We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form (-1)^s g, where the polynomial s determines the sign of the weight and the non-negative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is in polynomial time if either every weight function can be defined by a "pure affine" magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of "product type" with a linear sign polynomial. In all other cases, computing the partition function is FP^#P-complete.
The case of mixed signs has been been considered previously by Goldberg, Grohe, Jerrum and Thurley @cite_12 , in the case of one symmetric binary function on an arbitrary finite domain. Their theorem generalizes that of Bulatov and Grohe @cite_21 for the non-negative case. @cite_12 give two examples, which can also be expressed as Boolean weighted @math , and fall within the scope of this paper. The first appeared as an open problem in @cite_21 . The complexity of these problems can be deduced from @cite_12 and from the results of this paper.
@cite_21 @cite_12
2282669851 2000931246 2151450116 2042993834
We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k for any positive integer k > 1. This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 General random coding theorems for lattices are derived from the Minkowski-Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski-Hlawka theorem itself is obtained as the limit, for p spl rarr spl infin , of a simple lemma for linear codes over GF(p) used with p-level amplitude modulation. The relation between the combinatorial packing of solid bodies and the information-theoretic "soft packing" with arbitrarily small, but positive, overlap is illuminated. The "soft-packing" results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda-Poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 (P N). The following theorem is proved. Let f(x_1, , x_m) be a binary nonzero polynomial of m variables of degree . H the number of binary m -tuples (a_1, , a_m) with f(a_1, , a_m) = 1 is less than 2^ m- +1 , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. where m + and 3 . This theorem completely characterizes the codewords of the th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results.