László Lovász

Factoring Polynomials with Rational Coefficients.

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

The ellipsoid method and its consequences in combinatorial optimization

L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

Cones of Matrices and Set-Functions and 0–1 Optimization

A system, method, and apparatus for facilitating a self-organizing workforce of one or more workers through payment and recognition incentives, a set of configurable operating rules, and a set of credentials to represent the reputations and organizational capital of individual workers. The system includes a worksite having one or more configurable worksite rules where the one or more workers may work on an idea. Work includes posting to a discussion about the idea, voting on the idea, and recommending an outcome for the idea. Worker credentials for each worker are updated based on a workers work on the idea within the worksite. The worker credentials include merit, which is a measure of the quantity, quality, and significance of work done; links, which is a function of what one worker thinks of another workers work; wisdom, which reflects the workers ability to spot a good idea; and, influence, which is a function of merit, links, and wisdom, and which reflects a workers overall organizational capital within the system.

Submodular functions and convexity

In “continuous” optimization convex functions play a central role. Besides elementary tools like differentiation, various methods for finding the minimum of a convex function constitute the main body of nonlinear optimization. But even linear programming may be viewed as the optimization of very special (linear) objective functions over very special convex domains (polyhedra). There are several reasons for this popularity of convex functions: Convex functions occur in many mathematical models in economy, engineering, and other sciencies. Convexity is a very natural property of various functions and domains occuring in such models; quite often the only non-trivial property which can be stated in general.

Random walks, universal traversal sequences, and the complexity of maze problems

It is well known that the reachability problem for directed graphs is logspace-complete for the complexity class NSPACE(log n) , and thus holds the key to the open question of whether DSPACE(logn)= NSPACE(logn) ([3,4,5,6]). Here as usual OSPACE(logn) is the class of languages that are accepted in logn space by deterministic Turing Ma chi nes, wh i 1eNSPACE( log n) i s the c1ass 0 f 1anguages that are accepted in log n space by nondeterministic ones. The reachability problem for undirected graphs has also been considered ([5]), but it has remained an open question whether undirected graph reachability is logspace-complete for NSPACE(logn). Here we derive results suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version. These results are an affirmative answer to a question of S. Cook.

Kneser's conjecture, chromatic number, and homotopy

Abstract If the simplicial complex formed by the neighborhoods of points of a graph is (k − 2)-connected then the graph is not k-colorable. As a corollary Knesers conjecture is proved, asserting that if all n-subsets of a (2n − k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets.

Interactive proofs and the hardness of approximating cliques

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.

Approximating clique is almost NP-complete

The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P.<<ETX>>

A Characterization of Perfect Graphs

Throughout this note, graph means finite, undirected graph without loops and multiple edges.

Polynomial algorithms for perfect graphs

We show that the weighted versions of the stable set problem, the clique problem, the coloring problem and the clique covering problem are solvable in polynomial time for perfect graphs. Our algorithms are based on the ellipsoid method and a polynomial time separation algorithm for a certain class of positive semidefinite matrices related to Lovaszs bound θ( G ) on the Shannon capacity of a graph. We show that θ G ) can be computed in polynomial time for all graphs G and also give a new characterization of perfect graphs in terms of this number θ( G ). In addition we prove that the problem of verifying that a graph is imperfect is in NP. Moreover, we show that the computation of the stability number and the fractional stability number of a graph are unrelated with respect to hardness (if P ≠ NP ).