Seiji Kataoka

Some exact algorithms for the knapsack sharing problem

Abstract The knapsack sharing problem (KSP) is formulated as an extension to the ordinary knapsack problem. The KSP is N P -hard. We present a branch-and-bound algorithm and a binary search algorithm to solve this problem to optimality. These algorithms are implemented and computational experiments are carried out to analyse the behavior of the developed algorithms. As a result, we find that the binary search algorithm solves KSPs with up to 20 000 variables in less than a minute in our computing environment.

A heuristic algorithm for the mini-max spanning forest problem☆

Abstract In this paper we formulate the mini-max spanning forest problem for undirected graphs as a generalization of the minimum spanning tree problem. We prove that the former problem is NP -hard. Then we develop a heuristic algorithm to solve this problem approximately, and give an upper bound for relative errors. Through a series of numerical tests, we examine the performance of the developed algorithm.

On some LP problems for performance evaluation of timed marked graphs

Three linear programming (LP) formulations are known for performance evaluation of timed marked graphs. Two of these are dual. This paper shows that the third problem is isomorphic to one of the dual problems. An illustrative example is also given. >

Algorithms to solve the knapsack constrained maximum spanning tree problem

The knapsack problem and the minimum spanning tree problem are both fundamental in operations research and computer science. We are concerned with a combination of these two problems. That is, we are given a knapsack of a fixed capacity, as well as an undirected graph where each edge is associated with profit and weight. The problem is to fill the knapsack with a feasible spanning tree such that the tree profit is maximized. We prove this problem 𝒩𝒫-hard, present upper and lower bounds, develop a branch-and-bound algorithm to solve the problem to optimality and propose a shooting method to accelerate computation. We evaluate the developed algorithm through a series of numerical experiments for various types of test problems.

A branch-and-bound algorithm for the mini-max spanning forest problem

The mini-max spanning forest problem requires to find a spanning forest of an undirected graph that minimizes the maximum of the costs of constituent trees. In a previous work we proved this problem NP-hard. In the current paper we present three lower bounds for this problem and develop a branch-and-bound algorithm to solve the problem exactly. The algorithm is implemented and numerical experiments are conducted on a series of test problems. The experiments compare the performances of the proposed bounds and search strategies in the algorithm as well as investigate the effects of instance characteristics on the behavior of the algorithm. Also, extension of the problem to the case of more than two root vertices as well as to the problem of determining the root locations are discussed.

Minimum directed 1-subtree relaxation for score orienteering problem

Score orienteering is a variant of the orienteering game where competitors start at a specified control point, visit as many control points as possible and thereby collect prizes, and return to the starting point within a prescribed amount of time. The competitor with the highest amount of prizes wins the game. In this paper, we propose the minimum directed 1-subtree problem as a new relaxation to this problem and develop two methods to improve its lower bound. We describe a cut and dual simplex method and a Lagrangian relaxation method, and construct an algorithm that combines these two methods in an appropriate way. Computational experiments are carried out to analyse the behavior of the proposed relaxation with respect to the characteristics of the test problems. Especially for large-scale instances, we find that the proposed relaxation is superior to the ordinary assignment relaxation.

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.

Upper and lower bounding procedures for minimum rooted k-subtree problem

Abstract Given an undirected graph and a fixed root node on it, the minimum rooted k-subtree problem is the problem of finding a minimum-cost connected subtree with exactly k edges that includes the root node. Applications of this problem appear in several fields. The problem is proved to be NP -hard. Due to the existence of a root node, we are able to derive a greedy procedure that gives a stronger lower-bound to this problem than those in earlier studies. Heuristics are also discussed to obtain an upper bound. Computational results indicate that the proposed lower bounding procedure is effective when k is small. Especially for grid-type graphs, we observe that the upper and lower bounds frequently coincide with each other.

A virtual pegging approach to the max-min optimization of the bi-criteria knapsack problem

We are concerned with a variation of the knapsack problem, the bi-objective max–min knapsack problem (BKP), where the values of items differ under two possible scenarios. We have given a heuristic algorithm and an exact algorithm to solve this problem. In particular, we introduce a surrogate relaxation to derive upper and lower bounds very quickly, and apply the pegging test to reduce the size of BKP. We also exploit this relaxation to obtain an upper bound in the branch-and-bound algorithm to solve the reduced problem. To further reduce the problem size, we propose a ‘virtual pegging’ algorithm and solve BKP to optimality. As a result, for problems with up to 16,000 items, we obtain a very accurate approximate solution in less than a few seconds. Except for some instances, exact solutions can also be obtained in less than a few minutes on ordinary computers. However, the proposed algorithm is less effective for strongly correlated instances.

Listing all the minimum spanning trees in an undirected graph

Efficient polynomial time algorithms are well known for the minimum spanning tree problem. However, given an undirected graph with integer edge weights, minimum spanning trees may not be unique. In this article, we present an algorithm that lists all the minimum spanning trees included in the graph. The computational complexity of the algorithm is O(N(mn+n 2 log n)) in time and O(m) in space, where n, m and N stand for the number of nodes, edges and minimum spanning trees, respectively. Next, we explore some properties of cut-sets, and based on these we construct an improved algorithm, which runs in O(N m log n) time and O(m) space. These algorithms are implemented in C language, and some numerical experiments are conducted for planar as well as complete graphs with random edge weights.