Affiliation:
1. Aalto University and KTH Royal Institute of Technology
2. KTH Royal Institute of Technology
Abstract
Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible.
To be more precise, we introduce the following notion of a predicate
P
being
useful
for a (real-valued) objective
Q
: given an almost satisfiable Max-
P
instance, there is an algorithm that beats a random assignment on the corresponding Max-
Q
instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate
P
is exactly the same as saying that
P
is useful for
P
itself.
We say that
P
is useless if it is not useful for any
Q
. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-
P
, it is hard to find an assignment such that the induced distribution on
k
-bit strings defined by the instance is not essentially uniform.
Under the
unique games conjecture
, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.
Funder
Natural Sciences and Engineering Research Council of Canada
European Research Council
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Reference17 articles.
1. Randomly Supported Independence and Resistance
2. On the Usefulness of Predicates
3. Approximation Resistant Predicates from Pairwise Independence
4. Chan S. O. 2012. Approximation resistance from pairwise independent subgroups. Tech. rep. 110 ECCC. Chan S. O. 2012. Approximation resistance from pairwise independent subgroups. Tech. rep. 110 ECCC.
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