Ankur Moitra: Pareto Optimal Solutions for Smoothed Analysts
Channel: Stanford Computer Science Theory
Date: 16/10/2011

Video from Beyond Worst Case Analysis, 
Stanford, CA 
Sept 19-21, 2011

Ankur Moitra: Pareto Optimal Solutions for Smoothed Analysts

Consider an optimization problem with $n$ binary variables and $d+1$ linear objective functions. Each valid solution $x \in \{0,1\}^n$ gives rise to an objective vector in $\R^{d+1}$, 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 $n$ (Roeglin, Teng, FOCS 2009). Unfortunately, the bound obtained had a rather bad dependence on $d$; roughly $n^{d^d}$. In this paper we show a significantly improved bound of $n^{2d}$.

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 $B$.

This is joint work with Ryan O'Donnell.