Theory Of Average Case Complexity Research Articles

A new algorithm design technique for hard problems

Heuristic algorithms have been used for a long time to tackle problems that are known or conjectured intractable. A heuristic algorithm is one that provides a correct decision for most inputs, but may fail on some. We focus on the case when failure means that the algorithm does not return any answer, rather than returning a wrong result. These algorithms are called errorless heuristics.A reasonable quality measure for heuristics is the failure rate over the set of n-bit instances. When no efficient exact algorithm is available for a problem, then, ideally, we would like one with vanishing failure rate. We show, however, that this is hard to achieve: unless a complexity theoretic hypothesis fails, some NP-complete problems cannot have a polynomial-time errorless heuristic algorithm with any vanishing failure rate.On the other hand, we prove that vanishing, even exponentially small, failure rate is achievable, if we use a somewhat different accounting scheme to count the failures. This is based on special sets, that we call α-spheres, which are the images of the n-bit strings under a bijective, polynomial-time computable and polynomial-time invertible encoding function α. Our main result is that polynomial-time errorless heuristic algorithms exist, with exponentially low failure rates on the α-spheres, for a large class of decision problems. This class includes, surprisingly, all known intuitively natural NP-complete problems. We also explore some connections to the theory of average case complexity.

Matrix Transformation Is Complete for the Average Case

In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the average case reduction theory is still at an early stage, and only a few average case complete problems are known. The first algebraic problem complete for the average case under a natural probability distribution is presented. The problem is this: Given a unimodular matrix X of integers, a set S of linear transformations of such unimodular matrices and a natural number n, decide if there is a product of $\leq n$ (not necessarily different) members of S that takes X to the identity matrix.

Randomizing Reductions of Search Problems

This paper closes a gap in the foundations of the theory of average-case complexity. First, it clarifies the notion of a feasible solution for a search problem and proves its robustness. Second, it gives a general and usable notion of many–one randomizing reductions of search problems and proves that it has desirable properties. All reductions of search problems to search problems in the literature on average-case complexity can be viewed as such many–one randomizing reductions, including those reductions in the literature that use iterations and therefore do not look many–one. As an illustration, this paper presents a careful proof of a theorem of Impagliazzo and Levin in the framework of the present work.

On the theory of average case complexity

This paper takes the next step in developing the theory of average case complexity initiated by Leonid A. Levin. Previous works have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. Our results include: 1. ⊎|the equivalence of search and decision problems in the context of average case complexity; 2. ⊎|an initial analysis of the structure of distributional-NP (i.e., NP problems coupled with “simple distributions”) under reductions which preserve average polynomial-time; 3. ⊎|a proof that if all of distributional-NP is in average polynomial-time then non-deterministic exponential-time equals deterministic exponential time (i.e., a collapse in the worst case hierarchy); 4. ⊎|definitions and basic theorems regarding other complexity classes such as average log-space. An exposition of the basic definitions suggested by Levin and suggestions for some alternative definitions are provided as well.