Approximation Algorithms For Optimization Problems Research Articles

The fundamental theorem of linear programming: extensions and applications

The fundamental theorem of linear programming (LP) states that every feasible linear program that is bounded below has an optimal solution in a zero-dimensional face (a vertex) of the feasible polyhedron. We extend this result in two directions. We find a larger class of objective functions for which vertex optimality holds, and we give conditions guaranteeing the existence of an optimal solution in a larger family of faces of the feasible polyhedron. Our results also extend, with a very simple proof, the Frank–Wolfe theorem on the existence of an optimal solution to a Quadratic Program that is bounded below. We then apply our results to build up a general framework for obtaining upper and lower bounds and constant factor approximation algorithms for optimization problems. Furthermore, we show that several known results providing continuous formulations for discrete optimization problems can be easily derived and generalized with our extension of the fundamental theorem of Linear Programming. Finally, by exploiting the equivalence between continuous and discrete formulations of the problem of minimizing a function on a polyhedron, we prove polynomial-time solvability and present efficient algorithms for several new classes of continuous optimization problems.

Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 + a 1 x 1 + . . . + a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form (a 0 + a 1 x 1 + . . . + a n x n )/(b 0 + b 1 x 1 + . . . + b n x n ) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo’s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo’s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.

A faster fully polynomial approximation scheme for the single-machine total tardiness problem

Abstract Lawler [E.L. Lawler, A fully polynomial approximation scheme for the total tardiness problem, Operations Research Letters 1 (1982) 207–208] proposed a fully polynomial approximation scheme for the single-machine total tardiness problem which runs in O n 7 e time (where n is the number of jobs and e is the desired level of approximation). A faster fully polynomial approximation scheme running in O ( n 5 log n + n 5 e ) time is presented in this note by applying an alternative rounding scheme in conjunction with implementing Kovalyov’s [M.Y. Kovalyov, Improving the complexities of approximation algorithms for optimization problems, Operations Research Letters 17 (1995) 85–87] bound improvement procedure.

Approximation algorithms for optimization problems in graphs with superlogarithmic treewidth

We present a generic scheme for approximating NP-hard problems on graphs of treewidth k = ω ( log n ) . When a tree-decomposition of width ℓ is given, the scheme typically yields an ℓ / log n -approximation factor; otherwise, an extra log k factor is incurred. Our method applies to several basic subgraph and partitioning problems, including the maximum independent set problem.

Improving the complexities of approximation algorithms for optimization problems

A procedure to improve the lower and upper bound for the optimal objective function value of a minimization problem is presented. The procedure is based on an approximation scheme. Properties of this approximation scheme are established on which the correctness and effectiveness of the bound improvement procedure are based. The procedure is applied to improve the computational complexities of approximation algorithms for several single and parallel machine scheduling problems.