On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one.     

Chip-firing based methods in the Riemann–Roch theory of directed graphs

Baker and Norine proved a Riemann–Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann–Roch-type results on directed graphs. We give a simple proof for a Riemann–Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann–Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.     

A Geometric Characterization of Poly-antimatroids

The concept of "antimatroid with repetition" was coined by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages [Björner A., L. Lovász, and P. R. Shor, Chip-firing games on graphs, European Journal of Combinatorics 12 (1991), 283–291]. There are some equivalent ways to define antimatroids. They may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. For poly-antimatroids we use the set system approach. In this research we concentrate on interrelations between geometric, algorithmic, and lattice properties of poly-antimatroids. Much to our surprise it turned out that even the two-dimensional case is not trivial.     

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.     

Matchings in regular graphs

We consider k-regular graphs with specified edge connectivity and show how some classical theorems and some new results concerning the existence of matchings in such graphs can be proved by using the polyhedral characterization of Edmonds. In addition, we show that lower bounds of Lovász and Plummer on the number of perfect matchings in bicritical graphs can be improved for cubic bicritical graphs.     

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.     

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.     

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     

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     

Chip-Firing and Riemann-Roch Theory for Directed Graphs

Baker and Norine developed a graph theoretic analogue of the classical Riemann-Roch theorem. Amini and Manjunath extended their criteria to all full-dimensional lattices orthogonal to the all ones vector. We show that Amini and Manjunath?s criteria holds for all full-dimensional lattices orthogonal to some positive vector and study some combinatorial examples of such lattices. Two distinct generalizations of the chip-firing game of Baker and Norine to directed graphs are provided. We describe how the “row” chip-firing game is related to the sandpile model and the “column” chip-firing game is related to directed G-parking functions. We finish with a discussion of arithmetical graphs, introduced by Lorenzini, viewing them as a class of vertex weighted graphs whose Laplacian is orthogonal to a positive vector and describe how they may be viewed as a special class of unweighted strongly connected directed graphs.     

Topological bounds on the dimension of orthogonal representations of graphs

An orthogonal representation of a graph is an assignment of nonzero real vectors to its vertices such that distinct non-adjacent vertices are assigned to orthogonal vectors. We prove general lower bounds on the dimension of orthogonal representations of graphs using the Borsuk–Ulam theorem from algebraic topology. Our bounds strengthen the Kneser conjecture, proved by Lovász in 1978, and some of its extensions due to Bárány, Schrijver, Dol’nikov, and Kriz. As applications, we determine the integrality gap of fractional upper bounds on the Shannon capacity of graphs and the quantum one-round communication complexity of certain promise equality problems.     

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     

Regularity lemmas in a Banach space setting

Szemerédi’s regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lovász and Szegedy (2007) gave a Hilbert space interpretation of the lemma and an interpretation in terms of compactness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lovász and Szegedy (2007) as well as a result of Borgs et al. (2014).     

A branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

On the subdifferential of a submodular function

The author recently introduced a concept of a subdifferential of a submodular function defined on a distributive lattice. Each subdifferential is an unbounded polyhedron. In the present paper we determine the set of all the extreme points and rays of each subdifferential and show the relationship between subdifferentials of a submodular function and subdifferentials, in an ordinary sense of convex analysis, of Lovász's extension of the submodular function. Furthermore, for a modular function on a distributive lattice we give an algorithm for determining which subdifferential contains a given vector and finding a nonnegative linear combination of extreme vectors of the subdifferential which expresses the given vector minus the unique extreme point of the subdifferential.     

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     

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.     

Chip-firing via open covers

We investigate chip-firing with respect to open covers of discrete graphs and metric graphs. For the case of metric graphs we show that given an open cover and a sink q, stabilization of a divisor D is unique and that there is a distinguished configuration equivalent to D, which we call the critical configuration. Also, we show that given a double cover of the metric graph by stars, which is the continuous analogue of the sandpile model, the critical configurations are in bijection with reduced divisors. Passing to the discrete case, we interpret open covers of a graph as simplicial complexes on the vertex and observe that chip-firing with respect to a simplicial complex is equivalent to the model introduced by Paoletti [G. Paoletti. July 11 2007: Master in Physics at University of Milan, defending thesis “Abelian sandpile models and sampling of trees and forests”; supervisor: Prof. S. Caracciolo. http://pcteserver.mi.infn.it/caraccio/index.html]. We generalize this setup for directed graphs using weighted simplicial complexes on the vertex set and show that the fundamental results extend. In the undirected case we present a generalization of the Cori-Le Borgne algorithm for chip-firing models via open covers, giving an explicit bijection between the critical configurations and the spanning trees of a graph.(http://www.elsevier.com/locate/endm)     

On the facets of the lift-and-project relaxations of graph subdivisions

We study the behavior of lift-and-project procedures for solving combinatorial optimization problems as described by Lovász and Schrijver (1991), in the context of the stable set problem on graphs. Following the work of Wolsey (1976), we investigate how to generate facets of the relaxations obtained by these procedures from facets of the relaxations of the original graph, after applying fundamental graph operations. We show our findings for the odd subdivision of an edge and its generalization, the stretching of a vertex operation.     

Forbidden subgraphs and the Kőnig property

A graph has the Kőnig property if its matching number equals its transversal number. Lovász proved a characterization of graphs having the Kőnig property by forbidden subgraphs, restricted to graphs with a perfect matching. Korach, Nguyen, and Peis proposed an extension of Lovászʼs result to a characterization of all graphs having the Kőnig property in terms of forbidden configurations (certain arrangements of a subgraph and a maximum matching). In this work, we prove a characterization of graphs having the Kőnig property in terms of forbidden subgraphs which is a strengthened version of the characterization by Korach et al. As a consequence of our characterization of graphs with the Kőnig property, we prove a forbidden subgraph characterization for the class of edge-perfect graphs.