a polynomial time
C
approximation algorithm.
Posed
in this generality this is a large research area with many positive and negative
results.
In this paper we concentrate on negative results, i.e.
results of the
form that for some
C >
1 a certain problem cannot be approximated within
C
in polynomial time. These results are invariably based on plausible complexity
theoretic assumptions, the weakest possible being NP
negationslash
=P since if NP=P, all
considered problems can be solved exactly in polynomial time.
The most basic NPcomplete problem is satisfiability of CNFformulas and
probably the most used variant of this is 3SAT where each clause contains at
most 3 variables. For simplicity, let us assume that each clause contains exactly
1
MaxE
k
Sat is the variant of CNFSat where each clause is of length exactly
k
.
1
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3 variables.
The optimization variant of this problem is to satisfy as many
clauses as possible.
It is not hard to see that a random assignment satisfies
each clause with probability 7/8 and hence if there are
m
clauses it is not hard
(even deterministically) to find an assignment that satisfies 7
m/
8 clauses. Since
we can never satisfy more than all the clauses this gives a 8
/
7approximation
algorithm. This was one of the first approximation algorithms considered [17]
and one of the main results of this paper is that this is optimal to within an
arbitrary additive constant
ǫ
.
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