eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2014-09-04
433
448
10.4230/LIPIcs.APPROX-RANDOM.2014.433
article
Parity is Positively Useless
Wenner, Cenny
We give the first examples of non-trivially positively-useless predicates subject only to P != NP. In particular, for every constraint function Q : {-1,1}^4 -> R, we construct Contraint-Satisfaction-Problem (CSP) instances without negations which have value at least 1-eps when evaluted for the arity-four odd-parity predicate, yet it is NP-hard to find a solution with value significantly better than a random biased assignment when evaluated for Q. More generally, we show that all parities except one are positively useless.
Although we are not able to exhibit a single protocol producing hard instances when evaluated for every Q, we show that two protocols do the trick. The first protocol is the classical one used by Håstad with a twist. We extend the protocol to multilayered Label Cover and employ a particular distribution over layers in order to limit moments of table biases. The second protocol is a modification of Chan's multi-question protocol where queried tuples of Label Cover vertices are randomized in such a way that the tables can be seen as being independently sampled from a common distribution and in effect having identical expected biases. We believe that our techniques may prove useful in further analyzing the approximability of CSPs without negations.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol028-approx-random2014/LIPIcs.APPROX-RANDOM.2014.433/LIPIcs.APPROX-RANDOM.2014.433.pdf
Approximation hardness
approximation resistance
parity
usefulness
negations
monotone
constraint satisfaction problems
smoothness
multilayer
L