Multicriteria combinatorial optimization problems on a set of polypermutations

The studies on multicriteria combinatorial optimization are continued. A possible approach to solving multicriterion problems is developed and substantiated. An algorithm is developed and implemented. Some peculiarities of efficient solutions to multicriterion problems are described. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 152–160, March–April 2008.     

A nonlinear compromise scheme in multicriteria evaluation and optimization problems

The concept of a nonlinear compromise scheme in multicriteria problems of evaluation and optimization is presented. It is shown that the problem is to approximate correctly the utility function and construct a substantial mathematical model (scalar convolution) adequate to the given situation to solve various multicriteria problems. In analysis problems, this convolution is an objective function. Its extremization results in a compromise-optimal vector of arguments. An illustrative example is given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 106–114, July–August 2009.     

Combinatorial optimization under uncertainty

In this paper, the general problem of Euclidean combinatorial optimization under uncertainty is formulated for the first time and the concepts of a stochastic multiset, a multiset of fuzzy numbers, a stochastic Euclidean combinatorial set, and general Euclidean combinatorial set of fuzzy stochastic numbers that combines the properties of both types of uncertainty are introduced. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 35–44, September–October 2008.     

Analysis of mathematical models and methods of solving combinatorial optimization problems on game-type permutations

The paper is concerned with an optimization problem on game-type permutations, where one or both players have combinatorial constraints on their strategies. A mathematical model of such problems is constructed and analyzed. A modified graphical method is proposed to solve (2xn)-and (mx2)-dimensional problems. High-dimensional problems are reduced to linear programming and combinatorial optimization problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 103–114, November–December 2007.     

Subclasses of solvable problems from classes of combinatorial optimization problems

Well-known subclasses of solvable problems from classes of combinatorial optimization are reviewed. For solvable problems such as the traveling salesman problem, location problem, assignment problem, and clustering problem, the changes in the objective function on a given ordering of combinatorial configurations are analyzed. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 97–105, March–April 2009.     

Four-layer framework for combinatorial optimization problems domain

Four-layer framework for combinatorial optimization problems/models domain is suggested for applied problems structuring and solving: (1) basic combinatorial models and multicriteria decision making problems (e.g., clustering, knapsack problem, multiple choice problem, multicriteria ranking, assignment/allocation); (2) composite models/procedures (e.g., multicriteria combinatorial problems, morphological clique problem); (3) basic (standard) solving frameworks, e.g.: (i) Hierarchical Morphological Multicriteria Design (HMMD) (ranking, combinatorial synthesis based on morphological clique problem), (ii) multi-stage design (two-level HMMD), (iii) special multi-stage composite framework (clustering, assignment/location, multiple choice problem); and (4) domain-oriented solving frameworks, e.g.: (a) design of modular software, (b) design of test inputs for multi-function system testing, (c) combinatorial planning of medical treatment, (d) design and improvement of communication network topology, (e) multi-stage framework for information retrieval, (f) combinatorial evolution and forecasting of software, devices. The multi-layer approach covers ‘decision cycle’, i.e., problem statement, models, algorithms/procedures, solving schemes, decisions, decision analysis and improvement.     

Method of ordering the values of a linear function on a set of permutations

The paper deals with a new method of solving a combinatorial problem with account for the properties of the set of permutations and its structure. Using this method, the values of the linear objective function are sequenced and the set of permutations is decomposed over hyperplanes, with account of element recurrences. This makes it possible to develop an algorithm of finding the point (an element of the set of permutations) at which the objective function attains a given value. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 50–61, March–April 2009.     

Finding Total Unimodularity in Optimization Problems Solved by Linear Programs

A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with the same cost from the fractional solution. Examples are two scheduling problems (Baptiste and Schieber, J. Sched. 6(4):395–404, 2003; Brucker and Kravchenko, J. Sched. 11(4):229–237, 2008) and the single disk prefetching/caching problem (Albers et al., J. ACM 47:969–986, 2000). We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.     

Algorithmic soluyion of two parametric optimization problems on a set of complete combinations

Algorithmic solutions of parametric problems of two types with a parameter in the objective function and with a parameter in the system of constraints are considered in a Euclidean combinatorial set of combinations with repetitions. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 160–165, November–December, 1999.     

Attainability of optimal solutions to a linear problem of multicriterion optimization with respect to the weighed sum of variously important and transitively subordinated criteria

This paper presents methods of finding criteria coefficients such that optimal solutions are obtained to a linear problem of multicriteria optimization with respect to the weighed sum of variously important and transitively subordinated criteria. The case of a partial transitive subordination is also considered, and a method is founded that finds coefficients such that optimal solutions to the problem of multicriteria optimization are attainable with respect to the weighed sum of variously important and partially transitively subordinated criteria. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 135–138, September–October 2008.     

