Markus Bläser

On the complexity of the multiplication of matrices of small formats

We prove a lower bound of 2mn + 2n - m - 2 for the bilinear complexity of the multiplication of n × m-matrices with m × n-matrices using the substitution method (m ≥ n ≥ 3). In particular, we obtain the improved lower bound of 19 for the bilinear complexity of 3 × 3-matrix multiplication.

A 3/4-approximation algorithm for maximum ATSP with weights zero and one

We present a polynomial time 3/4-approximation algorithm for the maximum asymmetric TSP with weights zero and one.

Computing small partial coverings

We study the generalization of covering problems such as the set cover problem to partial covering problems. Here we only want to cover a given number k of elements rather than all elements. For instance, in the k-partial (weighted) set cover problem, we wish to compute a minimum weight collection of sets that covers at least k elements. As a main result, we show that the k-partial set cover problem and its special cases like the k-partial vertex cover problem are all fixed parameter tractable (with parameter k). As a second example, we consider the minimum weight k-partial t-restricted cycle cover problem.

Complexity of the cover polynomial

The cover polynomial introduced by Chung and Graham is a two-variate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is #Phard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomial: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial is #Phard under polynomial-time Turing reductions, while only three points are easy. Our construction uses a gadget which is easier to analyze and more general than the XOR-gadget used by Valiant in his proof that the permanent is #P-complete.

Deterministically testing sparse polynomial identities of unbounded degree

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 Linniks 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.

Computing Cycle Covers without Short Cycles

A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cycle cover with minimum weight. We present factor 4=3 approximation algorithms for Min-k-DCC with running time O(n5/2) (independent of k). Specifically, we obtain a factor 4/3 approximation algorithm for the asymmetric travelling salesperson problem with distances one and two and a factor 2/3 approximation algorithm for the directed path packing problem with the same running time. On the other hand, we show that k-DCC is NP-complete for k ≥ 3 and that Min-k-DCC has no PTAS for k ≥ 4, unless P = NP. Furthermore, we design a polynomial time factor 7/6 approximation algorithm for Min-k-UCC. As a lower bound, we prove that Min-k-UCC has no PTAS for k ≥ 12, unless P = NP.

ON THE COMPLEXITY OF THE INTERLACE POLYNOMIAL

We consider the two-variable interlace polynomial introduced by Arratia, Bollobas and Sorkin (2004). We develop graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except on one line, where it is trivially polynomial time computable, and four lines, where the complexity is still open. This solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at any point except at 0.

A 5/2n/sup 2/-lower bound for the rank of n/spl times/n-matrix multiplication over arbitrary fields

We prove a lower bound of 5/2n/sup 2/-3n for the rank of n/spl times/n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.We prove a lower bound of 5/2n/sup 2/-3n for the rank of n/spl times/n-matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

Approximating Maximum Weight Cycle Covers in Directed Graphs with Weights Zero and One

Abstract A cycle cover of a graph is a spanning subgraph, each node of which is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. Given a complete directed graph with edge weights zero and one, Max-k-DDC(0,1) is the problem of finding a k-cycle cover with maximum weight. We present a 2/3 approximation algorithm for Max-k-DDC(0,1) with running time O(n 5/2). This algorithm yields a 4/3 approximation algorithm for Max-k-DDC(1,2) as well. Instances of the latter problem are complete directed graphs with edge weights one and two. The goal is to find a k-cycle cover with minimum weight. We particularly obtain a 2/3 approximation algorithm for the asymmetric maximum traveling salesman problem with distances zero and one and a 4/3 approximation algorithm for the asymmetric minimum traveling salesman problem with distances one and two. As a lower bound, we prove that Max-k-DDC(0,1) for k ≥ 3 and Max-k-UCC(0,1) (finding maximum weight cycle covers in undirected graphs) for k ≥ 7 are \APX-complete.

Lower bounds for the multiplicative complexity of matrix multiplication

Abstract. We prove a lower bound of km + mn + k—m + n— 3 for the multiplicative complexity of the multiplication of