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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18 Oct 2014
TL;DR: In this article, the authors show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k) and distributional communication complexity ≥ 2k.
Abstract: We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥ 2k. This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [1], our gap is the largest possible. By a result of Braverman and Rao [2], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold.
59 citations
Posted Content•
TL;DR: The many open questions and the few things that are known about the average-case complexity of computational problems are reviewed, and a theory of completeness for distributional problems under reductions that preserveaverage-case tractability is initiated.
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 whether the existence of hard-on-average problems in NP can be based on the P$
eq$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.
59 citations
TL;DR: It is shown that determining minimal size tries is an NP-complete problem for several variants of tries and that, for tries m which leaf chains are deleted, determining the trie for which average access time is minimal is also an NP -complete problem.
Abstract: Trle structures are a convenient way of indexing files in which a key consists of a number of attributes Records correspond to leaves in the trle Retrieval proceeds by following a path from the root to a leaf, the choice of edges being determined by attribute values The size of a trle for a file depends on the order in which attributes are tested It is shown that determining minimal size tries IS an NP-complete problem for several variants of tries and that, for tries m which leaf chains are deleted, determining the trie for which average access time is minimal is also an NP-complete problem These results hold even for files in which attribute values are chosen from a binary or ternary alphabet KE] WORDS AND PHRASES reformation retrieval, trle indexes, trte size, average search t i m e , complexity CR CATEGORIES 3 74, 4 33, 5 25
59 citations
TL;DR: Simulation results show that the extended min-sum (EMS) algorithm, combined with factor correction techniques, perform almost as well as the Q-ary sum-product algorithm (QSPA) but with a much lower complexity.
Abstract: The extended min-sum (EMS) algorithm is redescribed as a reduced-search trellis algorithm (called M-EMS algorithm). Two variants of the M-EMS algorithm, called T-EMS algorithm and D-EMS algorithm, are presented. Simulation results show that, these three algorithms (referred to as X-EMS algorithms for convenience), combined with factor correction techniques, perform almost as well as the Q-ary sum-product algorithm (QSPA) but with a much lower complexity.
59 citations
TL;DR: This is a survey on the computational complexity of nonlinear mixedinteger optimization, highlighting a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to stronglyPolynomialtime algorithms for special cases.
Abstract: This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomialtime algorithms for special cases.
59 citations