Abstract

In this paper, we survey some recent progress in the smoothed analysis of algorithms and heuristics in mathematical programming, combinatorial optimization, computational geometry, and scientific computing. Our focus will be more on problems and results rather than on proofs. We discuss several perturbation models used in smoothed analysis for both continuous and discrete inputs. Perhaps more importantly, we present a collection of emerging open questions as food for thought in this field.

Prelinminaries

The quality of an algorithm is often measured by its time complexity (Aho, Hopcroft & Ullman (1983) and Cormen, Leiserson, Rivest & Stein (2001)). There are other performance parameters that might be important as well, such as the amount of space used in computation, the number of bits needed to achieve a given precision (Wilkinson (1961)), the number of cache misses in a system with a memory hierarchy (Aggarwal et al. (1987), Frigo et al. (1999), and Sen et al. (2002)), the error probability of a decision algorithm (Spielman & Teng (2003a)), the number of random bits needed in a randomized algorithm (Motwani & Raghavan (1995)), the number of calls to a particular “oracle” program, and the number of iterations of an iterative algorithm (Wright (1997), Ye (1997), Nesterov & Nemirovskii (1994), and Golub & Van Loan (1989)). The quality of an approximation algorithm could be its approximation ratio (Vazirani (2001)) and the quality of an online algorithm could be its competitive ratio (Sleator & Tarjan (1985) and Borodin & El-Yaniv (1998)).