Eitan Zemel

An Algorithm for Large Zero-One Knapsack Problems

We describe an algorithm for the 0-1 knapsack problem (KP), which relies mainly on three new ideas. The first one is to focus on what we call the core of the problem, namely, a knapsack problem equivalent to KP, defined on a particular subset of the variables. The size of this core is usually a small fraction of the full problem size, and does not seem to increase with the latter. While the core cannot be identified without solving KP, a satisfactory approximation can be found by solving the associated linear program (LKP). The second new ingredient is a binary search-type procedure for solving LKP which, unlike earlier methods, does not require any ordering of the variables. The computational effort involved in this procedure is linear in the number of variables. Finally, the third new feature is a simple heuristic which under certain conditions finds an optimal solution with a probability that increases with the size of KP. Computational experience with an algorithm based on the above ideas, on several ...

Nash and correlated equilibria: Some complexity considerations

This paper deals with the complexity of computing Nash and correlated equilibria for a finite game in normal form. We examine the problems of checking the existence of equilibria satisfying a certain condition, such as “Given a game G and a number r, is there a Nash (correlated) equilibrium of G in which all players obtain an expected payoff of at least r?” or “Is there a unique Nash (correlated) equilibrium in G?” etc. We show that such problems are typically “hard” (NP-hard) for Nash equilibria but “easy” (polynomial) for correlated equilibria.

Totally Balanced Games and Games of Flow

A class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games. The proof relies on the max flow-min cut theorem of Ford and Fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation.

Facets of the Knapsack Polytope From Minimal Covers

In this paper we give easily computable best upper and lower bounds on the coefficients of facets of the knapsack polytope associated with minimal covers. For some coefficients the upper bounds are equal to the lower bounds; for the others the two bounds differ by 1. We give a necessary and sufficient condition for all lower bounds to be equal to the corresponding upper bounds, i.e. for the facet associated with the given minimal cover to be unique. Also, we define a partial order on the set of minimal covers and show that all facets associated with minimal covers can be obtained from weak covers; but that each facet obtainable from several ordered minimal covers is easiest to compute from the strongest one.Further, we characterize the class of all facets associated with minimal covers, and show that the facets obtainable by Padberg’s sequential lifting procedure are precisely those members of the class which have integer coefficients for a certain right-hand side. We then give a procedure for generating ...

Generalized Network Problems Yielding Totally Balanced Games

A class of multiperson mathematical optimization problems is considered and is shown to generate cooperative games with nonempty cores. The class includes, but is not restricted to, numerous versions of network flow problems. It was shown by Owen that for games generated by linear programming optimization problems, optimal dual solutions correspond to points in the core. We identify a special class of network flow problems for which the converse is true, i.e., every point in the core corresponds to an optimal dual solution.

The Maximum Coverage Location Problem

In this paper we define and discuss the following problem which we call the maximum coverage location problem. A transportation network is given together with the locations of customers and facilities. Thus, for each customer i, a radius

An

r_i

O(n\log ^2 n)

is known such that customer i can currently be served by a facility which is located within a distance of

Algorithm for the kth Longest Path in a Tree with Applications to Location Problems

r_i

Hashing vectors for tabu search

from the location of customer i. We consider the problem from the point of view of a new company which is interested in establishing new facilities on the network so as to maximize the company’s “share of the market.” Specifically, assume that the company gains an amount of