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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09 Jul 2006
TL;DR: In this article, it was shown that for the set of all d times n binary matrices with entries 0 or 1 and d < n, MKCS exactly recovers the input for an overwhelming fraction of the matrices provided the Kolmogorov complexity of the input is O(d).
Abstract: Consider a d times n matrix A, with d < n. The problem of solving for x in y = Ax is underdetermined, and has infinitely many solutions (if there are any). Given y, the minimum Kolmogorov complexity solution (MKCS) of the input x is defined to be an input z (out of many) with minimum Kolmogorov-complexity that satisfies y = Az. One expects that if the actual input is simple enough, then MKCS will recover the input exactly. This paper presents a preliminary study of the existence and value of the complexity level up to which such a complexity-based recovery is possible. It is shown that for the set of all d times n binary matrices (with entries 0 or 1 and d < n), MKCS exactly recovers the input for an overwhelming fraction of the matrices provided the Kolmogorov complexity of the input is O(d). A weak converse that is loose by a log n factor is also established for this case. Finally, we investigate the difficulty of finding a matrix that has the property of recovering inputs with complexity of O(d) using MKCS
45 citations
22 Jul 2013
TL;DR: In this article, the authors proposed a new algorithm that incorporates another important feature of the biological system: the probability value at each node is adapted using local feedback from neighbouring nodes, which achieves the optimal efficiency of O(log n) expected time.
Abstract: Maximal Independent Set selection is a fundamental problem in distributed computing. A novel probabilistic algorithm for this problem has recently been proposed by Afek et al, inspired by the study of the way that developing cells in the fly become specialised. The algorithm they propose is simple and robust, but not as efficient as previous approaches: the expected time complexity is O(log2n). Here we first show that the approach of Afek et al cannot achieve better efficiency than this across all networks, no matter how the global probability values are chosen.However, we then propose a new algorithm that incorporates another important feature of the biological system: the probability value at each node is adapted using local feedback from neighbouring nodes. Our new algorithm retains all the advantages of simplicity and robustness, but also achieves the optimal efficiency of O(log n) expected time. The new algorithm also has only a constant message complexity per node.
44 citations
TL;DR: In this paper, quantities are discussed which can serve as measures of the complexity of complex dynamical systems, and some of the most interesting patterns have zero randomness but infinite complexity in the present sense.
Abstract: In an increasing number of simple dynamical systems, patterns arise which are judged as “complex” in some naive sense. In this talk, quantities are discussed which can serve as measures of this complexity. They are measure-theoretic constructs. In contrast to the Kolmogorov complexity, they are small both for completely ordered and for completely random patterns. Some of the most interesting patterns have indeed zero randomness but infinite complexity in the present sense.
43 citations
TL;DR: It is shown that no monotone circuit of size $O(n^{k/4})$ solves the k-clique problem with high probability on $\ER(n,p)$ for two sufficiently far-apart threshold functions $p(n)$ and $2n^{-2/(k-1)}$.
Abstract: We present lower and upper bounds showing that the average-case complexity of the $k$-Clique problem on monotone circuits is $n^{k/4 + O(1)}$. Similar bounds for $\mathsf{AC}^0$ circuits were shown in Rossman [Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 721--730] and Amano [Comput. Complexity, 19 (2010), pp. 183--210].
43 citations
01 Mar 2013
TL;DR: The S-adic conjecture as mentioned in this paper states that a sequence has linear complexity if and only if it is an Sadic sequence satisfying C for some finite set S of morphisms, and provides an overview of the factor complexity of Sadic sequences and some examples that illustrate some interesting properties.
Abstract: The S-adic conjecture postulates the existence of a condition C such that a sequence has linear complexity if and only if it is an S-adic sequence satisfying C for some finite set S of morphisms. We present an overview of the factor complexity of S-adic sequences and we give some examples that either illustrate some interesting properties, or that are counterexamples to what might seem to be a "good" condition C.
43 citations