Topic
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.
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Book•
30 Jun 2011
TL;DR: Error and complexity in Numerical Methods and Error-Free, Parallel, and Probabilistic Computations as mentioned in this paper was studied in error-free, parallel, and probabilistic computations.
Abstract: Chapter 1: Introduction Chapter 2: Error: Precisely What, Why, and How Chapter 3: Complexity: What, Why, and How Chapter 4: Errors and Approximations in Digital Computers Chapter 5: Error and Complexity in Numerical Methods Chapter 6: Error and Complexity in Error-Free, Parallel, and Probabilistic Computations Index
19 citations
Journal Article•
TL;DR: This work shows that assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the envy-free pricing problem cannot be approximated in polynomial time within O(log |C|) for some e > 0.
Abstract: We consider the envy-free pricing problem, in which we want to compute revenue maximizing prices for a set of products P assuming that each consumer from a set of consumer samples C will buy the product maximizing her personal utility, i.e., the difference between her respective budget and the product’s price. We show that assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the envy-free pricing problem cannot be approximated in polynomial time within O(log |C|) for some e > 0. This is the first result giving evidence that envy-free pricing might be hard to approximate within essentially better ratios than the logarithmic ratio obtained so far. Additionally, it gives another example of how average case complexity is connected to the worst case approximation complexity of notorious optimization problems.
19 citations
Posted Content•
TL;DR: In this paper, the complexity of the conjugacy problem in the Baumslag-Solitar group BS(1,2) was investigated, and it was shown that the problem can be solved in polynomial time in a strongly generic setting.
Abstract: The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G(1,2) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G(1,2) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G(1,2) by reducing the division problem in power circuits to the conjugacy problem in G(1,2). The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.
18 citations
TL;DR: This paper disproves Tiwari's conjecture, by exhibiting an infinite family of functions for which is essentially at most , and leads to progress on another major problem in this area.
Abstract: network consists of k+1 processors with links only between and (0≤i
18 citations
01 Jan 1976
TL;DR: In this chapter, a non-asymptotic theory of iterative computational complexity is constructed with strict lower and upper bounds for nonfinite cost problems analytic.
Abstract: Publisher Summary In this chapter, a non-asymptotic theory of iterative computational complexity is constructed with strict lower and upper bounds. Complexity is a measure of cost. The relevant costs depend on the model under analysis. The costs can be taken as units of time, number of comparisons, size of storage, or number of arithmetics. A number of different costs can be relevant to a model. The complexity of an algorithm, of a class of algorithms, or of a problem can be analyzed. The subject dealing with the analysis of a class of algorithms or of a problem is called computational complexity. Computational complexity comes in many flavors depending on the class of algorithms, the problem, and the costs. There are three types of computational complexity. In each of these, the costs are taken as the arithmetic operations. Algebraic computational complexity deals with a problem and a class of algorithms that solve the problems at finite cost. The branch of complexity theory that deals with nonfinite cost problems analytic is called computational complexity.
18 citations