Abstract. Constraints have played a central role in cp because they capture key substructures of a problem and efficiently exploit them to boost inference. This paper intends to do the same thing for search, proposing constraint-centered heuristics which guide the exploration of the search space toward areas that are likely to contain a high number of solutions. We first propose new search heuristics based on solution counting information at the level of individual constraints. We then describe efficient algorithms to evaluate the number of solutions of two important families of constraints: occurrence counting constraints, such as alldifferent, and sequencing constraints, such as regular. In both cases we take advantage of existing filtering algorithms to speed up the evaluation. Experimental results on benchmark problems show the effectiveness of our approach. 1

We present a simple randomized algorithm for approximating permanents. The algorithm with inputs A, # > 0 produces an output XA with (1-#)per(A) XA (1+#)per(A) for almost all (0-1) matrices A. For any positive constant # > 0, and almost all (0-1) matrices the algorithm runs in time O(n #), i.e., almost linear in the size of the matrix, where # = #(n) is any function satisfying #(n) as n # #.

A number of combinatorial problems known to be of NP time complexity can be reduced to class P within a particular algebraic context. For example, the problem of determining whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ k, where Λ is an appropriate nilpotent

Subgraph counting is a fundamental problem in algorithm design and has many applications in data mining, biology, social networks, and many other domains. Over the past years this problem has been studied extensively from a theoretical point of view. Because of the intensive computational resources required, traditional algorithms are infeasible even for medium sized graphs. A natural way to address this problem in a massive graph is to use the data streaming model, where edges arrive in an arbitrary order and the algorithm is required to use limited memory to approximate the number of subgraphs. Prior to our work, most subgraph counting algorithms are based on edge sampling. In this paper we develop a novel approach for counting cycles of an arbitrary but fixed size in the turnstile model, i. e., the input stream is a sequence of edge insertions and deletions. Our algorithm is based on the idea of computing instances of complex-valued random variables over the given stream, and improves drastically upon the naïve sampling algorithms. In contrast to most existing approaches, our algorithm can also be easily applied in the distributed setting. We believe that the idea of using complex-valued random variables will find further applications, in particular with respect to also counting more general subgraphs.

Let An = (aij) n i,j=1 be an n × n positive matrix with entries in [a,b], 0 < a ≤ b. Let Xn = ( √ aijxij) n i,j=1 be a random matrix, where {xij} are i.i.d. N(0,1) random variables. We show that for large n, det(X T n Xn) concentrates sharply at the permanent of An, in the sense that n −1 log(det(X T n Xn)/perAn) →n→ ∞ 0 in probability. 1. Introduction. For a set F ⊂ R and integers n ≥ m, denote by M(n,m,F) the set of n × m matrices with entries in F. Put M(n,F) = M(n,n,F). Let Sn be the symmetric group of permutations acting on {1,...,n}. For A ∈ M(n,C), the permanent of A is defined as perA = ∑

Since the celebrated work of Jerrum, Sinclair, and Vigoda, we have known that the permanent of a {0, 1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain to sample perfect matchings of a bipartite graph. A separate strand of the literature has pursued the possibility of an alternate, algebraic polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. In the case where A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first—and therefore the number of trials we need to obtain a good estimate of the permanent—is (1 + O(1/d)) n when A is the algebra of d × d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, if d is constant—the only case in which an efficient algorithm is known—we show that the critical ratio exceeds 2 n /n O(d). Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group. 1