Superpolynomial Lower Bounds for Monotone Span Programs

monotone span programs computing explicit functions. The best previous lower bound was by Beimel, Gál, Paterson [7]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. We prove an lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields. Received: August 1, 1996     

The Combinatorics of Biased Riffle Shuffles

Received: September 23, 1997     

Separation of the Monotone NC Hierarchy

, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC ≠ monotone-P. 2. For every i≥1, monotone-≠ monotone-. 3. More generally: For any integer function D(n), up to (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const). Only a separation of monotone- from monotone- was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds: 1.  For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result. 2.  For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known. Received: December 19, 1997     

On a Degree Property of Cross-Intersecting Families

Motivated by some applications in computational complexity, Razborov and Vereshchagin proved a degree bound for cross-intersecting families in [1]. We sharpen this result and show that our bound is best possible by constructing appropriate families. We also consider the case of cross-t-intersecting families. Received October 28, 1999     

Equivalence in Finite-Variable Logics is Complete for Polynomial Time

of first-order logic whose formulas contain at most k variables (for some ). We show that for each , equivalence in the logic is complete for polynomial time. Moreover, we show that the same completeness result holds for the powerful extension of with counting quantifiers (for every ). The k-dimensional Weisfeiler–Lehman algorithm is a combinatorial approach to graph isomorphism that generalizes the naive color-refinement method (for ). Cai, Fürer and Immerman [6] proved that two finite graphs are equivalent in the logic if, and only if, they can be distinguished by the k-dimensional Weisfeiler-Lehman algorithm. Thus a corollary of our main result is that the question of whether two finite graphs can be distinguished by the k-dimensional Weisfeiler–Lehman algorithm is P-complete for each . Received: March 23, 1998     

Non-Expansive Hashing

non-expansive hashing scheme, similar inputs are stored in memory locations which are close. We develop a non-expansive hashing scheme wherein any set of size from a large universe may be stored in a memory of size (any , and ), and where retrieval takes operations. We explain how to use non-expansive hashing schemes for efficient storage and retrieval of noisy data. A dynamic version of this hashing scheme is presented as well. Received: February 5, 1996     

Bounds on Pairs of Families with Restricted Intersections

such that for any , . We are interested in the maximum product , given r and L. We give asymptotically optimal bounds for L containing only elements of s<q residue classes modulo q, where q is arbitrary (even non-prime) and s is a constant. As a consequence, we obtain a version of the Frankl–R?dl result about forbidden intersections for the case of two forbidden intersections. We also give tight bounds for L={0,...,k}. Received: August 5, 1998     

Algorithms for Measuring Perturbability in Matroid Optimization

perturbability function of a matroid measures the maximum increase in the weight of its minimum weight bases that can be produced by increases of a given total cost on the weights of its elements. We present an algorithm for computing this function that runs in strongly polynomial time for matroids in which independence can be tested in strongly polynomial time. Furthermore, for the case of transversal matroids we are able to take advantage of their special structure to design a faster algorithm for computing the perturbability function. Received: June 13, 1997     

On the Signed Domination in Graphs

G on n vertices with minimum degree r, there exists a two-coloring of the vertices of G with colors +1 and -1, such that the closed neighborhood of each vertex contains more +1's than -1's, and altogether the number of 1's does not exceed the number of -1's by more than . As a construction by Füredi and Mubayi shows, this is asymptotically tight. The proof uses the partial coloring method from combinatorial discrepancy theory. Received May 12, 1998     

Quick Approximation to Matrices and Applications

m ×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, R?dl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. From our matrix approximation, the above graph algorithms and the Regularity Lemma and several other results follow in a simple way. We generalize our approximations to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems. Received: July 25, 1997     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

On Linear Combinatorics III. Few Directions and Distorted Lattices

Our main tools will be certain ``distorted' lattices which contain many rich lines (i.e. straight lines incident upon many points of the lattice). Received: November 28, 1996     

List Homomorphisms and Circular Arc Graphs

G , H, and lists , a list homomorphism of G to H with respect to the lists L is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Received: July 22, 1996/Revised: Revised June 10, 1998     

A Bound on the Total Chromatic Number

. The proof is probabilistic. Received: November 26, 1996     

Decomposing Hypergraphs into Simple Hypertrees

T be a simple k-uniform hypertree with t edges. It is shown that if H is any k-uniform hypergraph with n vertices and with minimum degree at least , and the number of edges of H is a multiple of t then H has a T-decomposition. This result is asymptotically best possible for all simple hypertrees with at least two edges. Received December 28, 1998     

Adaptive Multicolouring

k -colourable graphs, both for the case when no recolouring is allowed, and for a case where limited recolouring is allowed. Received November 14, 1996     

Graphs Whose Circular Chromatic Number Equals the Chromatic Number

Received: October 24, 1997     

Isomorphism of Coloured Graphs with Slowly Increasing Multiplicity of Jordan Blocks

n -vertex edge coloured graphs with multiplicity of Jordan blocks bounded by k can be done in time . Received: November 29, 1994     

Testing Monotonicity

at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). The complexity of the test is O(n/ε). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions. Received March 29, 1999     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997