The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial time. However, until recently, in order to obtain constant-round

The strive for efficiency is ancient and universal, as time is always short for humans. Computational Complexity is a mathematical study of the what can be achieved when time (and other resources) are scarce. In this

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant—optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a “frequently self-knowingly correct algorithm. ” We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein’s junta distributions [PR07]. Key words: approximation, Dodgson elections, election systems, frequently selfknowingly correct algorithms, greedy algorithms, optimal lobbying, preference aggregation.

In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of ”efficiency-onaverage”, simple distributions and efficiency-on-average preserving reductions, Levin developed a theory analogous to the theory of N P-completeness. In particular, he showed that there exists a simple distributional problem that is complete under these reductions. But since then very few distributional problems were shown to be complete in this sense. In this paper we show a simple sufficient condition for an N P-complete decision problem to have a distributional version that is complete under these reductions (and thus to be ”hard on the average ” with respect to some simple probability distribution). Apparently all known N P-complete decision problems meet this condition.

“P versus N P – a gift to mathematics from Computer Science”

Abstract. More than two decades elapsed since Levin set forth a theory of average-case complexity. In this survey we present the basic aspects of this theory as well as some of the main results regarding it. The current presentation deviates from our old “Notes on Levin’s Theory of Average-Case Complexity ” (ECCC, TR97-058, 1997) in several aspects. In particular: – We currently view average-case complexity as referring to the performance on “average ” (or rather typical) instances, and not as the average performance on random instances. (Thus, it may be more justified to refer to this theory by the name typical-case complexity, but we retain the name average-case for historical reasons.) – We include a treatment of search problems, and a presentation of the reduction of “NP with sampleable distributions ” to “NP with P-computable distributions ” (due to Impagliazzo and Levin, 31st FOCS, 1990). – We include Livne’s result (ECCC, TR06-122, 2006) by which all natural NPC-problems have average-case complete versions. This result seems to shed doubt on the association of P-computable distributions with natural distributions. Keywords: Average-Case Complexity. This text has been revised based on [6, Sec. 10.2].