Vladimir G. Deineko

A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem

Abstract.This paper deals with exponential neighborhoods for combinatorial optimization problems. Exponential neighborhoods are large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods.¶First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NP-complete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved in polynomial time, the corresponding situation for the QAP looks pretty hopeless: Every exponential neighborhood that is considered in this paper provably leads to an NP-complete optimization problem for the QAP.

Hardness of approximation of the discrete time-cost tradeoff problem

We consider the discrete version of the well-known time-cost tradeoff problem for project networks, which has been extensively studied in the project management literature. We prove a strong in-approximability result with respect to polynomial time bicriteria approximation algorithms for this problem.

The approximability of MAX CSP with fixed-value constraints

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. In this article, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x = a), is either solvable exactly in polynomial time or else is APX-complete, even if the number of occurrences of variables in instances is bounded. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description relies on the well-known algebraic combinatorial property of supermodularity.

The convex-hull-and-line traveling salesman problem: a solvable case

Abstract We solve the special case of the Euclidean Traveling Salesman Problem where n − m cities lie on the boundary of the convex hull of all n cities, and the other m cities lie on a line segment inside this convex hull by an algorithm which needs O( mn ) time and O( n ) space.

Exact algorithms for the Hamiltonian cycle problem in planar graphs

We construct an exact algorithm for the Hamiltonian cycle problem in planar graphs with worst case time complexity O(c^n), where c is some fixed constant that does not depend on the instance. Furthermore, we show that under the exponential time hypothesis, the time complexity cannot be improved to O(c^o^(^n^)).

On the robust assignment problem under a fixed number of cost scenarios

We investigate the complexity of the min-max assignment problem under a fixed number of scenarios. We prove that this problem is polynomial-time equivalent to the exact perfect matching problem in bipartite graphs, an infamous combinatorial optimization problem of unknown computational complexity.

Polynomially solvable cases of the traveling salesman problem and a new exponential neighborhood

LetA=(aij) be the distance matrix of an arbitrary (asymmetric) traveling salesman problem and let τ=τ1τ2...τm be the optimal solution of the corresponding assignment problem with the subtours τ=τ1τ2...τm. By choosing (m−1) transpositions (k, l) withk ∈ τi−1,l ∈ τi (i=2, ...,m) and patching the subtours by using these transpositions in any order, we get a set of cyclic permutations. It will be shown that within this set of cyclic permutations a tour with minimum distance can be found by O(n2|τ|* operations, where |τ|* is the maximum number of nodes in a subtour of τ. Moreover, applying this result to the case whenA=(aij) is a permuted distribution matrix (Monge-matrix) and thepatching graph Gτ is a multipath, a result of Gaikov can be improved: By combining the above theory with a result of Park alinear algorithm for finding an optimal TSP solution can be derived, provided τ is already known.ZusammenfassungEs sieA=(aij) die Entfernungsmatrix eines (asymmetrischen) Rundreiseproblems und τ=τ1τ2...τm die Optimallösung des zugehörigen Zuordnungsproblems mit den Teilzyklen τ=τ1τ2...τm. Wählt man (m−1) Transpositionen (k, l) mitk ∈ τi−1,l ∈ τi (i=2, ...,m) und verknüpft man die Teilzyklen unter Zuhilfenahme dieser Transpositionen in beliebiger Ordnung, so erhält man eine Menge zyklischer Permutationen. Es wird gezeigt, daß man die kürzeste Tour in dieser Menge von Rundreisen mit einem Rechenaufwand von O(n2|τ|* Operationen bestimmen kann, wobei |τ|* die maximale Anzahl von Städten in einem Teilzyklus von τ ist. Im Falle, daß die EntfernungsmatrixA=(aij) eine permutierte Verteilungsmatrix (Monge-Matrix) und der VerknüpfungsgraphGτ ein mehrfacher Weg ist, kann ein Resultat von Gaikov verbessert werden. In Verbindung mit einem Resultat von Park führt die oben entwickelte Theorie in diesem Fall zu einemlinearen Verfahren zur Bestimmung einer optimalen Rundreise.

A new family of scientific impact measures: The generalized Kosmulski-indices

This article introduces the generalized Kosmulski-indices as a new family of scientific impact measures for ranking the output of scientific researchers. As special cases, this family contains the well-known Hirsch-index h and the Kosmulski-index h(2). The main contribution is an axiomatic characterization that characterizes every generalized Kosmulski-index in terms of three axioms.

Four point conditions and exponential neighborhoods for symmetric TSP

In most of the known polynomially solvable cases of the symmetric travelling salesman problem (TSP) which result from restrictions on the underlying distance matrices, the restrictions have the form of so-called four-point conditions (the inequalities involve four cities). In this paper we treat all possible (symmetric) four-point conditions and investigate whether the corresponding TSP can be solved in polynomial time. As a by-product of our classification we obtain new families of exponential neighborhoods for the TSP which can be searched in polynomial time and for which conditions on the distance matrix can be formulated so that the search for an optimal TSP solution can be restricted to these exponential neighborhoods.

Another well-solvable case of the QAP: Maximizing the job completion time variance

We analyze a special case of the maximum quadratic assignment problem where one matrix is a monotone anti-Monge matrix and the other matrix has a multi-layered structure that is built on top of certain Toeplitz matrices. To demonstrate an application of our main result, we derive a (simple and concise) alternative proof for a recent result on the scheduling problem of maximizing the variance of job completion times.