A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem

Dyer, Martin; Frieze, Alan; Kannan, Ravi; Kapoor, Ajai; Perković, Ljubomir; Vazirani, Umesh

doi:10.1017/s0963548300000675

Cited by 48 publications

(35 citation statements)

References 14 publications

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“…The problem of efficiently computing a multiplicative (1 ± )-approximation of p has received much attention (Dyer et al 1993;Jerrum & Sinclair 1997;Kannan 1994); the first polynomial-time algorithm was given by Morris & Sinclair (1999) using sophisticated Monte Carlo Markov Chain techniques, and more recently a simpler randomized algorithm based on dynamic programming and "dart throwing" was given by Dyer (2003).…”

Section: Application To Deterministic Approximate Countingmentioning

confidence: 99%

Every Linear Threshold Function has a Low-Weight Approximator

Servedio

2007

comput. complex.

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Abstract. Given any linear threshold function f on n Boolean variables, we construct a linear threshold function g which disagrees with f on at most an fraction of inputs and has integer weights each of magnitude at most √ n · 2Õ (1/ 2 ) . We show that the construction is optimal in terms of its dependence on n by proving a lower bound of Ω( √ n) on the weights required to approximate a particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately counting the fraction of satisfying assignments to an instance of the zero-one knapsack problem to within an additive ± . The algorithm runs in time polynomial in n (but exponential in 1/ 2 ). In our second application, we show that any linear threshold function f is specified to within error by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive ±1/(n · 2Õ (1/ 2 ) ). This is the first such accuracy bound which is inverse polynomial in n, and gives the first polynomial bound (in terms of n) on the number of examples required for learning linear threshold functions in the "restricted focus of attention" framework.

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Section: Application To Deterministic Approximate Countingmentioning

confidence: 99%

Every Linear Threshold Function has a Low-Weight Approximator

Servedio

2007

comput. complex.

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“…We have that the Subset Sum problem admits a solution of cost d if and only if the Gap Filling problem has an s-t path of cost in the interval [d + 2n, d + 2n]. Since the #Subset Sum problem is #P-complete (Dyer et al (1993)), this implies that also the #Gap Filling problem is #P-complete. Nykänen and Ukkonen (2002) …”

Section: Complexity and The Pseudo-polynomial Algorithmmentioning

confidence: 99%

Gap Filling as Exact Path Length Problem

Salmela

Sahlin

Mäkinen

et al. 2015

Lecture Notes in Computer Science

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One of the last steps in a genome assembly project is filling the gaps between consecutive contigs in the scaffolds. This problem can be naturally stated as finding an s-t path in a directed graph whose sum of arc costs belongs to a given range (the estimate on the gap length). Here s and t are any two contigs flanking a gap. This problem is known to be NP-hard in general. Here we derive a simpler dynamic programming solution than already known, pseudo-polynomial in the maximum value of the input range. We implemented various practical optimizations to it, and compared our exact gap filling solution experimentally to popular gap filling tools. Summing over all the bacterial assemblies considered in our experiments, we can in total fill 76% more gaps than the best previous tool and the gaps filled by our method span 136% more sequence. Furthermore, the error level of the newly introduced sequence is comparable to that of the previous tools. The experiments also show that our exact approach does not easily scale to larger genomes, where the problem is in general difficult for all tools.

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“…Since the partition problem is reducible from the #P-complete knapsack problem [6] and its own reduction as well as ours is parsimonious [9], the problem of counting all s-t paths of length at most L is #P-complete. Note that the existence of parallel edges is not necessary for the reduction; we could bisect each parallel edge creating an auxiliary vertex to form a graph of the same functionality but without parallel edges.…”

Section: Counting Approximate Solutions Is #P-completementioning

confidence: 99%

Counting Approximately-Shortest Paths in Directed Acyclic Graphs

Mihaľák

Šrámek

Widmayer

2014

Approximation and Online Algorithms

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Abstract. Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fullypolynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L1 and L2, we show how to approximately count the s-t paths that have length at most L1 in the first graph and length not much larger than L2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph.

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A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem

Cited by 48 publications

References 14 publications

Every Linear Threshold Function has a Low-Weight Approximator

Every Linear Threshold Function has a Low-Weight Approximator

Gap Filling as Exact Path Length Problem

Counting Approximately-Shortest Paths in Directed Acyclic Graphs

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