Vector optimization problems with linear criteria over a fuzzy combinatorial set of alternatives

Vector optimization problems over a fuzzy combinatorial set of permutations are investigated. Based on the properties of the convex hull of a fuzzy combinatorial set of permutations, modifications of multicriteria choice methods are developed and substantiated for a fuzzy feasible combinatorial set. Mathematical models of some application problems are presented.     

Remarks on stability of solutions of the property domination problem

A class of combinatorial problems is considered whose investigation and solution require the notions of the theory of fuzzy sets. The necessary and sufficient conditions of stability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 36–40, July–August, 1999.     

Approximation algorithms for combinatorial multicriteria optimization problems

The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and # P -completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well-known tree and Christofides' heuristics for the traveling salesman problem is investigated in the multicriteria case with respect to the two definitions of approximability.     

New means of cybernetics,informatics, computer engineering,and systems analysis

The class of combinatorial problems equivalent to the problem of determination of relative positions of n interval sequences is formulated. It is shown that an adequate mathematical model of solving a stated problem is a finite dynamic memoryless automaton and that the adequate mathematical apparatus is continuous logic. Algorithms that solve the problem are constructed. Misc  Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 173–181, May-June 2009. Original article submitted July 4, 2007.     

Multicriteria Global Minimum Cuts

We consider two multicriteria versions of the global minimum cut problem in undirected graphs. In the k-criteria setting, each edge of the input graph has k non-negative costs associated with it. These costs are measured in separate, non-interchangeable, units. In the AND-version of the problem, purchasing an edge requires the payment of all the k costs associated with it. In the OR-version, an edge can be purchased by paying any one of the k costs associated with it. Given k bounds b₁,b₂,. . . ,b_k, the basic multicriteria decision problem is whether there exists a cut C of the graph that can be purchased using a budget of b_i units of the ith criterion, for 1 ≤ i ≤ k. We show that the AND-version of the multicriteria global minimum cut problem is polynomial for any fixed number k of criteria. The OR-version of the problem, on the other hand, is NP-hard even for k = 2, but can be solved in pseudo-polynomial time for any fixed number k of criteria. It also admits an FPTAS. Further extensions, some applications, and multicriteria versions of two other optimization problems are also discussed.     

Using computational learning strategies as a tool for combinatorial optimization

In this paper, we describe how a basic strategy from computational learning theory can be used to attack a class of NP‐hard combinatorial optimization problems. It turns out that the learning strategy can be used as an iterative booster: given a solution to the combinatorial problem, we will start an efficient simulation of a learning algorithm which has a “good chance” to output an improved solution. This boosting technique is a new and surprisingly simple application of an existing learning strategy. It yields a novel heuristic approach to attack NP‐hard optimization problems. It does not apply to each combinatorial problem, but we are able to exactly formalize some sufficient conditions. The new technique applies, for instance, to the problems of minimizing a deterministic finite automaton relative to a given domain, the analogous problem for ordered binary decision diagrams, and to graph coloring. This revised version was published online in June 2006 with corrections to the Cover Date.     

Description of some classes of equiprobable Boolean functions with no zero Fourier coefficients

Theory of groups and methods of combinatorial analysis are used to obtain some classes of equiprobable Boolean functions with no zero Fourier coefficients. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 95–111, November–December 2006.     

Combinatorial properties of noninteger vertices of a polytope in a three-index axial assignment problem

It is proved that, for any r ∈ { 2n, 2n + 1,…, 3n−2} and only for such r, the polytope of a three-index axial assignment problem of order n, n ≥ 2, contains completely r-noninteger vertices (r-CNVs), i.e., vertices such that all their positive components are fractional and their number equals r. For each r ∈ {2n, 2n + 1,…, 3n −2}, all the types of r-CNVs are characterized and the combinatorial properties of completely r-noninteger vertices of the polytope are studied. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 33–44, January–February 2007.     

Antibody-based computing: an application to the stable marriage problem

Antibodies, among other things, are important components of the immune system. This paper proposes using the specific recognition capability exhibited by antibodies for computation, in particular, for solving the stable marriage problem, which has been studied as a combinatorial computational problem. Antibody-based computation is proposed by integrating the recognition capabilities of antibodies. The computation is carried out on an array form that is suitable not only for expressing stable marriage problems, but also for further integration to antibody microarrays. This work was presented in part at the 12th International Symposium on Artificial Life and Robotics, Oita, Japan, January 25–27, 2007     

Solving a continuous nonlinear problem of optimal set partition with arrangement of subset centers in the case of a convex objective functional

A continuous nonlinear single-commodity problem of optimal partition of a set Ω in an n-measurable Euclidean space into disjoint subsets with arrangement of their centers is analyzed using equality and inequality constraints in the case of a convex objective functional. A method and algorithm are proposed to solve this problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 134–152, March–April 2008.