Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

Statistic complexity: combining kolmogorov complexity with an ensemble approach.

Abstract: Background The evaluation of the complexity of an observed object is an old but outstanding problem. In this paper we are tying on this problem introducing a measure called statistic complexity. Methodology/Principal Findings This complexity measure is different to all other measures in the following senses. First, it is a bivariate measure that compares two objects, corresponding to pattern generating processes, on the basis of the normalized compression distance with each other. Second, it provides the quantification of an error that could have been encountered by comparing samples of finite size from the underlying processes. Hence, the statistic complexity provides a statistical quantification of the statement ‘ is similarly complex as ’. Conclusions The presented approach, ultimately, transforms the classic problem of assessing the complexity of an object into the realm of statistics. This may open a wider applicability of this complexity measure to diverse application areas.

The Complexity of Generating and Checking Proffs of Membership

Abstract: We consider the following questions: 1. Can one compute satisfying assignments for satisfiable Boolean formulas in polynomial time with parallel queries to NP? 2. Is the unique optimal clique problem (UOCLIQUE) complete for PNP[O(log n)]? 3. Is the unique satisfiability problem (USAT) NP hard? We define a framework that enables us to study the complexity of generating and checking proofs of membership. We connect the above three questions to the complexity of generating and checking proofs of membership for sets in NP and PNP[O(log n)]. We show that an affirmative answer to any of the three questions implies the existence of coNP checkable proofs for PNP[O(log n)] that can be generated in FP ∥ NP . Furthermore, we construct an oracle relative to which there do not exist coNP checkable proofs for NP that are generated in FP ∥ NP . It follows that relative to this oracle all of the above questions are answered negatively.

On average complexity of global optimization problems

Abstract: We discuss the average case complexity of global optimization problems. By the average complexity, we roughly mean the amount of work needed to solve the problem with the expected error not exceeding a preassigned error demand. The expectation is taken with respect to a probability measure on a classF of objective functions. Since the distribution of the maximum, maxxf(x), is known only for a few nontrivial probability measures, the average case complexity of optimization is still unknown. Although only preliminary results are available, they indicate that on the average, optimization is not as hard as in the worst case setting. In particular, there are instances, where global optimization is intractable in the worst case, whereas it is tractable on the average. We stress, that the power of the average case approach is proven by exhibiting upper bounds on the average complexity, since the actual complexity is not known even for relatively simple instances of global optimization problems. Thus, we do not know how much easier global optimization becomes when the average case approach is utilized.

Average-Case Complexity.

Abstract: We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question is whether the existence of hard-on-average problems in NP can be based on the P ≠ NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case complexity. We discuss some of these "hardness amplification" results.