Linear programming without the matrix

Abstract

We study the problem facing a set of decision-makers who must select values for the variables of a linear program, when only parts of the matrix are available to each of them. The goal is to find a feasible solution that is as close to the true optimum as possible, When each decision-maker decides one variable and knows all constraints involving this variable, we show that the worst-case ratio is related tc,the maximum number of variables appearing in each constraint, and a simple “safe” heuristic is optimal. Since this problem involves constrained optimization, there is a novel criterion, besides the competitive ratio, comparing the performance of a heuristic with the best feasible distributed algorithm, perhaps specializing on the current inst ante; we show different bounds for this parameter. When the constraint structure (the zero-nonzero pattern of the matrix) is known in advance, and the variables are partitioned bet ween decision-makers, then the optimum ratio is a complicated parameter of the associated hypergraph, which we bound from above and below in terms of variants of clique and graph coloring; but several interesting special cases are characterized completely. 1 Department of Computer Science and Engineering, University of California at San Diego. Research supported by the National Science Foundation. 2 AT&T Bell Laboratories, Murray Hill, NJ 07974. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee arrd/or specific permission. 25th ACM STOC ‘93-51931CA, LE3A e 1993 ACM 0-89791-591-7/93/0005/0121 . ..$l .50