Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .

We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well-understood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX, and show that every problem in APX can be “placed ” (i.e. has approximation preserving reduction to a problem) in MAX SNP. Our methods introduce a general technique for creating approximation-preserving reductions which show that any “well ” approximable problem can be reduced in an approximation-preserving manner to a problem which is hard to approximate to corresponding factors. We demonstrate this technique by applying it to the classes RMAX(2) and MIN F+Π2(1)which have the clique problem and the set cover problem, respectively, as complete problems. We use the syntactic nature of MAX SNP to define a general paradigm, non-oblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the best-known for a variety of specific MAX SNP-hard problems. Non-oblivious local search provably out-performs standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.

Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a constant factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension. 1

this paper we obtain new results on the structure of several computationally-defined approximation classes. In particular, after defining a new approximation preserving reducibility to be used for as many approximation classes as possible, we give the first examples of natural NPO-complete problems and the first examples of natural APX-intermediate problems. Moreover, we state new connections between the approximability properties and the query complexity of NPO problems.

We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di#erent computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used.

This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a collection�of “constraints” (i.e., functions����������) and an instance of a problem is constraints drawn from�applied to specified subsets of Boolean variables. We study the two minimization analogs of classes studied in [15]: in one variant, namely MIN CSP�, the objective is to find an assignment to minimize the number of unsatisfied constraints, while in the other, namely MIN ONES�, the goal is to find a satisfying assignment with minimum number of ones. These two classes together capture an entire spectrum of important minimization problems including- Min Cut, vertex cover, hitting set with bounded size sets, integer programs with two variables per inequality, graph bipartization, clause deletion in CNF formulae, and nearest codeword. Our main result is that there exists a finite partition of the space of all constraint sets such that for any given�, the approximability of MIN CSP�and MIN ONES� is completely determined by the partition containing it. Moreover, we present a compact set of rules that determines which partition contains a given family�. Our classification identifies the central elements governing the approximability of problems in these classes, by unifying a large collection algorithmic and hardness of approximation results. When contrasted with the work of [15], our results also serve to formally highlight inherent differences between maximization and minimization problems.

In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner---a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound--- NP-hardness---on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, i.e., it is complete for \Theta p 2 , for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses. Email: edith@bamboo.lemoyne.edu. Supported in part by grant NSF-INT-9513368/DAAD-315-PROfo -ab. Work done in part while visiting Friedrich-Schiller-Universitat Jena and the University of Amst...

The class PTAS is defined to consist of all NP optimization problems that permit polynomial-time approximation schemes. This paper explores the possibility that a core of PTAS may be characterized through syntactic classes endowed with restrictions on the structure of the input instances. Recent work in approximability of NP-hard problems has led to the identification of a syntactic class called MAX SNP as the core of APX, the class of constant-factor approximable NP optimization problems. This has enhanced our understanding of these classes from both an algorithmic and a complexity-theoretic point of view. Our work is motivated by the hope that a similar understanding can be attained for PTAS. We argue that while the core of APX is the purely syntactic class MAX SNP, in the case of PTAS we must identify the core in terms of syntactic prescriptions for the problem definition augmented with structural restrictions on the input instances. Specifically, we propose such a unified framework...

We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) definition of niceness, we show that if a nice weighted problem is hard to approximate within r, then its polynomially bounded weighted version is hard to approximate within r \Gamma o(1). Then we turn our attention to specific problems, and we show that the unweighted

A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)-approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new, and sometimes harder, techniques. For some interesting problems, only n”(“E) approximation schemes are known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time f ( l/n)nO(‘), no matter how fast function f grows. Our result relies on a connection with Parameterized Complexity Theory, and we show that this connection is necessary.