A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents

Abstract

We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n×n matrix to within a multiplicative factor of e n . To this end we develop the first strongly polynomial-time algorithm for matrix scaling -an important nonlinear optimization problem with many applications. Our work suggests a simple new (slow) polynomial time decision algorithm for bipartite perfect matching, conceptually different from classical approaches. A DETERMINISTIC STRONGLY POLYNOMIAL ALGORITHM 547Our approach to this problem is completely different from the previously taken routes. It involves a natural reduction technique between problem instances: scaling. Observe the following linearity of permanents: Multiplying a row or column by a constant c, multiplies the permanent by c as well. More generally, we say that a matrix B is a scaling of A (by positive vectors x, y ∈ ( + ) n ) if B = XAY , where X = diag(x) and Y = diag(y) are diagonal matrices with x (resp. y) on their diagonal (these being the factors that multiply the rows and the columns respectively). As observed,per(A). Thus scaling reductions not only allow us to compute per(A) from per(B), but in fact any k-factor approximation of per(B) efficiently yields the same approximation for per(A).The idea, then, is to scale an input matrix A into a matrix B whose permanent we can efficiently approximate. A natural strategy is to seek an efficient algorithm for scaling A to a doubly stochastic B. For suppose we succeed: the permanent of B is clearly at most 1, and per(B) ≥ n!/n n > e −n by the lower bound of [7,8]. Consequently, per(A) is also approximated to within an e n factor, as claimed.Note that such a scaling may not always exist -when per(A)= 0. Moreover, even if scaling exists, the scaling vectors x, y may have irrational coordinates, so we may have to settle for an approximately doubly stochastic matrix. The scaling algorithm must, therefore, be accompanied by approximate versions of the van der Waerden bound, and indeed we prove results of the following type (see also proposition 5.1).Lemma 1.2. Let B be a nonnegative n × n matrix, in which all row sums are 1, and where no column sum exceeds 1+ 1 n 2 , then per(B) ≥ e −(n+1) .So we want to efficiently scale A to an almost doubly stochastic matrix. This scaling problem that so naturally arose from our considerations, turned out to have been studied in other contexts as well. The next subsection briefly describes these as well as scaling algorithms -old and new. Matrix scaling Background.Our discussion will be restricted to square matrices, though everything generalizes to rectangular matrices (and some of it even to multidimensional arrays).Let r, c ∈ ( + ) n be two positive vectors with r i = c j . A matrix B is an (r, c)-matrix if r and c are the vectors of row and columns sums of B

Hamiltonian cycles in Dirac graphs

“…This has algorithmic implications: since h(G), the maximum of a concave function subject to linear constraints, can be estimated efficiently, we have an efficient algorithm for estimating both Ψ and Φ for Dirac graphs to within subexponential factors. This is reminiscent of a beautiful result of Linial et al [12] on approximating permanents of nonnegative matrices. They show in particular that one can approximate such a permanent to within a factor e n (actually meaning to within e n/2 ) in deterministic polynomial time.…”

“…(That (45) implies (46) follows from -and was the reason for -the definition of δ m given in (12). Since the coefficient of g(V ) on the r.h.s.…”

“…(Noting that δ >2κρ (see (12)) we find that the r.h.s. of (63) is at least 2κ 2 ρf (V )/n ≥ 2κ 2 a/n > n 1/3 a (whereas qa < a log O(1) n).…”

The Complexity of Factors of Multivariate Polynomials

“…This hypothesis should be compared with the known work on polynomial time deterministic or randomized approximation algorithms for the permanent of nonnegative matrices [31], [2], [21]. Based on the Markov chain approach, Jerrum, Sinclair, and Vigoda [21] have recently established a fully polynomial randomized approximation scheme for computing the permanent of an arbitrary real matrix with nonnegative entries.…”

On the hardness of approximating the permanent of structured matrices

“…Recall that the class P is the class cc 11 (2002) Approximating the permanent of structured matrices 159 of functions counting the number of accepting computations in a nondeterministic polynomial time Turing machine (see Valiant 1979a,b,c), while the class BPP is the analogue of the class P for probabilistic computations (with bounded error). The inapproximability results mentioned above apply to arbitrary matrices and take advantage of the random self-reducibility properties of the permanent (see Cai et al 1999;Feige & Lund 1996Linial et al 1998 and references therein). Such results should be contrasted with the randomized polynomial time approximation schemes, which apply to matrices with nonnegative entries (Jerrum et al 2000).…”