Random Max 2-sat and Max Cut
نویسندگان
چکیده
Given a 2-SAT formula F consisting of n variables and b n random clauses, what is the largest number of clauses maxF satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of 34 n and n. We prove that for < 1, the expected number of clauses satisfiable is b n o(1); for large , it is ( 34 + (p ))n; for = 1 + ", it is at least (1 + " O("3))n and at most (1+ " ("3= ln "))n; and in the “scaling window” = 1+ (n 1=3), it is n (1). In all cases, maxF is concentrated. In addition, we consider two “online” versions of the MAX 2-SAT problem. For one, we show that a “lazy” algorithm is optimal, and we analyze it exactly. Finally, we prove analogous results for “random MAX CUT”, the size of a maximum cut in a sparse random graph.
منابع مشابه
Solving Sparse Semi-Random Instances of Max Cut and Max CSP in Linear Expected Time
We show that a maximum cut of a random graph below the giantcomponent threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable-constraint satisfaction problems, or Max 2-CSPs. In addition to Max Cut, Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum ...
متن کاملSolving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time
We show that a maximum cut of a random graph below the giantcomponent threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independen...
متن کاملNew Inference Rules for Max-SAT
Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of origina...
متن کاملun 2 00 3 Random max sat , Random max cut , and Their Phase Transitions
Given a 2-sat formula F consisting of n variables and ⌊cn⌋ random clauses, what is the largest number of clauses maxF satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of 3 4 cn and cn. We prove that for c < 1, the expected number of clauses satisfiable is ⌊cn⌋ −Θ(1/n); for large c, it is ( 4 c+Θ( √ c))n; for c = 1+ε, it is at least (1+ε−O(ε3)...
متن کاملA Comparison of Methods for Solving MAX-SAT Problems
We compare the performance of 3 approaches to the solution of MAX-SAT problems, including a version of the DavisPutnam-Loveland algorithm extended to solve MAX-SAT, an integer programming branch-and-cut algorithm and an algorithm for MAX-2-SAT problems based on a semide nite programming relation.
متن کامل