On the solution of concave knapsack problems

We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.Work supported in part by a Fannie and John Hertz Foundation graduate fellowship.     

An improved algorithm for selecting <Emphasis Type="Italic">p</Emphasis> items with uncertain returns according to the minmax-regret criterion

We consider the problem of selecting a subset of p investments of maximum total return out of a set of n available investments with uncertain returns, where uncertainty is represented by interval estimates for the returns, and the minmax regret objective is used. We develop an algorithm that solves this problem in O(min{p,n–p}n) time. This improves the previously known complexity O(min{p,n–p}²n).This research has been supported by the Spanish Science and Technology Ministry and FEDER Grant No. BFM2002-04525-C02-02.Received: October 2002 / Accepted: September 2003     

A quadratically constrained minimization problem arising from PDE of Monge–Ampère type

This note develops theory and a solution technique for a quadratically constrained eigenvalue minimization problem. This class of problems arises in the numerical solution of fully-nonlinear boundary value problems of Monge–Ampère type. Though it is most important in the three dimensional case, the solution method is directly applicable to systems of arbitrary dimension. The focus here is on solving the minimization subproblem which is part of a method to numerically solve a Monge–Ampère type equation. These subproblems must be evaluated many times in this numerical solution technique and thus efficiency is of utmost importance. A novelty of this minimization algorithm is that it is finite, of complexity O(n³)\mathcal{O}(n^3), with the exception of solving a very simple rational function of one variable. This function is essentially the same for any dimension. This result is quite surprising given the nature of the constrained minimization problem.     

New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems

We consider the bounded integer knapsack problem (BKP) , subject to: , and x_j{0,1,…,m_j},j=1,…,n. We use proximity results between the integer and the continuous versions to obtain an O(n³W²) algorithm for BKP, where W=max_j=1,…,nw_j. The respective complexity of the unbounded case with m_j=∞, for j=1,…,n, is O(n²W²). We use these results to obtain an improved strongly polynomial algorithm for the multicover problem with cyclical 1’s and uniform right-hand side.     

A new implementation of an algorithm for the optimal assignment problem: An improved version of Munkres' algorithm

Existing implementations of Munkres' algorithm for the optimal assignment problem are shown to requireO(n ⁴) time in the worstn×n case. A new implementation is presented which runs in worst-case timeO(n ³) and compares favorably in performance with the algorithm of Edmonds and Karp for this problem.The results of this paper were obtained by the author while at the Department of Computer Science, Cornell University. This work was supported in part by a Vanderbilt University Research Council Grant.     

Algorithms for approximate calculation of the minimum of a convex function from its values

The paper deals with a numerical minimization problem for a convex function defined on a convexn-dimensional domain and continuous (but not necessarily smooth). The values of the function can be calculated at any given point. It is required to find the minimum with desired accuracy. A new algorithm for solving this problem is presented, whose computational complexity asn is considerably less than that of similar algorithms known to the author. In fact, the complexity is improved fromCn ⁷ ln²(n+1) [4] toCn ² ln(n+1).Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 95–102, January, 1996.     

Linear-size nonobtuse triangulation of polygons

We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than π/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n ²), and the running time isO(n log² n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm. The research of S. Mitchell was supported by the Applied Mathematical Sciences program, U.S. Department of Energy Research and by the U.S. Department of Energy under Contract DE-AC04-76DP00789. J. Ruppert's work was performed while he was at the NASA Ames Research Center as an employee of Computer Sciences Corporation, under NASA Contrast NAS 2-12961.     

Scheduling unit time jobs with integer release dates to minimize the weighted number of tardy jobs

Consider a set of n unit time jobs, each one having a release date, a due date, both nonnegative integers, and a weight, a positive real number. Given a set of m parallel machines, we describe an algorithm for finding schedules with minimum weighted number of tardy jobs. The complexity of the proposed algorithm is O(n²\frac(1+logm)m)O(n^{2}\frac{(1+\log m)}{m}) . The best previous algorithm for this problem has complexity O(mn ³) and employs network flow techniques. Our method is based on a characterization for schedules of this type and employs graph theoretic tools.     

Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs

Motivated by just-in-time manufacturing, we consider a single machine scheduling problem with dual criteria, i.e., the minimization of the total weighted earliness subject to minimum number of tardy jobs. We discuss several dominance properties of optimal solutions. We then develop a heuristic algorithm with time complexity O(n³) and a branch and bound algorithm to solve the problem. The computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium-size problems.     

About strongly polynomial time algorithms for quadratic optimization over submodular constraints

We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(NM log(N ²/M)) algorithm for the problemNetwork defined on a network onM arcs andN nodes; an O(n logn) algorithm for thetree problem onn variables; an O(n logn) algorithm for theNested problem, and a linear time algorithm for theGeneralized Upper Bound problem. These algorithms are the best known so far for these problems. The status of the general problem and open questions are presented as well.This research has been supported in part by ONR grant N00014-91-J-1241.Corresponding author.     

