Edward Gimadi

Efficient Randomized Algorithm for a Vector Subset Problem

We introduce a randomized approximation algorithm for NP-hard problem of finding a subset of m vectors chosen from n given vectors in multidimensional Euclidean space \(\mathbb {R}^k\) such that the norm of the corresponding sum-vector is maximum. We derive the relation between algorithm’s time complexity, relative error and failure probability parameters. We show that the algorithm implements Polynomial-time Randomized Approximation Scheme (PRAS) for the general case of the problem. Choosing particular parameters of the algorithm one can obtain asymptotically exact algorithm with significantly lower time complexity compared to known exact algorithm. Another set of parameters provides polynomial-time 1 / 2-approximation algorithm for the problem. We also show that the algorithm is applicable for the related (minimization) clustering problem allowing to obtain better performance guarantees than existing algorithms.

Multi-index assignment problem: an asymptotically optimal approach

The multi-index axial assignment problem is considered. It is known to be NP-hard in general case. In the paper a sublinear approximation algorithm is proposed. Performance and probability-of-failure bounds of the algorithm are established, and conditions of asymptotical optimality of the algorithm described are proved in the case of matrices with random elements distributed independently with a common distribution function which is minorised by the uniform distribution.

On Asymptotically Optimal Approach to the m-Peripatetic Salesman Problem on Random Inputs

We study the m-Peripatetic Salesman Problem on random inputs. In earlier papers we proposed a polynomial asymptotically optimal algorithm for the m-PSP with different weight functions on random inputs. The probabilistic analysis carried out for that algorithm is not suitable in the case of the m-PSP with identical weight functions.

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).

On Vector Summation Problem in the Euclidean Space

We consider a problem of finding a subset of the smallest size in the given set of vectors such that the norm of sum vector is greater or equal to some given value. We show that the problem can be solved optimally with the same complexity as the problem of finding the subset of given cardinality with minimum norm of sum vector.

Approximation Algorithms for the Maximum m-Peripatetic Salesman Problem

We consider the maximum m-Peripatetic Salesman Problem (MAX m-PSP), which is a natural generalization of the classic Traveling Salesman Problem. The problem is strongly NP-hard. In this paper we propose two polynomial approximation algorithms for the MAX m-PSP with different and identical weight functions, correspondingly. We prove that for random inputs uniformly distributed on the interval [a, b] these algorithms are asymptotically optimal for \(m=o(n)\). This means that with high probability their relative errors tend to zero as the number n of the vertices of the graph tends to infinity. The results remain true for the distributions of inputs that minorize the uniform distribution.

An Exact Polynomial Algorithm for the Outerplanar Facility Location Problem with Improved Time Complexity

The Unbounded Facility Location Problem on outerplanar graphs is considered. The algorithm with time complexity \( O(n m^3)\) was known for solving this problem, where \( n\) is the number of vertices, \( m\) is the number of possible plant locations. Using some properties of maximal outerplanar graphs (binary 2-trees) and the existence of an optimal solution with a family of centrally-connected service areas, the recurrence relations are obtained allowing to design an algorithm which can solve the problem in \( O(n m^{2.5})\) time.

The TSP-approach to approximate solving the m-Cycles Cover Problem

In the m-Cycles Cover problem it is required to find a collection of m vertex-disjoint cycles that covers all vertices of the graph and the total weight of edges in the cover is minimum (or maximum). The problem is a generalization of the Traveling salesmen problem. It is strongly NP-hard. We discuss a TSP-approach that gives polynomial approximate solutions for this problem. It transforms an approximation TSP algorithm into an approximation m-CCP algorithm. In this paper we present a number of successful transformations with proven performance guarantees for the obtained solutions.

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.