Dual Problems: Attribute Selection and Example Selection

Since the theory of NP-completeness was developed in the 1970s, a huge number of important optimisation problems have been shown to be NP-hard. This means—according to a widely held conjecture—that no exact solution algorithm exists whose running time is bounded by a polynomial in the size of the problem description. When faced with an instance of an NP-hard problem, one therefore has two options: either run an exact algorithm and hope that it terminates within a reasonable amount of time, or resort to some kind of heuristic method. The problem with many heuristics (such as local search) is that one cannot tell a priori whether they are going to produce a solution of good quality. However, for some heur-istics it is possible to prove a 'performance guarantee'. For a minimisation problem this would mean a theorem stating that the cost of the solution produced by the heuristic will never be more than x% above the optimum, where x depends on the problem and the heuristic concerned. For example, the well-known heuristic of Christofides for the Travelling Salesman Problem (TSP) always returns a solution whose cost is no more than 50% above the optimum, provided that the costs satisfy the triangle inequality. A heuristic whose running time is polynomially bounded and for which such a performance guarantee is known is called an approximation algorithm. In the past 15 years or so, the field of approximation algorithms has blossomed from a few isolated ad hoc arguments into a well-developed and mature field. This book by Vazirani therefore appears timely. The main body of the book, excluding the introduction and appendices, is divided into three parts. The first part, consisting of 11 chapters, shows how approximation algorithms may sometimes be devised using relatively simple combinatorial arguments. The second part, consisting of 15 chapters, introduces some more sophisticated techniques, such as randomised rounding, semi-definite programming and the primal–dual schema. The third section, which is only four chapters, contains some miscellaneous results and open problems, and also a nice survey of recent results on inapproximability, including the celebrated PCP theorem. The ideas in Parts 1 and 2 are nicely illustrated on a wide variety of problemsmum Satisfiability, various problems involving flows and cuts (including Max-Cut), and certain problems arising in machine scheduling. Some problems which are hard to approximate, such as the Maximum Clique problem, are dealt with in Part 3. The appendices give self-contained …