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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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TL;DR: It is proved that the $K_{a,b}$ counting problem admits an $n^{a+o(1)}$-time algorithm if $a\geq 8$, while any-n-a-\epsilon algorithm fails to solve it even on random bipartite graph for any constant $\epsilons>0$ under the Strong Exponential Time Hypotheis.
Abstract: In this paper, we seek a natural problem and a natural distribution of instances such that any $O(n^{c-\epsilon})$-time algorithm fails to solve most instances drawn from the distribution, while the problem admits an $n^{c+o(1)}$-time algorithm that correctly solves all instances. Specifically, we consider the $K_{a,b}$ counting problem in a random bipartite graph, where $K_{a,b}$ is a complete bipartite graph for constants $a$ and $b$. We proved that the $K_{a,b}$ counting problem admits an $n^{a+o(1)}$-time algorithm if $a\geq 8$, while any $n^{a-\epsilon}$-time algorithm fails to solve it even on random bipartite graph for any constant $\epsilon>0$ under the Strong Exponential Time Hypotheis. Then, we amplify the hardness of this problem using the direct product theorem and Yao's XOR lemma by presenting a general framework of hardness amplification in the setting of fine-grained complexity.

15 citations

Proceedings Article
25 Jan 2015
TL;DR: This work examines the inferential complexity of Bayesian networks specified through logical constructs, and first considers simple propositional languages, and then move to relational languages.
Abstract: We examine the inferential complexity of Bayesian networks specified through logical constructs. We first consider simple propositional languages, and then move to relational languages. We examine both the combined complexity of inference (as network size and evidence size are not bounded) and the data complexity of inference (where network size is bounded); we also examine the connection to liftability through domain complexity. Combined and data complexity of several inference problems are presented, ranging from polynomial to exponential classes.

15 citations

Posted Content
TL;DR: The course "Communication complexity (for Algorithm Designers) as discussed by the authors was the first one to consider the problem of proving lower bounds for communication complexity lower bounds in the context of algorithms.
Abstract: This document collects the lecture notes from my course "Communication Complexity (for Algorithm Designers),'' taught at Stanford in the winter quarter of 2015. The two primary goals of the course are: 1. Learn several canonical problems that have proved the most useful for proving lower bounds (Disjointness, Index, Gap-Hamming, etc.). 2. Learn how to reduce lower bounds for fundamental algorithmic problems to communication complexity lower bounds. Along the way, we'll also: 3. Get exposure to lots of cool computational models and some famous results about them --- data streams and linear sketches, compressive sensing, space-query time trade-offs in data structures, sublinear-time algorithms, and the extension complexity of linear programs. 4. Scratch the surface of techniques for proving communication complexity lower bounds (fooling sets, corruption bounds, etc.).

15 citations

Journal ArticleDOI
TL;DR: It is proved that any LMP which satisfies (A.1) of Part I is tractable and its exponent is at most 2.1, and it is shown that optimal or nearly optimal sample points can be derived from hyperbolic cross points, and exhibit nearly optimal algorithms.

15 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
20216
202010
20199
201810
201732