155 research outputs found
Computing an Approximately Optimal Agreeable Set of Items
We study the problem of finding a small subset of items that is
\emph{agreeable} to all agents, meaning that all agents value the subset at
least as much as its complement. Previous work has shown worst-case bounds,
over all instances with a given number of agents and items, on the number of
items that may need to be included in such a subset. Our goal in this paper is
to efficiently compute an agreeable subset whose size approximates the size of
the smallest agreeable subset for a given instance. We consider three
well-known models for representing the preferences of the agents: ordinal
preferences on single items, the value oracle model, and additive utilities. In
each of these models, we establish virtually tight bounds on the approximation
ratio that can be obtained by algorithms running in polynomial time.Comment: A preliminary version appeared in Proceedings of the 26th
International Joint Conference on Artificial Intelligence (IJCAI), 201
Definable inapproximability: New challenges for duplicator
AbstractWe consider the hardness of approximation of optimization problems from the point of view of definability. For many -hard optimization problems it is known that, unless , no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.</jats:p
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