Eli Berger

Dynamic Monopolies of Constant Size

The paper deals with a polling game on a graph. Initially, each vertex is colored white or black. At each round, each vertex is colored by the color shared by the majority of vertices in its neighborhood, at the previous round. (All recolorings are done simultaneously.) We say that a set W0 of vertices is a dynamic monopoly or dynamo if starting the game with the vertices of W0 colored white, the entire system is white after a finite number of rounds. D. Peleg (1998, Discrete Appl. Math.86, 262?273) asked how small a dynamic monopoly may be as a function of the number of vertices. We show that the answer is O(1).

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets Vi of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each Vi at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s.

On non-approximability for quadratic programs

This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ that maximizes x/sup T/Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 - /spl epsi/ for all /spl epsi/ > 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log/sup /spl gamma// n)for some /spl gamma/ > 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be /spl Theta/(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is /spl Omega/ (log n/log log n), essentially answering one of the open problems of Alon et al. [AMMN].

A Tree Version of Kőnig's Theorem

Kőnigs theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Halls theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed.

Forcing large transitive subtournaments

The Erd?s-Hajnal Conjecture states that for every given H there exists a constant c ( H ) 0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least | V ( G ) | c ( H ) . The conjecture is still open. However some time ago its directed version was proved to be equivalent to the original one. In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. Both the directed and the undirected versions of the conjecture are known to be true for small graphs (or tournaments), and there are operations (the so-called substitution operations) allowing to build bigger graphs (or tournaments) for which the conjecture holds. In this paper we prove the conjecture for an infinite class of tournaments that is not obtained by such operations. We also show that the conjecture is satisfied by every tournament on at most 5 vertices.

Eulerian edge sets in locally finite graphs

In a finite graph, an edge set Z is an element of the cycle space if and only if every vertex has even degree in Z. We extend this basic result to the topological cycle space, which allows infinite circuits, of locally finite graphs. In order to do so, it becomes necessary to attribute a parity to the ends of the graph.

KKM—A Topological Approach For Trees

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems.A 2-tree hypergraph is a family of nonempty subsets of T ∪ R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets.For each such hypergraph H we denote by τ (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by ν(H) the maximal number of disjoint sets in it.In this paper we prove that in a 2-tree hypergraph τ(H)≤2ν(H) and in a homogeneous 2-tree hypergraph τ(H)≤3ν(H). This improves the result of Alon [3], that τ(H)≤8ν(H) in both cases.Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.

Note: A note on the edge cover number and independence number in hypergraphs

Consider a hypergraph of rank r>2 with m edges, independence number @a and edge cover number @r. We prove the inequality@r=<(r-2)m+@ar-1.One application of this inequality is a special case of a conjecture of Aharoni and the first author extending Rysers Conjecture to matroids.

An Optimized Symbolic Bounded Model Checking Engine

It has been shown that bounded model checking using a SAT solver can solve many verification problems that would cause BDD based symbolic model checking engines to explode. However, no single algorithmic solution has proven to be totally superior in resolving all types of model checking problems. We present an optimized bounded model checker based on BDDs and describe the advantages and drawbacks of this model checker as compared to BDD-based symbolic model checking and SAT-based model checking. We show that, in some cases, this engine solves verification problems that could not be solved by other methods.

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of