Limits of Local Search: Quality and Efficiency

Abstract

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set $$\mathcal {D}$$D of geometric objects, compute the minimum-sized subset of P that hits all objects in $$\mathcal {D}$$D. For the case where $$\mathcal {D}$$D is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum (SIAM J Comput 11:555---556, 1982), was finally achieved in Mustafa and Ray (Discret Comput Geom 44:883---895, 2010). Surprisingly, the algorithm to achieve the PTAS is simple: local-search. In particular, the algorithm starts with any hitting set, and iteratively tries to decrease its size by trying to replace some k points by $$k-1$$k-1 points; call such an algorithm a $$(k, k-1)$$(k,k-1)-local search algorithm. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. In particular, the current best work shows that if $$k \ge 30$$kź30, then local-search is able to give a constant factor (as a function of k) approximation ratio (Fraser in Algorithms for Geometric Covering and Piercing Problems, 2012). Unfortunately this then implies that the running time for local-search to provably work at all is $$\Omega (n^{30})$$Ω(n30) using the current framework. As currently local search is the only known method that gives approximation factors that could be useful in practice, it becomes important to explore the limits--in both efficiency and quality--of local search. Simple examples show that (1, 0) and (2, 1) local search cannot give constant factor approximations. In this paper, we show that, surprisingly, just (3, 2) local search is able to give a constant-factor approximation; in fact we are able to get the precise quality limit of (3, 2)-local search: factor 8 approximation. This simplest working instance of local search already gives an approximation factor that is better than all known other methods! In fact, our improvement applies to all algorithms that use local-search for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others. Finding efficient (3, 2)-local search algorithms then becomes the key bottleneck in efficient and good-quality algorithms. In this paper we present such improved algorithms.