Are stable instances easy?

arXiv:0906.3162v1 [cs.CC] 17 Jun 2009 Are stable instances easy? Yonatan Bilu ∗ Mobileye Vision Technologies Ltd. 12 Hartom Street, PO Box 45157 Jerusalem, 91450 Israel. yonatan.bilu@mobileye.com Nathan Linial ∗ Institute of Computer Science Hebrew University Jerusalem 91904, Israel. nati@cs.huji.ac.il October 22, 2018 Abstract We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP–hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polynomial time all sufficiently stable instances of some NP–hard problem. The paper focuses on the Max–Cut problem, for which we show that this is indeed the case. 1 Introduction Computational complexity theory as we know it today is concerned mostly with worst-case analysis of computational problems. For example, we say that a problem is NP-hard if the existence of an algorithm that correctly decides every instance of the problem implies that SAT can be decided in a polynomially equivalent time complexity. However, the study of decision and optimization problems is motivated not merely by theoretical considerations. Much of our interest in such problems arises because they formalize certain real-world tasks. From this perspective, we are not interested in all problem instances, but only in those which can actually occur in reality. This is often the case with clustering problems, which are ubiquitous in most fields of engineering, exper- imental and applied science. Any concrete formulation of the clustering problem is likely to be NP-hard. However this does not preclude the possibility that the problem can be solved efficiently in practice. In fact, in numerous application areas, large-scale clustering problems are solved on a regular basis. As mentioned above, we are only interested in instances where the data is actually made up of fairly well-defined clusters - the instances where solving the problem is interesting from the practical perspective. Put differently, the usual way for proving that clustering is NP-hard is by a reduction to, say, SAT. This reduction entails the construction of instances for the clustering problem, such that the existence of an algorithm that can solve all of them efficiently implies the existence of an algorithm that efficiently solves SAT. However, it may well be the case that all these instances are clearly artificial, and solving them is of no practical interest. ∗This research is supported by grants from the binational Science Foundation Israel-US and the Israel Science Foundation. 1 As a concrete example, consider the problem of clustering protein sequences into families. Out of the enormous space of all possible sequences, only a tiny fraction is encountered in nature, and it is only about these (or slight modifications thereof) that we actually care. Our case in point is the Max-Cut problem, which can be thought of as a clustering into two clusters. It is well known that this problem is NP-complete, and so it is believed that there is no algorithm that solves it correctly on all graphs, in polynomial time. In this work we strive to identify properties of instances of the Max-Cut problem (i.e., of weighted graphs), which capture the notion that the input has a well-defined structure w.r.t Max-Cut (i.e., the maximal cut “stands out” among all possible cuts). Our goal is to show that Max-Cut can be solved efficiently on inputs that have such properties. Consideration of a similar spirit have led to the development of Smoothed Analysis initiated in [15], (see [16] for some of the exciting developments in that area. The similarity has two main facets: (i) Both lines of research attempt to investigate the computational complexity of problems from a non-worst-case perspective, (ii) Both are investigations of the geometry of the instance space of the problem under consideration. The goal being to discover interesting parts of this space in which the instances have complexity lower than the worst case. Viewed from this geometric perspective, the set-up that we study here is very different than what is done in the theory of smoothed analysis. There one shows that the hard instances form a discrete and isolated subset of the input space. Consequently, for every instance of the problem, a small random perturbation is very likely to have low computational complexity. In the problems that we study here the situation is radically different. The “interesting” instances (stable instances as we shall call them) are very rare. Indeed, it is not hard to show that under reasonable models of random instances the probability that a random instance be stable is zero, or at least tends to zero as the problem size grows. What we wish to accomplish is to efficiently solve all instances within this subspace. We claim that this tiny set is inte

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