Pareto optimal solutions for smoothed analysts

  • Authors:

  • Ankur Moitra;Ryan O'Donnell

  • Affiliations:

  • MIT, Cambridge, USA;CMU, Pittsburgh, USA

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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Abstract

Consider an optimization problem with n binary variables and d+1 linear objective functions. Each valid solution x ∈{0,1}n gives rise to an objective vector in Rd+1, and one often wants to enumerate the Pareto optima among them. In the worst case there may be exponentially many Pareto optima; however, it was recently shown that in (a generalization of) the smoothed analysis framework, the expected number is polynomial in~n. Unfortunately, the bound obtained had a rather bad dependence on d; roughly ndd. In this paper we show a significantly improved bound of n2d. Our proof is based on analyzing two algorithms. The first algorithm, on input a Pareto optimal x, outputs a "testimony" containing clues about x's objective vector, x's coordinates, and the region of space B in which x's objective vector lies. The second algorithm can be regarded as a speculative execution of the first --- it can uniquely reconstruct x from the testimony's clues and just some of the probability space's outcomes. The remainder of the probability space's outcomes are just enough to bound the probability that x's objective vector falls into the region B.