NOTE – Edge-Coloring Cliques with Many Colors on Subcliques

with colors in which the edges of every copy of together receive at least 3 colors. We prove that this construction also has the property that at least colors appear on the edges of every copy of for . Received August 11, 1998     

Note – Edge-Coloring Cliques with Three Colors on All 4-Cliques

is constructed such that every copy of has at least three colors on its edges. As , the number of colors used is . This improves upon the previous probabilistic bound of due to Erdős and Gyárfás. Received: March 12, 1998     

Fast Algorithms for Finding O(Congestion + Dilation) Packet Routing Schedules

O (c+d) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path. The proof, however, used the Lovász Local Lemma and was not constructive. In this paper, we show how to find such a schedule in time, with probability , for any positive constant β, where is the sum of the lengths of the paths taken by the packets in the network, and m is the number of edges used by some packet in the network. We also show how to parallelize the algorithm so that it runs in NC. The method that we use to construct the schedules is based on the algorithmic form of the Lovász Local Lemma discovered by Beck. Received: July 8, 1996     

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     

Deterministic Routing With Bounded Buffers: Turning Offline Into Online Protocols

In this paper we present a deterministic protocol for routing arbitrary permutations in arbitrary networks. The protocol is analyzed in terms of the size of the network and the routing number of the network. Given a network H of n nodes, the routing number of H is defined as the maximum over all permutations on {1, ..., n} of the minimal number of steps to route offline in H. We show that for any network H of size n with routing number R our protocol needs time to route any permutation in H using only constant size edge buffers. This significantly improves all previously known results on deterministic routing. In particular, our result yields optimal deterministic routing protocols for arbitrary networks with diameter or bisection width , constant. Furthermore we can extend our result to deterministic compact routing. This yields, e.g., a deterministic routing protocol with runtime O(R logn) for arbitrary bounded degree networks if only O(logn) bits are available at each node for storing routing information. Our protocol is a combination of a generalized ``routing via simulation' technique with an new deterministic protocol for routing h-relations in an extended version of a multibutterfly network. This protocol improves upon all previous routing protocols known for variants of the multibutterfly network. The ``routing via simulation' technique used here extends a method previously introduced by the authors for designing compact routing protocols. Received July 18, 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     

On Bipartite Graphs with Linear Ramsey Numbers

Dedicated to the memory of Paul Erdős We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most is less than . This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all and there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than for some absolute constant c>1. Received December 1, 1999 RID="*" ID="*" Supported by NSF grant DMS-9704114 RID="**" ID="**" Supported by KBN grant 2 P03A 032 16     

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     

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     

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     

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     

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     

NOTE Spectral Characterization of Tree-Width-Two Graphs

) of a graph G, similar in spirit to his now-classical invariant . He showed that is minor-monotone and is related to the tree-width la(G) of G: and, moreover, , i.e. G is a forest. We show that and give the corresponding forbidden-minor and ear-decomposition characterizations. Received October 9, 1997/Revised July 27, 1999     

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     

Concentration of Multivariate Polynomials and Its Applications

are independent random variables which take values either 0 or 1, and Y is a multi-variable polynomial in 's with positive coefficients. We give a condition which guarantees that Y concentrates strongly around its mean even when several variables could have a large effect on Y. Some applications will be discussed. Received March 29, 1999     

Bicliques in Graphs I: Bounds on Their Number

Bicliques are inclusion-maximal induced complete bipartite subgraphs in graphs. Upper bounds on the number of bicliques in bipartite graphs and general graphs are given. Then those classes of graphs where the number of bicliques is polynomial in the vertex number are characterized, provided the class is closed under induced subgraphs. Received January 27, 1997     

On Large Systems of Sets with No Large Weak Δ-subsystems

weak Δ-system if the cardinality of the intersection of any two sets is the same. We elaborate a construction by R?dl and Thoma [9] and show that for large n, there exists a family ℱ of subsets of without weak Δ-systems of size 3 with . Received: October 1, 1997     

On the Relation Between Two Minor-Monotone Graph Parameters

, where μ and λ are minor-monotone graph invariants introduced by Colin de Verdière [3] and van der Holst, Laurent, and Schrijver [5]. It is also shown that a graph G exists with . The graphs G with maximal planar complement and , characterised by Kotlov, Lovász, and Vempala, are shown to be forbidden minors for . Received: June 13, 1997     

The Distance-regular Graphs with Intersection Number and with an Eigenvalue

and with an eigenvalue . Received: October 2, 1995/Revised: Revised November 26, 1997