Shlomo Hoory

Expander Graphs and their Applications

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even

The Moore Bound for Irregular Graphs

Abstract. What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10).

Rank bounds and integrality gaps for cutting planes procedures

We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovasz and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: first, we prove near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank resolution proofs. Thirdly, we give examples where the CP rank is O(log n) but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomial-size CP/LS proofs, but requiring linear rank.

The size of bipartite graphs with a given girth

What is the maximum number of edges in a bipartite graph of girth g whose left and right sides are of size nL, nR? We generalize the known results for g = 6, 8 to an arbitrary girth.

Linear-Time Reductions of Resolution Proofs

DPLL-based SAT solvers progress by implicitly applying binary resolution. The resolution proofs that they generate are used, after the SAT solvers run has terminated, for various purposes. Most notable uses in formal verification are: extracting an unsatisfiable core , extracting an interpolant , and detecting clauses that can be reused in an incremental satisfiability setting (the latter uses the proof only implicitly, during the run of the SAT solver). Making the resolution proof smaller can benefit all of these goals. We suggest two methods that are linear in the size of the proof for doing so. Our first technique, called Recycle-Units , uses each learned constant (unit clause) (x ) for simplifying resolution steps in which x was the pivot, prior to when it was learned. Our second technique, called Recycle-Pivots , simplifies proofs in which there are several nodes in the resolution graph, one of which dominates the others, that correspond to the same pivot. Our experiments with industrial instances show that these simplifications reduce the core by ≈ 5% and the proof by ≈ 13%. It reduces the core less than competing methods such as run-till-fix , but whereas our algorithms are linear in the size of the proof, the latter and other competing techniques are all exponential as they are based on SAT runs. If we consider the size of the proof graph as being polynomial in the number of variables (it is not necessarily the case in general), this gives our method an exponential time reduction comparing to existing tools for small core extraction. Our experiments show that this result is evident in practice more so for the second method: rarely it takes more than a few seconds, even when competing tools time out, and hence it can be used as a cheap proof post-processing procedure.

Universal traversal sequences for expander graphs

Graph reachability is a key problem in the study of various logarithmic space complexity classes. Its version for directed graphs is logspace complete for NSPACE(logn), and hence if proved to be in DSPACE(logn), the open question DSPACE(logn) = NSPACE(log n) will be settled. Seemingly the problem is easier for undirected graphs. In [1] it was shown to be in RLP (1-sided error, logspace, polynomial expected time). Recently it was shown by [3] to be in ZPLP (no-error, logspace, polynomial expected time).

A Note on Unsatisfiable k -CNF Formulas with Few Occurrences per Variable

The (k,s)-SAT problem is the satisfiability problem restricted to instances where each clause has exactly k literals and every variable occurs at most s times. It is known that there exists a function f such that for s \leq f(k) all (k,s)-SAT instances are satisfiable, but (k,f(k)+1)-SAT is already NP-complete (k \geq 3). We prove that f(k) = O(2k \cdot log k/k), improving upon the best known upper bound O(2k/kalpha), where alpha=log3 4 - 1 \approx 0.26. The new upper bound is tight up to a log k factor with the best known lower bound Omega(2k/k).

Simple permutations mix well

We study the random composition of a small family of O(n3) simple permutations on {0, 1}n. Specifically, we ask what is the number of compositions needed to achieve a permutation that is close to k-wise independent. We improve on a result of Gowers [An almost m-wise independent random permutation of the cube, Combin. Probab. Comput. 5(2) (1996) 119-130] and show that up to a polylogarithmic factor, n3k3 compositions of random permutations from this family suffice. We further show that the result applies to the stronger notion of k-wise independence against adaptive adversaries. This question is essentially about the rapid mixing of the random walk on a certain graph, and we approach it using a new technique to construct canonical paths. We also show that if we are willing to use a much larger family of simple permutations then we can guarantee closeness to k-wise independence with fewer compositions and fewer random bits.

Computing unsatisfiable k -SAT instances with few occurrences per variable

(k, s)-SAT is the propositional satisfiability problem restricted to instances where each clause has exactly k distinct literals and every variable occurs at most s times. It is known that there exists an exponential function f such that for s ≤ f(k) all (k, s)-SAT instances are satisfiable, but (k, f (k) + 1)- SAT is already NP-complete (k ≥ 3). Exact values of f are only known for k = 3 and 4, and it is open whether f is computable. We introduce a computable function f1 which bounds f from above and determine the values of f1 by means of a calculus of integer sequences. This new approach enables us to improve the best known upper bounds for f(k), generalizing the known constructions for unsatisfiable (k, s)-SAT instances for small k.

Monotone circuits for the majority function

We present a simple randomized construction of size O(n3) and depth 5.3logn+O(1) monotone circuits for the majority function on n variables. This result can be viewed as a reduction in the size and a partial derandomization of Valiants construction of an O(n5.3) monotone formula, [15]. On the other hand, compared with the deterministic monotone circuit obtained from the sorting network of Ajtai, Komlos, and Szemeredi [1], our circuit is much simpler and has depth O(logn) with a small constant. The techniques used in our construction incorporate fairly recent results showing that expansion yields performance guarantee for the belief propagation message passing algorithms for decoding low-density parity-check (LDPC) codes, [3]. As part of the construction, we obtain optimal-depth linear-size monotone circuits for the promise version of the problem, where the number of 1s in the input is promised to be either less than one third, or greater than two thirds. We also extend these improvements to general threshold functions. At last, we show that the size can be further reduced at the expense of increased depth, and obtain a circuit for the majority of size and depth about