A new algorithm for the quasi-assignment problem

The quasi-assignment problem can be used to solve the bus scheduling problem, the tourist guide problem, and the minimum number of chains in a partially ordered set. A successive shortest path algorithm for the assignment problem is extended to the quasiassignment problem. The algorithm is a variation of the primal-dual algorithm, and its computational complexity isO(n ³).The research for this paper was partly supported by the Chinese National Science Foundation.     

Recognition problems for special classes of polynomials in 0–1 variables

This paper investigates the complexity of various recognition problems for pseudo-Boolean functions (i.e., real-valued functions defined on the unit hypercubeB ⁿ = {0, 1} ⁿ ), when such functions are represented as multilinear polynomials in their variables. Determining whether a pseudo-Boolean function (a) is monotonic, or (b) is supermodular, or (c) is threshold, or (d) has a unique local maximum in each face ofB ⁿ , or (e) has a unique local maximum inB ⁿ , is shown to be NP-hard. A polynomial-time recognition algorithm is presented for unimodular functions, previously introduced by Hansen and Simeone as a class of functions whose maximization overB ⁿ is reducible to a network minimum cut problem.     

Finding minimal nested polygons

An algorithm for finding a polygon with minimum number of edges nested in two simplen-sided polygons is presented. The algorithm solves the problem in at mostO(n logn) time, and improves the time complexity of two previousO(n ²) algorithms.The work was supported by NSERC grant OPG0041629.     

The Iterative Convex Minorant Algorithm for Nonparametric Estimation

Abstract

The problem of minimizing a smooth convex function over a specific cone in IRⁿ is frequently encountered in nonparametric statistics. For that type of problem we suggest an algorithm and show that this algorithm converges to the solution of the minimization problem. Moreover, a simulation study is presented, showing the superiority of our algorithm compared to the EM algorithm in the interval censoring case 2 setting.     

Multivariate Polynomial Minimization and Its Application in Signal Processing

We make a conjecture that the number of isolated local minimum points of a 2n-degree or (2n+1)-degree r-variable polynomial is not greater than n ^r when n 2. We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of r variables may have at most one local minimum point though it may have 2^r critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.     

Dynamic Euclidean minimum spanning trees and extrema of binary functions

We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n ^1/2 log² n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in timeO(n ^e ) per update. Our algorithm uses a novel construction, theordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. This research was supported in part by NSF Grant CCR-9258355     

A capacity-rounding algorithm for the minimum-cost circulation problem: A dual framework of the Tardos algorithm

Recently, É. Tardos gave a strongly polynomial algorithm for the minimum-cost circulation problem and solved the open problem posed in 1972 by J. Edmonds and R.M. Karp. Her algorithm runs in O(m ² T(m, n) logm) time, wherem is the number of arcs,n is the number of vertices, andT(m, n) is the time required for solving a maximum flow problem in a network withm arcs andn vertices. In the present paper, taking an approach that is a dual of Tardos's, we also give a strongly polynomial algorithm for the minimum-cost circulation problem. Our algorithm runs in O(m ² S(m, n) logm) time and reduces the computational complexity, whereS(m, n) is the time required for solving a shortest path problem with a fixed origin in a network withm arcs,n vertices, and a nonnegative arc length function. The complexity is the same as that of Orlin's algorithm, recently developed by efficiently implementing the Edmonds-Karp scaling algorithm.     

An approximation algorithm for the minimum two peripatetic salesmen problem with different weight functions

We present a polynomial algorithm with time complexity O(n ⁵) and approximation ratio 4/3 (plus some additive constant) for the minimum 2-peripatetic salesman problem in a complete n-vertex graph with different weight functions valued 1 and 2 (abbreviated to as 2-PSP(1, 2)-min-2w). This result improves the available algorithm for this problem with approximation ratio 11/7.     

Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality

The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NP-complete, even when all variables are binary. This paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality, and with all variables bounded between 0 andU. For such systems, a 2-approximation algorithm is presented that runs in time O(mnU ² log(Un ² m)), so it is polynomial in the input size if the upper boundU is polynomially bounded. The algorithm works by finding first a super-optimal feasible solution that consists of integer multiples of 1/2. That solution gives a tight bound on the value of the minimum. It furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, in particular, a 2-approximation algorithm for the problem of minimizing the total weight of true variables, among all truth assignments to the 2-satisfiability problem.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported in part by ONR contracts N00014-88-K-0377 and N00014-91-J-1241.Research supported in part by ONR contract N00014-91-C-0026.Part of this work was done while the author was visiting the International Computer Science Institute in Berkeley, CA and DIMACS, Rutgers University, New Brunswick, NJ.     

Polynomial time algorithms for some classes of constrained nonconvex quadratic problems

In this paper we consider different classes of noneonvex quadratic problems that can be solved in polynomial time. We present an algorithm for the problem of minimizing the product of two linear functions over a polyhedron P in R ⁿ The complexity of the algorithm depends on the number of vertices of the projection of P onto the R ² space. In the worst-case this algorithm requires an exponential number of steps but its expected computational time complexity is polynomial. In addition, we give a characterization for the number of isolated local minimum areas for problems on this form.

Furthermore, we consider indefinite quadratic problems with variables restricted to be nonnegative. These problems can be solved in polynomial time if the number of negative eigenvalues of the associated symmetric matrix is fixed.