TY - JOUR
T1 - Satisfiability decay along conjunctions of pseudo-random clauses
AU - Shamir, Eli
PY - 2006/10
Y1 - 2006/10
N2 - k-SAT is a fundamental constraint satisfaction problem. It involves S(m), the satisfaction set of the conjunction of m clauses, each clause a disjunction of k literals. The problem has many theoretical, algorithmic and practical aspects. When the clauses are chosen at random it is anticipated (but not fully proven) that, as the density parameter m/n (n the number of variables) grows, the transition of S(m) to being empty, is abrupt: It has a "sharp threshold", with probability 1 - o(1). In this articlewe replace the random ensemble analysis by a pseudo-random one: Derive the decay rule for individual sequences of clauses, subject to combinatorial conditions, which in turn hold with probability 1 - o(1). This is carried out under the big relaxation that k is not constant but k = γ log n, or even r log log n. Then the decay of S is slow, "near-perfect" (like a radioactive decay), which entails sharp thresholds for the transition-time of S below any given level, down to S = 0.
AB - k-SAT is a fundamental constraint satisfaction problem. It involves S(m), the satisfaction set of the conjunction of m clauses, each clause a disjunction of k literals. The problem has many theoretical, algorithmic and practical aspects. When the clauses are chosen at random it is anticipated (but not fully proven) that, as the density parameter m/n (n the number of variables) grows, the transition of S(m) to being empty, is abrupt: It has a "sharp threshold", with probability 1 - o(1). In this articlewe replace the random ensemble analysis by a pseudo-random one: Derive the decay rule for individual sequences of clauses, subject to combinatorial conditions, which in turn hold with probability 1 - o(1). This is carried out under the big relaxation that k is not constant but k = γ log n, or even r log log n. Then the decay of S is slow, "near-perfect" (like a radioactive decay), which entails sharp thresholds for the transition-time of S below any given level, down to S = 0.
KW - Constraint satisfaction
KW - k-SAT
KW - Learning from examples
KW - Pseudorandom analysis
KW - Sharp thresholds
KW - Unique k-SAT representations
UR - http://www.scopus.com/inward/record.url?scp=33845715946&partnerID=8YFLogxK
U2 - 10.1093/jigpal/jzl011
DO - 10.1093/jigpal/jzl011
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AN - SCOPUS:33845715946
SN - 1367-0751
VL - 14
SP - 815
EP - 825
JO - Logic Journal of the IGPL
JF - Logic Journal of the IGPL
IS - 5
ER -