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     

On the Existence of Disjoint Cycles in a Graph

G is a graph of order at least 3k with . Then G contains k vertex-disjoint cycles. Received: April 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     

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     

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     

Neighborhoods and Covering Vertices by Cycles

G =(V,E) is a 2-connected graph, and X is a set of vertices of G such that for every pair x,x' in X, , and the minimum degree of the induced graph <X> is at least 3, then X is covered by one cycle. This result will be in fact generalised by considering tuples instead of pairs of vertices. Let be the minimum degree in the induced graph <X>. For any , . If , and , then X is covered by at most (p-1) cycles of G. If furthermore , (p-1) cycles are sufficient. So we deduce the following: Let p and t () be two integers. Let G be a 2-connected graph of order n, of minimum degree at least t. If , and , then V is covered by at most cycles, where k is the connectivity of G. If furthermore , (p-1) cycles are sufficient. In particular, if and , then G is hamiltonian. Received April 3, 1998     

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     

Homomorphisms of Products of Graphs into Graphs Without Four Cycles

Given two graphs A and G, we write if there is a homomorphism of A to G and if there is no such homomorphism. The graph G is -free if, whenever both a and c are adjacent to b and d, then a = c or b = d. We will prove that if A and B are connected graphs, each containing a triangle and if G is a -free graph with and , then (here " denotes the categorical product). Received August 31, 1998/Revised April 19, 2000 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.     

A Minimum Degree Result for Disjoint Cycles and Forests in Graphs

F on s edges and k disjoint cycles. The main result is the following theorem. Let F be a forest on s edges without isolated vertices and let G be a graph of order at least with minimum degree at least , where k, s are nonnegative integers. Then G contains the disjoint union of the forest F and k disjoint cycles. This theorem provides a common generalization of previous results of Corrádi & Hajnal [4] and Brandt [3] who considered the cases (cycles only) and (forests only), respectively. Received: October 13, 1995     

The Combinatorics of Biased Riffle Shuffles

Received: September 23, 1997     

Principally Unimodular Skew-Symmetric Matrices

principally unimodular (PU) if every principal submatrix has determinant 0 or ±1. Let A be a symmetric (0, 1)-matrix, with a zero diagonal. A PU-orientation of A is a skew-symmetric signing of A that is PU. If A′ is a PU-orientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A′. This construction is based on the fact that the PU-orientations of indecomposable matrices are unique up to negation and multiplication of certain rows and corresponding columns by −1. This generalizes the well-known result of Camion, that if a (0, 1)-matrix can be signed to be totally unimodular then the signing is unique up to multiplying certain rows and columns by −1. Camion's result is an easy but crucial step in proving Tutte's famous excluded minor characterization of totally unimodular matrices. Received: May 17, 1996     

Mixed and Isoperimetric Estimates on the Log-Sobolev Constants of Graphs and Markov Chains

Two types of lower bounds are obtained on the log-Sobolev constants of graphs and Markov chains. The first is a mixture of spectral gap and logarithmic isoperimetric constant, the second involves the Gaussian isoperimetric constant. The sharpness of both types of bounds is tested on some examples. Product generalizations of some of these results are also briefly given. Received March 1, 1999/Revised July 17, 2000 RID="*" ID="*" Research supported in part by the NSF Grant No. DMS–9803239. The author greatly enjoyed the hospitality of CIMAT, Gto, Mexico, where part of this work was done.     

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     

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

d , and the testing algorithm can perform queries of the form: ``who is the ith neighbor of vertex v'. The tester should determine with high probability whether the graph is bipartite or ε-far from bipartite for any given distance parameter ε. Distance between graphs is defined to be the fraction of entries on which the graphs differ in their incidence-lists representation. Our testing algorithm has query complexity and running time where N is the number of graph vertices. It was shown before that queries are necessary (for constant ε), and hence the performance of our algorithm is tight (in its dependence on N), up to polylogarithmic factors. In our analysis we use techniques that were previously applied to prove fast convergence of random walks on expander graphs. Here we use the contrapositive statement by which slow convergence implies small cuts in the graph, and further show that these cuts have certain additional properties. This implication is applied in showing that for any graph, the graph vertices can be divided into disjoint subsets such that: (1) the total number of edges between the different subsets is small; and (2) each subset itself exhibits a certain mixing property that is useful in our analysis. Received: February 6, 1998     

A Bound on the Total Chromatic Number

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

The Linear-Array Conjecture in Communication Complexity Is False

linear array network consists of k+1 processors with links only between and (0≤i<k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at and y is stored at . Let be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, , where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which is essentially at most . Our construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice as large as the best lower bound obtainable this way. Received: March 1, 1996     

Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

. The sensitivity of a point is dist, i.e. the number of neighbors of the point in the discrete cube on which the value of differs. The average sensitivity of is the average of the sensitivity of all points in . (This can also be interpreted as the sum of the influences of the variables on , or as a measure of the edge boundary of the set which is the characteristic function of.) We show here that if the average sensitivity of is then can be approximated by a function depending on coordinates where is a constant depending only on the accuracy of the approximation but not on . We also present a more general version of this theorem, where the sensitivity is measured with respect to a product measure which is not the uniform measure on the cube. Received: November 12, 1996     

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     

An Anti-Ramsey Theorem

Let be the Turán number which gives the maximum size of a graph of order containing no subgraph isomorphic to . In 1973, Erdős, Simonovits and Sós [5] proved the existence of an integer such that for every integer , the minimum number of colours , such that every -colouring of the edges of which uses all the colours produces at least one all whose edges have different colours, is given by . However, no estimation of was given in [5]. In this paper we prove that for . This formula covers all the relevant values of n and p. Received January 27, 1997/Revised March 14, 2000