Rutgers Discrete Mathematics Seminar

Title: Pareto Optimal Solutions for Smoothed Analysts

Speaker: Ankur Moitra, IAS

Date: Tuesday, February 21, 2012 2:00pm

Location: Hill Center, Room 124, Rutgers University, Busch Campus, Piscataway, NJ


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 talk 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.