4,668 research outputs found
A complexity dichotomy for poset constraint satisfaction
In this paper we determine the complexity of a broad class of problems that
extends the temporal constraint satisfaction problems. To be more precise we
study the problems Poset-SAT(), where is a given set of
quantifier-free -formulas. An instance of Poset-SAT() consists of
finitely many variables and formulas
with ; the question is
whether this input is satisfied by any partial order on or
not. We show that every such problem is NP-complete or can be solved in
polynomial time, depending on . All Poset-SAT problems can be formalized
as constraint satisfaction problems on reducts of the random partial order. We
use model-theoretic concepts and techniques from universal algebra to study
these reducts. In the course of this analysis we establish a dichotomy that we
believe is of independent interest in universal algebra and model theory.Comment: 29 page
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Let C be a depth-3 circuit with n variables, degree d and top fanin k (called
sps(k,d,n) circuits) over base field F. It is a major open problem to design a
deterministic polynomial time blackbox algorithm that tests if C is identically
zero. Klivans & Spielman (STOC 2001) observed that the problem is open even
when k is a constant. This case has been subjected to a serious study over the
past few years, starting from the work of Dvir & Shpilka (STOC 2005).
We give the first polynomial time blackbox algorithm for this problem. Our
algorithm runs in time poly(nd^k), regardless of the base field. The only field
for which polynomial time algorithms were previously known is F=Q (Kayal &
Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first
blackbox algorithm for depth-3 circuits that does not use the rank based
approaches of Karnin & Shpilka (CCC 2008).
We prove an important tool for the study of depth-3 identities. We design a
blackbox polynomial time transformation that reduces the number of variables in
a sps(k,d,n) circuit to k variables, but preserves the identity structure.Comment: 14 pages, 1 figure, preliminary versio
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