Hardness Of Approximations

Discrete Energy Minimization, beyond Submodularity: Applications and Approximations

In this thesis I explore challenging discrete energy minimization problems that arise mainly in the context of computer vision tasks. This work motivates the use of such “hard-to-optimize” non-submodular functionals, and proposes methods and algorithms to cope with the NP-hardness of their optimization. Consequently, this thesis revolves around two axes: applications and approximations. The app...

Sub-optimality Approximations

The sub-optimality approximation problem considers an optimization problem O, its optimal solution σ∗, and a variable x with domain {d1, . . . , dm} and returns approximations to O[x← d1], . . . ,O[x← dm], where O[x← d1] denotes the problem O with x assigned to di. The sub-optimality approximation problem is at the core of online stochastic optimization algorithms and it can also be used for so...

Triangular approximations of fuzzy number with value and ambiguity functions

Statistical Significance of the Maximum Hardness Principle Applied to Some Selected Chemical Reactions.

The validity of the maximum hardness principle (MHP) is tested in the cases of 50 chemical reactions, most of which are organic in nature and exhibit anomeric effect. To explore the effect of the level of theory on the validity of MHP in an exothermic reaction, B3LYP/6-311++G(2df,3pd) and LC-BLYP/6-311++G(2df,3pd) (def2-QZVP for iodine and mercury) levels are employed. Different approximations ...

Communication Complexity of Combinatorial Auctions with Submodular Valuations

We prove the first communication complexity lower bound for constant-factor approximation of the submodular welfare problem. More precisely, we show that a (1− 1 2e+ )-approximation (' 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NP-hardness of (1− 1 2e+ )approximation in a computational model where ea...