M. Yu. Khachai

Computational and approximational complexity of combinatorial problems related to the committee polyhedral separability of finite sets

AbstractIn [1, 2], results on the computational and approximational complexity of a minimum affine separating committee (MASC) problem were obtained for finite sets A, B ⊂ ℚn . In particular, it was shown that this problem is NP-hard and does not belong to the class Apx (under the assumption that P ≠ NP). Nevertheless, questions concerning the bounds for its effective approximability threshold and for the computational complexity of a number of practically important particular cases of the problem obtained by imposing additional constraints, for example, by fixing the dimension of the space, remained open. In this paper, a lower bound is presented for the polynomial approximability threshold of the problem in the general case, and the intractability of the problem in spaces of fixed dimension greater than unity is proved. In particular, it is shown that the problem of committee separability remains hard even when it is formulated on the plane (i.e., in the simplest non-trivial case). This result follows from the fact that the well-known PC problem on covering a finite planar set by straight lines, whose hardness was proved in [3], is polynomially reducible to the problem under consideration. The method of reduction represents a modification of the method that was described in [4] and was used there for proving the hardness of problems on piecewise linear separability of finite sets on the plane.

Committees of Systems of Linear Inequalities

The conceptual issues of the theory of committee decision rules were considered, and its close relationship with the theory of substantiation of collective decision making and teaching neural networks was demonstrated. The problem of the minimum committee of an incompatible system of constraints, which arises at the stage of constructing the committee decision rule with a small number of elements, was discussed by itself. The problem of the minimum committee is known to be NP-hard. Results concerning estimation of the computational complexity of allied problems were obtained. To solve the problem of minimum committee of an incompatible system of linear inequalities, an effective approximate algorithm was suggested, its correctness was proved and computational complexity and guaranteed precision were estimated.

A polynomial-time approximation scheme for the Euclidean problem on a cycle cover of a graph

We study the minimum-weight k-size cycle cover problem (Min-k-SCCP) of finding a partition of a complete weighted digraph into k vertex-disjoint cycles of minimum total weight. This problem is a natural generalization of the known traveling salesman problem (TSP) and has a number of applications in operations research and data analysis. We show that the problem is strongly NP-hard in the general case and preserves intractability even in the geometric statement. For the metric subclass of the problem, a 2-approximation algorithm is proposed. For the Euclidean Min-2-SCCP, a polynomial-time approximation scheme based on Arora’s approach is built.

2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

The problem of finding a minimum clique (with respect to the total weight of its vertices and edges) of fixed size in a complete undirected weighted graph is considered along with some of its important subclasses. Approximability issues are analyzed. The inapproximability of the problem is proved for the general case. A 2-approximation efficient algorithm with time complexity O(n2) is suggested for the cases when vertex weights are nonnegative and edge weights either satisfy the triangle inequality or are squared pairwise distances for some point configuration of Euclidean space.

Parallel computations and committee constructions

The paper reviewed the results bearing out the deep-seated relation between the parallel computations and learning procedures for the laminated neural networks one of whose formalizations is represented by the theory of committee constructions. Additionally, consideration was given to two combinatorial problems concerned with learning pattern recognition in the class of affine committees—the problem of verifying existence of a three-element affine separating committee and that of element-minimal affine separating committee. The first problem was shown to be N P-complete, whereas the second problem is N P-hard and does not belong to the Apx class.

Computational Complexity of the Minimum Committee Problem and Related Problems

is called a committee solution with q elements of system (1) (or a committee) [1]. Minimum committee (MC) problem. Given a set X and subsets D 1 , D 2 , ..., D m ≠ , find a committee solution to system (1) with the least possible q (or show that the system has no committee solutions). Following [2], it is convenient to restate the MC problem in terms of integer linear programming. Let J 1 , J 2 , ..., J T be the index set of all maximal (under inclusion) consistent subsystems (MCSs) of system (1). Obviously, the system is consistent if and only if T = 1; otherwise, 1 < T < 2 m . Define two m × T incidence matrices A and B according to the rule

An exact algorithm with linear complexity for a problem of visiting megalopolises

A problem of visiting megalopolises with a fixed number of “entrances” and precedence relations defined in a special way is studied. The problem is a natural generalization of the classical traveling salesman problem. For finding an optimal solution, we give a dynamic programming scheme, which is equivalent to a method of finding a shortest path in an appropriate acyclic oriented weighted graph. We justify conditions under which the complexity of the algorithm depends on the number of megalopolises polynomially, in particular, linearly.

The problem of fingerprint identification: A reference database indexing method based on Delaunay triangulation

The class of problems of biometrical identification using fingerprints taken by an optical scanner is studied. A new approach to developing highly efficient identification algorithms based on preliminary indexing of the reference fingerprint database using Delaunay triangulation is proposed. Analysis of numerical experiments proved the proposed method to have high generalization ability and be stable to small perturbations of the original fingerprints.

Computational complexity of recognition learning procedures in the class of piecewise-linear committee decision rules

Two new results concerned with the computational and approximation complexities of the combinatorial optimization problems arising at learning pattern recognition in the class of committee piecewise-linear decision rule were discussed.

Efficient Algorithms with Performance Guarantees for Some Problems of Finding Several Cliques in a Complete Undirected Weighted Graph

We consider the problem of finding a fixed number of vertex-disjoint cliques of given sizes in a complete undirected weighted graph so that the total weight of vertices and edges in the cliques would be minimal. We show that the problem is strongly NP-hard both in the general case and in two subclasses, which have important applications. An approximation algorithm for this problem is presented. We show that the algorithm finds a solution with a bounded approximation ratio for the considered subclasses of the problem, and the bound is attainable. In the case when the number of cliques to be found is fixed in advance (i.e., is a parameter), the time complexity of the algorithm is polynomial.