Discordant Compact Logic-Arithmetic Structures in Discrete Optimization Problems (1309.6078v1)

Abstract: In sphere of research of discrete optimization algorithms efficiency the important place occupies a method of polynomial reducibility of some problems to others with use of special purpose components. In this paper a novel method of compact representation for sets of binary sequences in the form of "compact triplets structures" (CTS) and "compact couples structures" (CCS) is stated, supposing both logic and arithmetic interpretation of data. It is shown that any non-empty CTS in dual interpretation represents some unique Boolean formula in 3-CNF and the tabular CTS contains all satisfyig sets of the formula as concatenations of the triplets chosen from the neighbouring tiers. In general, any 3-CNF formula is transformed by decomposition to a system of discordant CTS's, each being associated with an individual permutation of variables constructed by a polynomial algorithm. As a result the problem of the formula satisfiability is reduced to the following one: ascertain the fact of existence (or absence) of a "joint satisfying set" (JSS) for all discordant structures, based on the different permutations. Further transformation of each CTS to CCS is used; correctness of preservation of the allowed sets is reached by simple algorithmic restrictions on triplets concatenation. Then the procedure of "inverting of the same name columns" in the various structures is entered for the purpose of reducing the problem of JSS revealing to elementary detection of n-tuples of zeros in the CCS system. The formula is synthesized, being on the structure a variation of 2-CNF, associated with the calculation procedure realizing adaptation of the polynomial algorithm of constraints distribution (well-known in the optimization theory) to the efficient resolving Boolean formula coded by means of discordant compact structures.