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Register a new userThe Complexity of Computing Minimal Unidirectional Covering Sets
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Dorothea Baumeister
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischers result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.
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We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. In other words, no algorithm with time polynomial in n and 1/epsilon can compute an epsilon-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. Furthermore, we prove, unless PPAD is in RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/sigma under perturbations with magnitude sigma. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.
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In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML15) showed how to compute the optimal bound for composing $k$ arbitrary $(epsilon,delta)$-differentially private algorithms. We characterize the optimal composition for the more general case of $k$ arbitrary $(epsilon_{1},delta_{1}),ldots,(epsilon_{k},delta_{k})$-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is $#$P-complete. Since computing optimal composition exactly is infeasible (unless FP=$#$P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyers dynamic programming approach to approximately counting solutions to knapsack problems (STOC03).
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