A note on near‐optimal coloring of shift hypergraphs

As shown in the original work on the Lovász Local Lemma due to Erd?s & Lovász (Infinite and Finite Sets, 1975), a basic application of the Local Lemma answers an infinitary coloring question of Strauss, showing that given any integer set S, the integers may be k‐colored so that S and all its translates meet every color. The quantitative bounds here were improved by Alon, Kriz & Nesetril (Studia Scientiarum Mathematicarum Hungarica, 1995). We obtain an asymptotically optimal bound in this note, using the technique of iteratively applying the Lovász Local Lemma in order to prune dependencies. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 53–56, 2016     

On Lovász’ lattice reduction and the nearest lattice point problem

Answering a question of Vera Sós, we show how Lovász’ lattice reduction can be used to find a point of a given lattice, nearest within a factor ofc ^d (c = const.) to a given point in R ^d . We prove that each of two straightforward fast heuristic procedures achieves this goal when applied to a lattice given by a Lovász-reduced basis. The verification of one of them requires proving a geometric feature of Lovász-reduced bases: ac ₁ ^d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, wherec ₁ = √2/3. As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor ofC ^d . In another application, we improve the Grötschel-Lovász-Schrijver version of H. W. Lenstra’s integer linear programming algorithm. The algorithms, when applied to rational input vectors, run in polynomial time.     

An algorithmic approach to the Lovász local lemma. I

The Lovász Local Lemma is a remarkable sieve method to prove the existence of certain structures without supplying any efficient way of finding these structures. In this article we convert some of the applications of the Local Lemma into polynomial time sequential algorithms (at the cost of a weaker constant factor in the “exponent”). Our main example is the following: assume that in an n-uniform hypergraph every hyperedge intersects at most 2^n/48 other hyperedges, then there is a polynomial time algorithm that finds a two-coloring of the points such that no hyperedge is monochromatic.     

The Lovász Local Lemma and Its Applications to some Combinatorial Arrays

The Lovász Local Lemma is a useful tool in the “probabilistic method” that has found many applications in combinatorics. In this paper, we discuss applications of the Lovász Local Lemma to some combinatorial set systems and arrays, including perfect hash families, separating hash families, ?-free systems, splitting systems, and generalized cover-free families. We obtain improved bounds for some of these set sytems. Also, we compare some of the bounds obtained from the local lemma to those using the basic probabilistic method as well as the well-known “expurgation” method. Finally, we briefly consider a “high probability” variation of the method, wherein a desired object is obtained with high probability in a suitable space.     

Packet routing and job-shop scheduling inO(congestion+dilation) steps

In this paper, we prove that there exists a schedule for routing any set of packets with edge-simple paths, on any network, inO(c+d) steps, wherec is the congestion of the paths in the network, andd is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and job-shop scheduling.This research was conducted while the authors were at MIT. Support was provided by the Defense Advanced Research Projects Agency under Contract N00014-87-K-825, the Office of Naval Research under Contract N00014-86-K-0593, the Air Force under Contract OSR-86-0076, and the Army under Contract DAAL-03-86-K-0171. Tom Leighton is supported by an NSF Presidential Young Investigator Award with matching funds provided by IBM.     

On the Facial Thue Choice Index via Entropy Compression

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge coloring of a path is nonrepetitive if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Soták, and ?krabul'áková proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovász Local Lemma). Our approach is based on the Moser–Tardos entropy‐compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmovi?, Joret, Kozik, and Wood.     

The Johansson‐Molloy theorem for DP‐coloring

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound for triangle‐free graphs G, avoiding the technicalities of the entropy compression method and only using the usual “lopsided” Lovász Local Lemma (albeit in a somewhat unusual setting). On the other hand, we extend Molloy's result to DP‐coloring (also known as correspondence coloring), a generalization of list coloring introduced recently by Dvo?ák and Postle.     

An augmenting path algorithm for linear matroid parity

Linear matroid parity generalizes matroid intersection and graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm was given by Lovász. This paper presents an algorithm that uses timeO(mn ³), wherem is the number of elements andn is the rank. (The time isO(mn ^2.5) using fast matrix multiplication; both bounds assume the uniform cost model). For graphic matroids the time isO(mn ²). The algorithm is based on the method of augmenting paths used in the algorithms for all subcases of the problem. First author was supported in part by the National Science Foundation under grants MCS 78-18909, MCS-8302648, and DCR-8511991. The research was done while the second author was at the University of Denver and at the University of Colorado at Boulder.     

