1 research outputs found
Uniform-Circuit and Logarithmic-Space Approximations of Refined Combinatorial Optimization Problems
A significant progress has been made in the past three decades over the study
of combinatorial NP optimization problems and their associated optimization and
approximate classes, such as NPO, PO, APX (or APXP), and PTAS. Unfortunately, a
collection of problems that are simply placed inside the P-solvable
optimization class PO never have been studiously analyzed regarding their exact
computational complexity. To improve this situation, the existing framework
based on polynomial-time computability needs to be expanded and further refined
for an insightful analysis of various approximation algorithms targeting
optimization problems within PO. In particular, we deal with those problems
characterized in terms of logarithmic-space computations and uniform-circuit
computations. We are focused on nondeterministic logarithmic-space (NL)
optimization problems or NPO problems. Our study covers a wide range of
optimization and approximation classes, dubbed as, NLO, LO, APXL, and LSAS as
well as new classes NC1O, APXNC1, NC1AS, and AC0O, which are founded on uniform
families of Boolean circuits. Although many NL decision problems can be
naturally converted into NL optimization (NLO) problems, few NLO problems have
been studied vigorously. We thus provide a number of new NLO problems falling
into those low-complexity classes. With the help of NC1 or AC0
approximation-preserving reductions, we also identify the most difficult
problems (known as complete problems) inside those classes. Finally, we
demonstrate a number of collapses and separations among those refined
optimization and approximation classes with or without unproven
complexity-theoretical assumptions.Comment: (37 pages, A4, 10pt, 1 figure) This is a complete version of a
preliminary report, which appeared in the Proceedings of the 7th
International Conference on Combinatorial Optimization and Applications
(COCOA 2013), Chengdu, China, December 12--14, 2013, Lecture Notes in
Computer Science, Springer-Verlag, vol.8287, pp.318--329, 201