On the complexity of the multiplication of matrices of small formats

AbstractWe 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

The border support rank of two-by-two matrix multiplication is seven

We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format four-by-four-by-four and border rank at most six, but that do not vanish simultaneously on any tensor with the same support as the two-by-two matrix multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and Landsberg. We also give two proofs that the support rank of the two-by-two matrix multiplication tensor is seven over any field: one proof using a result of De Groote saying that the decomposition of this tensor is unique up to sandwiching, and another proof via the substitution method. These results answer a question asked by Cohn and Umans. Studying the border support rank of the matrix multiplication tensor is relevant for the design of matrix multiplication algorithms, because upper bounds on the border support rank of the matrix multiplication tensor lead to upper bounds on the computational complexity of matrix multiplication, via a construction of Cohn and Umans. Moreover, support rank has applications in quantum communication complexity

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

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-complet

A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree-r that are the limit of size-s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals

Adding cardinality constraints to integer programs with applications to maximum satisfiability

Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most ℓ literals, we obtain the problem Max-ℓSAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398–405] designed a (1−e−1)-approximation algorithm for Max-SAT-CC. This result is tight unless P=NP [U. Feige, J. ACM 45 (4) (1998) 634–652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-ℓSAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1-(1-1/ℓ)ℓ-ε. To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space

Private Computation: k-Connected versus 1-Connected Networks

We study the role of connectivity of communication networks in private computations under information theoretical settings in the honest-but-curious model. We show that some functions can be 1-privately computed even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be 1-privately computed on non-2-connected networks. Furthermore, we present a technique for simulating 1-private protocols that work on arbitrary (complete) networks on k-connected networks. For this simulation, at most $(1 - k/(n - 1)) \cdot L$ additional random bits are needed, where L is the number of bits exchanged in the original protocol and n is the number of players. Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks 1-privately, namely $\lceil (n - 2)/(k - 1) \rceil - 1$ random bits for networks consisting of n player