Bruno Codenotti

Leontief economies encode nonzero sum two-player games

We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria.We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in one-to-one correspondence with the equilibria of the corresponding economy.Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for two-player nonzero sum games.As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [17]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium.Perhaps more importantly, we also prove that it is NP-hard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [30], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwers Fixed Point Theorem.On the algorithmic side, we present an algorithm for finding an approximate equilibrium for some special Leontief economies, which achieves quasi-polynomial time whenever each trader does not demand too much more of any good than some other good.

Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities

We present a polynomial time algorithm for the computation of the market equilibrium in a version of Fisher’s model, where the traders have Leontief utility functions. These functions describe a market characterized by strict complementarity. Our algorithm follows from a representation of the equilibrium problem as a concave maximization problem, which is of independent interest. Our approach extends to a more general market setting, where the traders have utility functions from a wide family which includes CES utilities.

Perturbation: An Efficient Technique for the Solution of Very Large Instances of the Euclidean TSP

In this paper we introduce a technique for developing efficient iterated local search procedures and we apply it to solve very large instances of the Euclidean Traveling Salesman Problem (TSP). This technique, which we call perturbation, uses global information on TSP instances to speed-up the computation and to improve the quality of the tours found by heuristic methods. The main idea is to escape from local optima by introducing perturbations in the problem instance rather than in the solution. The performance of our algorithms has been tested and compared with known methods. To this end, we have executed a number of experiments both on available benchmarks, for which the optimal tour length is known, and on randomly generated instances, for which the comparison is done with the Held-Karp lower bound. The experimental results, performed on up to 100,000 cities, show that our algorithms outperform the known methods for iterating local search for very large instances.

Spectral analysis of Boolean functions as a graph eigenvalue problem

Several problems in digital logic can be conveniently approached in the spectral domain. In this paper we show that the Walsh spectrum of Boolean functions can be analyzed by looking at algebraic properties of a class of Cayley graphs associated with Boolean functions. We use this idea to investigate the Walsh spectrum of certain special functions.

Transitive Cellular Automata Are Sensitive

The notion of chaos is a very appealing one, and it has intrigued several scientists (see [1, 2, 5, 7] for some work on the properties that characterize a chaotic process). There are simple deterministic dynamical systems that exhibit unpredictable behavior. Though counter-intuitive, this fact has a very clear eEplanation. The lack of infinite precision causes a loss of information which is dramatic for some processes which quickly lose their deterministic nature to assume a non deterministic (unpredictable) one. This observation leads to the intuition that investigations about chaos are intrinsically of an interdisciplinary nature. Indeed, the study of chaos draws its deeper methods of analysis from mathematics, it owes to physics a treasure of important problems, and it brings challenges to the science of computing. The reason for this last fact relies on the above observation that a chaotic behavior lies in between two different modes of computation, determinism and nondeterminism, whose quantitative comparison is central to the main open questions in the theory of computing [6]. A chaotic phenomenon can indeed be viewed as a deterministic one, in the presence of infinite precision, and as a nondeterministic one, in the presence of finite precision constraints (see Figure 1). Thus one should look at chaotic processes as at processes merged into time, space, and precision bounds, which are the key resources in the science of computing.

Checking approximate computations over the reals

Checking Approximate Computations over the Reals S. Ar” M. Blumt B. Codenotti

HARDNESS RESULTS AND SPECTRAL TECHNIQUES FOR COMBINATORIAL PROBLEMS ON CIRCULANT GRAPHS

P. Gemmell~ This paper provides the first systematic investigation of checking approximate numerical computations over subsets of the reals. In most cases, approximate checking is more challenging than exact checking. Problem conditioning, i.e., the measure of sensitivity of the output to slight changes in the input, and the presence of approximate ion parameters foil the direct transformation of many exact checkers to the approximate setting. Furthermore, approximate checking over the reals is complicated by the lack of nice finite field properties such as the existence of a samplable distribution which is invariant under addition or multiplication by a scalar. We overcome the above problems by using such techniques as testing and checking over similar but distinct distributions, using functions’ random and downward self-reducibility properties, and taking advantage of the small variance of the sum of independent identically distributed random variables. We provide approximate checkers for a variety of computations, including matrix multiplication, linear system solution, matrix inversion, and computation of the determinant. We also present an approximate version of Beigel’s trick and extend the approximate linear self tester/corrector of [8] and the trigonometric selftester/corrector of [5] to more general computations. *Department of Computer Science, Princeton University, Princeton, NJ 08544-2087. Supported by NSF PYI grant CCR9057486 and a grant from MITL. t Computer Science Division, UC Berkeley, Berkeley, CA 94720, and International Computer Science Institute, Berkeley CA 94704. Supported by NSF grant CCR88-13632. t International Computer Science Institute, Berkeley, CA 94704, and IEI-CNR, Piss (Italy). Partially supported by the “ Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo”. Subproject 2. e-mail: codenotti@iei.pi .cnr.it ~Computer Science Division, UC Berkeley, Berkeley, CA 94720. Supported by GTE, Schlumberger Fellowships, and by NSF grant CCR88-13632.

Some structural properties of low-rank matrices related to computational complexity

Abstract We show that computing (and even approximating) maximum clique and minimum graph coloring for circulant graphs is essentially as hard as in the general case. In contrast, we show that, under additional constraints, e.g., prime order and/or sparseness, graph isomorphism and minimum graph coloring become easier in the circulant case, and we take advantage of spectral techniques for their efficient computation.

On the Permanent of Certain (0, 1) Toeplitz Matrices

Abstract We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection between these properties and the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used to give a constructive bound of the order of k 3/2 on the Ramsey number R(3,k) , which matches the best-known bound. On the other hand, we show that sublinear rank implies the existence of 4-cycles. Finally, we prove some partial results towards establishing lower bounds on matrix rigidity and consequently on the size of logarithmic depth arithmetic circuits for computing certain explicit linear transformations.

Parallel solution of block tridiagonal linear systems

Abstract We obtain convenient expressions and/or efficient algorithms for the permanent of certain very sparse (0, 1) Toeplitz matrices. The classes of matrices considered here include some nontrivial examples of circulants to which none of the previous approaches could be successfully applied.