A. E. Baburin

The problem of finding a subset of vectors with the maximum total weight

The NP-hardness is proved for the discrete optimization problems connected with maximizing the total weight of a subset of a finite set of vectors in Euclidean space, i.e., the Euclidean norm of the sum of the members. Two approximation algorithms are suggested, and the bounds for the relative error and time complexity are obtained. We give a polynomial approximation scheme in the case of a fixed dimension and distinguished a subclass of problems solvable in pseudopolynomial time. The results obtained are useful for solving the problem of choice of a fixed number of subsequences from a numerical sequence with a given number of quasiperiodical repetitions of the subsequence.

A 3/4-approximation algorithm for finding two disjoint Hamiltonian cycles of maximum weight

We study the problem in which, given a complete undirected edge-weighted graph, it is required to find two (edge) disjoint Hamiltonian cycles of maximum total weight. The problem is known to be NP-hard in the strong sense. We present a 3/4-approximation algorithm with the running time O(n3).

On the Asymptotic Optimality of an Algorithm for Solving the Maximum m-PSP in a Multidimensional Euclidean Space

An efficient algorithm A with a guaranteed error estimate is presented for solving the problem of finding several edge-disjoint Hamiltonian circuits (traveling salesman tours) of maximum weight in a complete weighted undirected graph in a multidimensional Euclidean space ℝk. The time complexity of the algorithm is O(n3). The number of traveling salesman tours for which the algorithm is asymptotically optimal is established.

APPROXIMATION ALGORITHMS FOR FINDING A MAXIMUM-WEIGHT SPANNING CONNECTED SUBGRAPH WITH GIVEN VERTEX DEGREES ⋆

In the paper a problem of finding a maximum-weight spanning connected subgraph with given vertex degrees is considered. The problem is MAX SNP-hard, because it is a generalization of a well-known Traveling Salesman Problem. Approximation algorithms are constructed for deterministic and random instances. Performance bounds of these algorithms are presented.

An Approximation Algorithm for Finding a d-Regular Spanning Connected Subgraph of Maximum Weight in a Complete Graph with Random Weights of Edges

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n2) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.

Certain Generalization of the Maximum Traveling Salesman Problem

The problem is considered of finding in a complete undirected weighted graph a connected spanning subgraph with the given degrees of the vertices having maximum total weight of the edges. An approximate polynomial algorithm is presented for this problem. The algorithm is analyzed, and some upper bounds of its error are established in the general case as well as in the metric and Euclidean cases.

Polynomial Algorithms for Some Hard Problems of Finding Connected Spanning Subgraphs of Extreme Total Edge Weight

Several hard optimization problems of finding spanning connected subgraphs with extreme total edge weight are considered. A number of results on constructing polynomial algorithms with performance guarantees for these problems is presented.1

Algorithms with Performance Guarantees for a Metric Problem of Finding Two Edge-Disjoint Hamiltonian Circuits of Minimum Total Weight

This paper aims at describing a metric problem of finding two minimum total weight edge-disjoint Hamiltonian circuits in a graph with two weight functions. The problem is NP-hard in strong sense if the weight functions w 1 and w 2 are different or equal. We construct two approximation 0(n 3) algorithms whose worstcase performance guarantees asymptotically equal 12/5 (in case of the different functions), and 9/4 (when w 1 and w 2 are equal).

Approximation algorithms for 2-Peripathetic Salesman Problem with edge weights 1 and 2

The invention concerns a machine for treating produce for peeling it, in particular a nut-husking machine. This machine comprises a tank (1) with a cylindrical wall, a rotary disk (10) dotted with a plurality of teeth (16) projecting above the disk by an adjustable height, means (4-9) for driving the disk into rotation, a sliding gate (24) to recover the processed produce, means for evacuating the wastes and liquid that include a passage (15 ) on the disk circumference, a ring (21) in this passage and a bottom (2) associated with a spout (3), and means distributing the liquid which include a distributor (12) fixed to the center of the disk (10) in order to eject the liquid at the disk in the centrifugal direction.

NP-hardness and approximation algorithms for solving Euclidean problem of finding a maximum total weight subset of vectors

A pick holder for a guitar is mounted in a cradle portion of a side wall of a guitar and contains a strip having a plurality of pick-receiving pockets. The pick holder has a first fastening portion affixed to the guitar body and a second fastening portion containing the pick-receiving pockets. The two fastening portions enable the strip of pick-receiving pockets to be detachably fastened to the guitar body.