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On the average case approximability of some NP optimization problems
Abstract
The main result in this dissertation is the following: We reformulate the definition of NP optimization problems, and then from this reformulation we define syntactically a subclass of NP optimization problems, whose decision versions contains NP complete problems. We prove that in the average case, each problem in this subclass is asymptotically approximable up to any given constant factor $\epsilon>0$ by a finite state machine. Precisely, let $\Pi$ be a problem in the subclass of NP optimization problems in our subclass. Suppose that the input I of $\Pi$ is a random sequence over all inputs according to a given probability measure $\mu$. Let n be the length of I. Then for any $\epsilon>0$, there exists a finite state machine which does the following: For input I this finite state machine produces a feasible solution whose objective value A(I) satisfies$$Pr \left({{\vert Q\sb{\sc{OPT}}(I) - A(I)\vert}\over{\max\{Q\sb{\sc{OPT}}(I), A(I)\}}}\ge\epsilon\right)\le K\exp(-hn),$$where $K>0$ is a universal constant and $h>0$ is a constant depending on $\Pi$ and $\epsilon$.
Subject Area
Computer science|Mathematics
Recommended Citation
Hong, Dawei, "On the average case approximability of some NP optimization problems" (1996). ETD collection for University of Nebraska-Lincoln. AAI9700089.
https://digitalcommons.unl.edu/dissertations/AAI9700089