The Local Cut Lemma

The Lovász Local Lemma is a very powerful tool in probabilistic combinatorics, that is often used to prove existence of combinatorial objects satisfying certain constraints. Moser and Tardos (2010) have shown that the LLL gives more than just pure existence results: there is an effective randomized algorithm that can be used to find a desired object. In order to analyze this algorithm, Moser and Tardos developed the so-called entropy compression method. It turned out that one could obtain better combinatorial results by a direct application of the entropy compression method rather than simply appealing to the LLL. The aim of this paper is to provide a generalization of the LLL which implies these new combinatorial results. This generalization, which we call the Local Cut Lemma, concerns a random cut in a directed graph with certain properties. Note that our result has a short probabilistic proof that does not use entropy compression. As a consequence, it not only shows that a certain probability is positive, but also gives an explicit lower bound for this probability. As an illustration, we present a new application (an improved lower bound on the number of edges in color-critical hypergraphs) as well as explain how to use the Local Cut Lemma to derive some of the results obtained previously using the entropy compression method.     

On the Beck‐Fiala conjecture for random set systems

Motivated by the Beck‐Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability ; and when |X| ? |Σ|^t the hereditary discrepancy of (X,Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.     

Solutions for non-negatively weighted TU games derived from extension operators

We suggest two alternatives to the Lovász-Shapley value for non-negatively weighted TU games, the dual Lovász-Shapley value and the Shapley² value. Whereas the former is based on the Lovász extension operator for TU games, the latter two are based on extension operators that share certain economically plausible properties with the Lovász extension operator, the dual Lovász extension operator and the Shapley extension operator, respectively.     

The 2‐dimensional rigidity of certain families of graphs

Laman's characterization of minimally rigid 2‐dimensional generic frameworks gives a matroid structure on the edge set of the underlying graph, as was first pointed out and exploited by L. Lovász and Y. Yemini. Global rigidity has only recently been characterized by a combination of two results due to T. Jordán and the first named author, and R. Connelly, respectively. We use these characterizations to investigate how graph theoretic properties such as transitivity, connectivity and regularity influence (2‐dimensional generic) rigidity and global rigidity and apply some of these results to reveal rigidity properties of random graphs. In particular, we characterize the globally rigid vertex transitive graphs, and show that a random d‐regular graph is asymptotically almost surely globally rigid for all d ≥ 4. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 154–166, 2007     

Ear Decompositions of Matching Covered Graphs

G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovász. We also give a simple proof of another theorem due to Lovász which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of or the fourth graph in the sequence is an odd-subdivision of . Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author. Received: November 3, 1997     

A (1 + ϵ)‐approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma*

In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2‐coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2‐coloring nonuniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a nonuniform hypergraph in which every edge e_i intersects at most |e_i|2^αk other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].     

Copositive programming motivated bounds on the stability and the chromatic numbers

The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovász theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs. Partial support by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438, is gratefully acknowledged.     

Greedy colorings of uniform hypergraphs

We give a very short proof of an Erd?s conjecture that the number of edges in a non‐2‐colorable n‐uniform hypergraph is at least f(n)2ⁿ, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) ≥ cn^1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) ≥ cn^1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k‐colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n‐uniform, n‐regular hypergraphs are 2‐colorable if n ≥ 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Gap,cosum and product properties of the θ′ bound on the clique number

In a paper published in 1978, McEliece, Rodemich and Rumsey improved the Lovász bound θ for the maximum clique problem. This strengthening has become well known under the name Lovász–Schrijver bound and is usually denoted by θ′. This article now deals with situations where this bound is not exact. To provide instances for which the gap between this bound and the actual clique number can be arbitrarily large, we establish homomorphy results for this bound under cosums and products of graphs. In particular we show that for circulant graphs of prime order there must be a positive gap between the clique number and the bound.     

Maximum stable set formulations and heuristics based on continuous optimization

 The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovász theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that restricting the matrix variable in the Lovász semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics. Received: December 13, 2000 / Accepted: September 3, 2002 Published online: December 19, 2002 RID="★" ID="★" Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. Key Words. maximum stable set – maximum clique – minimum vertex cover – semidefinite program – semidefinite relaxation – continuous optimization heuristics – nonlinear programming Mathematics Subject Classification (2000): 90C06, 90C27, 90C30     

Oriented matroids and Ky Fan’s theorem

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb. 1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C ^m,r .