A Pseudo-Polynomial Time Algorithm for Solving the Knapsack Problem in Polynomial Space

2016 ◽  
pp. 624-638
Author(s):  
Noriyuki Fujimoto
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Polynomial Space ◽  
Pseudo Polynomial Time

A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables

1976 ◽  
Vol 23 (1) ◽  
pp. 147-154 ◽  
Author(s):  
D. S. Hirschberg ◽  
C. K. Wong
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

scholarly journals Applications of Generating Functions to Stochastic Processes and to the Complexity of the Knapsack Problem

2021 ◽  
Author(s):  
Jorma Jormakka ◽  
Sourangshu Ghosh
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Polynomial Time ◽  
Knapsack Problem ◽  
Generating Functions ◽  
General Theorem ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Linear Transform

The paper describes a method of solving some stochastic processes using generating functions. A general theorem of generating functions of a particular type is derived. A generating function of this type is applied to a stochastic process yielding polynomial time algorithms for certain partitions. The method is generalized to a stochastic process describing a rather general linear transform. Finally, the main idea of the method is used in deriving a theoretical polynomial time algorithm to the knapsack problem.

scholarly journals Order Statistics for Probabilistic Graphical Models

2017 ◽  
Author(s):  
David Smith ◽  
Sara Rouhani ◽  
Vibhav Gogate
Keyword(s):  
Order Statistics ◽  
Graphical Models ◽  
Polynomial Time ◽  
Graphical Model ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Partition Problem ◽  
Polynomial Time Approximation Algorithm ◽  
Sampling Algorithms ◽  
Pseudo Polynomial Time

We consider the problem of computing r-th order statistics, namely finding an assignment having rank r in a probabilistic graphical model. We show that the problem is NP-hard even when the graphical model has no edges (zero-treewidth models) via a reduction from the partition problem. We use this reduction, specifically a pseudo-polynomial time algorithm for number partitioning to yield a pseudo-polynomial time approximation algorithm for solving the r-th order statistics problem in zero- treewidth models. We then extend this algorithm to arbitrary graphical models by generalizing it to tree decompositions, and demonstrate via experimental evaluation on various datasets that our proposed algorithm is more accurate than sampling algorithms.

scholarly journals A Framework for Sampling-Based XML Data Pricing

2017 ◽  
Author(s):  
Ruiming Tang ◽  
Antoine Amarilli ◽  
Pierre Senellart ◽  
Stéphane Bressan
Keyword(s):  
Data Quality ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Uniform Sampling ◽  
First Case ◽  
Sampling Problem ◽  
Uniform Random Sampling ◽  
Data Consumer ◽  
Pseudo Polynomial Time

While price and data quality should define the major tradeoff for consumers in data markets, prices are usually prescribed by vendors and data quality is not negotiable. In this paper we study a model where data quality can be traded for a discount. We focus on the case of XML documents and consider completeness as the quality dimension. In our setting, the data provider offers an XML document, and sets both the price of the document and a weight to each node of the document, depending on its potential worth. The data consumer proposes a price. If the proposed price is lower than that of the entire document, then the data consumer receives a sample, i.e., a random rooted subtree of the document whose selection depends on the discounted price and the weight of nodes. By requesting several samples, the data consumer can iteratively explore the data in the document.We present a pseudo-polynomial time algorithm to select a rooted subtree with prescribed weight uniformly at random, but show that this problem is unfortunately intractable. Yet, we are able to identify several practical cases where our algorithm runs in polynomial time. The first case is uniform random sampling of a rooted subtree with prescribed size rather than weights; the second case restricts to binary weights.As a more challenging scenario for the sampling problem, we also study the uniform sampling of a rooted subtree of prescribed weight and prescribed height. We adapt our pseudo-polynomial time algorithm to this setting and identify tractable cases.

M-Convex Function Minimization Under L1-Distance Constraint and Its Application to Dock Reallocation in Bike-Sharing System

2021 ◽  
Author(s):  
Akiyoshi Shioura
Keyword(s):  
Convex Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Steepest Descent ◽  
Nonlinear Integer Programming ◽  
Distance Constraint ◽  
Polynomial Time Algorithms ◽  
Bike Sharing ◽  
Pseudo Polynomial Time

In this paper, we consider a problem of minimizing an M-convex function under an L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given “center.” This is motivated by a nonlinear integer programming problem for reallocation of dock capacity in a bike-sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock reallocation problem through the connection with M-convexity and show its polynomial-time solvability using this connection. For this, we first show that the dock reallocation problem and its generalizations can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on the steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock reallocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. For this purpose, we develop a polynomial-time algorithm for a relaxation of the dock reallocation problem by using a proximity-scaling approach, which is of interest in its own right.

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