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Approximation of Multiobjective Optimization Problems
We study optimization problems with multiple objectives. Such problems are pervasive across many diverse disciplines -- in economics, engineering, healthcare, biology, to name but a few -- and heuristic approaches to solve them have already been deployed in several areas, in both academia and industry. Hence, there is a real need for a rigorous investigation of the relevant questions. In such problems we are interested not in a single optimal solution, but in the tradeoff between the different objectives. This is captured by the tradeoff or Pareto curve, the set of all feasible solutions whose vector of the various objectives is not dominated by any other solution. Typically, we have a small number of objectives and we wish to plot the tradeoff curve to get a sense of the design space. Unfortunately, typically the tradeoff curve has exponential size for discrete optimization problems even for two objectives (and is typically infinite for continuous problems). Hence, a natural goal in this setting is, given an instance of a multiobjective problem, to efficiently obtain a ``good'' approximation to the entire solution space with ``few'' solutions. This has been the underlying goal in much of the research in the multiobjective area, with many heuristics proposed for this purpose, typically however without any performance guarantees or complexity analysis. We develop efficient algorithms for the succinct approximation of the Pareto set for a large class of multiobjective problems. First, we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy the Pareto curve of a multiobjective optimization problem. We provide approximation algorithms with tight performance guarantees for bi-objective problems and make progress for the more challenging case of three and more objectives. Subsequently, we propose and study the notion of the approximate convex Pareto set; a novel notion of approximation to the Pareto set, as the appropriate one for the convex setting. We characterize when such an approximation can be efficiently constructed and investigate the problem of computing minimum size approximate convex Pareto sets, both for discrete and convex problems. Next, we turn to the problem of approximating the Pareto set as efficiently as possible. To this end, we analyze the Chord algorithm, a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and graphics
Bounded Independence Fools Degree-2 Threshold Functions
Let x be a random vector coming from any k-wise independent distribution over
{-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is
determined up to an additive epsilon for k = poly(1/epsilon). This answers an
open question of Diakonikolas et al. (FOCS 2009). Using standard constructions
of k-wise independent distributions, we obtain a broad class of explicit
generators that epsilon-fool the class of degree-2 threshold functions with
seed length log(n)*poly(1/epsilon).
Our approach is quite robust: it easily extends to yield that the
intersection of any constant number of degree-2 threshold functions is
epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if
the entries of x are k-wise independent standard normals, implying for example
that bounded independence derandomizes the Goemans-Williamson hyperplane
rounding scheme.
To achieve our results, we introduce a technique we dub multivariate
FT-mollification, a generalization of the univariate form introduced by Kane et
al. (SODA 2010) in the context of streaming algorithms. Along the way we prove
a generalized hypercontractive inequality for quadratic forms which takes the
operator norm of the associated matrix into account. These techniques may be of
independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was
wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem
8.1 to m^4, not m^
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