Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for testing if a given graph has a unique perfect matching ?" We present such an algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson [9] have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there is a deterministic NC algorithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see [1]). Rabin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in...

We present new algebraic approaches for several wellknown combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d − 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardness-randomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the the arithmetic Nisan-Wigderson generator of [KI04] together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f(x1,..., xn)) ≡ 0 then f has a circuit of depth d + 3 and size O(s · r + m r), where m is the total degree of f and r is the degree of y in P.

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f(¯x) = ∑ k i=1 hi(¯x) · gi(¯x), where each hi is a polynomial that depends on only ρ linear functions, and each gi is a product of linear functions (when hi = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi(deg(hi · gi)) = d we say that f is computable by a ΣΠΣ(k, d, ρ) circuit. We obtain the following results. 1. A deterministic black-box identity testing algorithm for ΣΠΣ(k, d, ρ) circuits that runs in quasi-polynomial time (for ρ = polylog(n + d)). In particular this gives the first black-box quasi-polynomial time PIT algorithm for depth-3 circuits with k multiplication gates. 2. A deterministic black-box identity testing algorithm for read-k ΣΠΣ circuits (depth-3 circuits where each variable appears at most k times) that runs in time n 2O(k2). In particular

Tutte introduced a V by V skew--symmetric matrix T = (t ij ), called the Tutte matrix, associated with a simple graph G = (V; E). He associates an indeterminate z e with each e 2 E, then defines t ij = \Sigmaz e when ij = e 2 E, and t ij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai--Edmonds decomposition, we describe a very simple yet efficient algorithm that replaces the indeterminates with constants without losing rank. Hence, by computing the rank of the resulting matrix, we can efficiently compute the size of a maximum matching of a graph. 1 Introduction Let G = (V; E) be a simple graph, and let (z e : e 2 E) be algebraically independent commuting indeterminates. We define a V by V skew--symmetric matrix T = (t ij ), called the Tutte matrix of G, such that t ij = \Sigmaz e if ij = e 2 E, and t ij = 0 otherwise. Tutte observed that T is nonsingular (that is, its determinant is not...

It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a P-time algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NC-algorithm for this problem. Unlike the Miller...

We present new algebraic approaches for two well-known combinatorial problems: non-bipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions.

We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests. 1

We focus on combinatorial problems arising from symmetric and skew-symmetric matrices. For much of the thesis we consider properties concerning the principal submatrices. In particular, we are interested in the property that every nonsingular principal submatrix is unimodular; matrices having this property are called principally unimodular. Principal unimodularity is a generalization of total unimodularity, and we generalize key polyhedral and matroidal results on total unimodularity. Highlights include a generalization of Hoffman and Kruskal's result on integral polyhedra, a generalization of Tutte's results on regular matroids, and partial results toward a decomposition theorem. Quite separate from the study of principal unimodularity we consider a particular skew-symmetric matrix of indeterminates associated with a graph. This matrix, called the Tutte matrix, was introduced by Tutte to study matchings. By considering the rank of an arbitrary submatrix of the Tutte matrix we disco...

We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with non-zero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2 s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only black-box access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linnik’s Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.