Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within $A$. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall expected time and space requirement in the average case are proven to be $O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.Comment: 11 pages, 1 figur

Exact algorithms for NP-hard problems: a survey

We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more

Knapsack Problems; Methods, Models and Applications

Knapsack problem (KP) has broad applications in different fields such as machine scheduling, space allocation, and asset optimization. Meanwhile, it is a hard problem due to its computational complexity, but numerous solution approaches have been developed for a variety of KP. In this dissertation, an extensive literature review is first provided. Then, the research focuses on methods, models, and applications for two variations of Knapsack problem: Multiple Knapsack Problem with Assignment Restrictions (MKAR) and Stochastic Knapsack Problem with Penalty Cost (SKPPC).A new procedure, Largest Unutilized Capacity First Algorithm (LUCF) is developed and tested on MKAR along with other assignment procedures available in the literature. It is concluded that LUCF performs very well and it returns the best initial feasible solution among all types of greedy algorithms for the solution of the MKAR. After the generation of initial feasible solutions, a tabu-search procedure is implemented to generate improved solutions. Three versions of intensification procedures are implemented within the tabu search procedure. Experimental results show significant improvement over the initial solution quality with the tabu search procedure. That is, this approach yields a high percentage of utilization for all combinations of problems, based on the initial solution provided by LUCF.For SKPPC, for each item of the knapsack, there are several possible processing times, each with certain probability of selection. For a given knapsack capacity, a strategy is developed to assign the optimal number of items to each the knapsack. Mathematical formulations are provided for both single knapsack and m-knapsack cases. The objective value function for the single knapsack problem exhibits a convex property, which leads to an optimal strategy to assign the number of items. For the m-knapsack case, the processing time of each item will be revealed after pre-scan operations. LUCF heuristic is combined here to obtain good solutions. This approach is finally adapted to the package security inspection problem. We discuss how one can determine the optimal number of items in each knapsack and the optimal number of operators needed for inspection with the objective of maximizing operator utilization and throughput

A Polynomial Algorithm for a NP-hard to solve Optimization Problem

Since Markowitz in 1952 described an efficient and practical way of finding the optimal portfolio allocation in the normal distributed case, a lot of progress in several directions has been made. The main objective of this thesis is to replace the original risk measure of the Markowitz setting by a more suitable one, Value-at-Risk. In adressing the optimal allocation problem in a slightly more general setting, thereby still allowing for a large number of different asset classes, an efficient algorithm is developed for finding the exact solution in the case of specially distributed losses. Applying this algorithm to even more general loss distributions results in a not necessarily exact matching of the VaR optimum. However, in this case, upper bounds for the euclidean distance between the exact optimum and the output of the proposed algorithm are given. An investigation of these upper bounds shows, that in general the algorithm results in quite good approximations to the VaR optimum. Finally, an application of a stochastic branch & bound algorithm to the current problem is discussed

An exact algorithm for the subset-sum problem

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）本文は図書館に配架されています。 / This material is stored in the JAIST library

An exact algorithm for the subset-sum problem

最適化：モデリングとアルゴリズム９In this paper we propose a new algorithm for solving the subset-sum problem. First we propose a new algorithm (xs-algorithm) for the partition problem. Then we describe a transformation of the subset-sum problem to the partition problem, and show how to solve the resulting problem by a slightly modified version of xs-algorithm. Finally, we present results of extensive computational experiments for several types of data instances. The paper is based on the master thesis of the first author