On the complexity of price equilibria

AbstractWe prove complexity, approximability, and inapproximability results for the problem of finding an exchange equilibrium in markets with indivisible (integer) goods, most notably a polynomial algorithm that approximates the market equilibrium arbitrarily close when the number of goods is bounded and the utilities linear. We also show a communication complexity lower bound in a model appropriate for markets. Our result implies that the ideal informational economy of a market with divisible goods and unique optimal allocations is unattainable in general

Experimental Aspects of Synthesis

We discuss the problem of experimentally evaluating linear-time temporal logic (LTL) synthesis tools for reactive systems. We first survey previous such work for the currently publicly available synthesis tools, and then draw conclusions by deriving useful schemes for future such evaluations. In particular, we explain why previous tools have incompatible scopes and semantics and provide a framework that reduces the impact of this problem for future experimental comparisons of such tools. Furthermore, we discuss which difficulties the complex workflows that begin to appear in modern synthesis tools induce on experimental evaluations and give answers to the question how convincing such evaluations can still be performed in such a setting.Comment: In Proceedings iWIGP 2011, arXiv:1102.374

On Planning while Learning

This paper introduces a framework for Planning while Learning where an agent is given a goal to achieve in an environment whose behavior is only partially known to the agent. We discuss the tractability of various plan-design processes. We show that for a large natural class of Planning while Learning systems, a plan can be presented and verified in a reasonable time. However, coming up algorithmically with a plan, even for simple classes of systems is apparently intractable. We emphasize the role of off-line plan-design processes, and show that, in most natural cases, the verification (projection) part can be carried out in an efficient algorithmic manner. 1. Introduction Suppose you find yourself in a complex labyrinth, with no recollection as to what brought you there or how to get out. You do have some knowledge as to the possible outcomes of your actions (e.g., gravitation works as usual). However, several basic characteristics of your surrounding are unknown (e.g., the map of th..

A Sub-Constant Error-Probability Low-Degree-Test, and a Sub-Constant Error-Probability PCP Characterization of NP

We introduce a new low-degree--test, a one that uses the restriction of low-degree polynomials to planes (i.e., affine sub-spaces of dimension 2), rather than the restriction to lines (i.e., affine sub-spaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, much smaller than a constant) . The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ffl ? 0, membership in any NP language can be verified with O(1) accesses, each reading logarithmic number of bits, and such that the error-probability is 2 \Gamma log 1\Gammaffl n . Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SET-COVER to within logarithmic factors is NP-hard. Previous analysis for low-degree--tests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, e..