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2 Foundations of Computing and Decision Sciences

2 Levin M. S.

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Common part of preference relations

Levin M. S.

Foundations of Computing and Decision Sciences

2003

Vol. 28, No. 4

223-246

The paper addresses a problem to build a common part for a set of preference relations (digraphs, posets) which are defined over the same set of vertices (e.g., alternatives). A special new contradiction is proposed for the vertices which are included into the common subgraph. A new problem formulation is: find the largest (by vertices and by arcs) common subgraph or its the ''best" approximation, while taking into account a contradiction. Thus the model is oriented to Pareto-effective solutions. The common subgraph can be considered as a structural measure for proximity of initial preference relations. The case of many initial digraphs allows various kinds of the aggregation for arc information. Our solving scheme is based on a combinatorial model (morphological clique problem). The following situations are described: (a) a general description for the two digraphs case and (b) the n digraph case. Numerical examples illustrate the problems and solving processes.

Digraph description of k-interchange technique for optimization over permutations and adaptive algorithm system

Levin M. S.

Foundations of Computing and Decision Sciences

2001

Vol. 26, No. 3

225-235

The paper describes a general glance to the use of element exchange techniques for optimization over permutations. A multi-level description of problems is proposed which is a fundamental to understand nature and complexity of optimization problems over permutations (e.g., ordering, scheduling, traveling salesman problem). The description is based on permutation neighborhoods of several kinds (e.g., by improvement of an objective function). Our proposed operational digraph and its kinds can be considered as a way to understand convexity and polynomial solvability for combinatorial optimization problems over permutations. Issues of an analysis of problems and a design of hierarchical heuristics are discussed. The discussion leads to a multi-level adaptive algorithm system which analyzes an individual problem and selects / designs a solving strategy (trajectory).