A compendium of NP optimization problems

Introduction 0.1 This is a continuously updated catalog of approximability results for NP optimization problems. The compendium will be printed as an appendix of the book Ausiello et al., 1998 that will be published in the end of 1998 or beginning of 1999. You can use WWW forms to report new problems, new results on existing problems or errors. Due to the fact that no NP-complete problem can be solved in polynomial time (unless P=NP), many approximability results (both positive and negative) of NP-hard optimization problems have appeared in the technical literature. In this compendium, we collect a large number of these results. In the following we refer to standard complexity classes (see [Johnson, 1990]). We recall that a function t(n) is 'quasi-polynomial' if a constant c exists such that t(n) ≤ n log c n and we denote by QP, QNP, and QR the analogues of the usual complexity classes in the quasi-polynomial time domain. The basic ingredients of an optimization problem are the set of instances or input objects, the set of feasible solutions or output objects associated with any instance, and the measure defined for any feasible solution. On the analogy of the theory of NP-completeness, we are interested in studying a class of optimization problems whose feasible solutions are short and easy-to-recognize. To this aim, suitable constraints have to be introduced. We thus give the following definition. Definition 0.1 An NP optimization problem A is a fourtuple (I, sol, m, goal) such that 1. I is the set of the instances of A and it is recognizable in polynomial time. 2. Given an instance x of I, sol(x) denotes the set of feasible solutions of x. These solutions are short, that is, a polynomial p exists such that, for any y ∈ sol(x), y ≤ p(x). Moreover, it is decidable in polynomial time whether, for any x and for any y such that y ≤ p(x), y ∈ sol(x). 3. Given an instance x and a feasible solution y of x, m(x, y) denotes the positive integer measure of y. The function m is computable in polynomial time and is also called the objective function. 4. goal ∈ {max, min}. The class NPO is the set of all NP optimization problems. The goal of an NPO problem with respect to an instance x is to find an optimum solution, that is, a feasible solution y such …