Anders Yeo

Traveling Salesman Should not be Greedy: Domination Analysis of Greedy-Type Heuristics for the TSP

Abstract Computational experiments show that the greedy algorithm (GR) and the nearest neighbor algorithm (NN), popular choices for tour construction heuristics, work at acceptable level for the Euclidean TSP, but produce very poor results for the general Symmetric and Asymmetric TSP (STSP and ATSP). We prove that for every n⩾2 there is an instance of ATSP (STSP) on n vertices for which GR finds the worst tour. The same result holds for NN. We also analyze the repetitive NN (RNN) that starts NN from every vertex and chooses the best tour obtained. We prove that, for the ATSP, RNN always produces a tour, which is not worse than at least n/2−1 other tours, but for some instance it finds a tour, which is not worse than at most n−2 other tours, n⩾4. We also show that, for some instance of the STSP on n⩾4 vertices, RNN produces a tour not worse than at most 2n−3 tours. These results are in sharp contrast to earlier results by Gutin and Yeo, and Punnen and Kabadi, who proved that, for the ATSP, there are tour construction heuristics, including some popular ones, that always build a tour not worse than at least (n−2)! tours.

The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

Kernel Bounds for Disjoint Cycles and Disjoint Paths

In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic assumptions. We build upon recent results by Bodlaender et al. [3] and Fortnow and Santhanam [13], that show that NP-complete problems that are or-compositional do not have polynomial kernels, unless NP ⊆ coNP/poly. To this machinery, we add a notion of transformation, and thus obtain that Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(k logk).

Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

When the greedy algorithm fails

We provide a characterization of the cases when the greedy algorithm may produce the unique worst possible solution for the problem of finding a minimum weight base in an independence system when the weights are taken from a finite range. We apply this theorem to TSP and the minimum bisection problem. The practical message of this paper is that the greedy algorithm should be used with great care, since for many optimization problems its usage seems impractical even for generating a starting solution (that will be improved by a local search or another heuristic).

Construction heuristics for the asymmetric TSP

Abstract Non-Euclidean traveling salesman problem (TSP) construction heuristics, and especially asymmetric TSP construction heuristics, have been neglected in the literature by comparison with the extensive efforts devoted to studying Euclidean TSP construction heuristics. This state of affairs is at odds with the fact that asymmetric models are relevant to a wider range of applications, and indeed are uniformly more general that symmetric models. Moreover, common construction approaches for the Euclidean TSP have been shown to produce poor quality solutions for non-Euclidean instances. Motivation for remedying this gap in the study of construction approaches is increased by the fact that such methods are a great deal faster than other TSP heuristics, which can be important for real time problems requiring continuously updated response. The purpose of this paper is to describe two new construction heuristics for the asymmetric TSP and a third heuristic based on combining the other two. Extensive computational experiments are performed for several different families of TSP instances, disclosing that our combined heuristic clearly outperforms well-known TSP construction methods and proves significantly more robust in obtaining (relatively) high quality solutions over a wide range of problems.

Level of repair analysis and minimum cost homomorphisms of graphs

Level of repair analysis (LORA) is a prescribed procedure for defense logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros [The optimisation of repair decisions using life-cycle cost parameters. IMA J. Management Math. 9 (1998) 403-413] and Barros and Riley [A combinatorial approach to level of repair analysis, European J. Oper. Res. 129 (2001) 242-251] developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.

Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number

Abstract Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n !/ p ( n ) tours for some polynomial p ( n ) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadratic assignment problem, is also considered with respect to the analogous question. Probabilistic, graph-theoretical, group-theoretical and number-theoretical methods and results are used.

On the Parameterized Complexity and Kernelization of the Workflow Satisfiability Problem

A workflow specification defines a set of steps and the order in which these steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of these steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications and for the construction of runtime reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li [2010] using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this article, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints. Finally, we consider preprocessing for the problem and prove that in an important special case, in polynomial time, we can reduce the given input into an equivalent one where the number of users is at most the number of steps. We also show that no such reduction exists for two natural extensions of this case, which bounds the number of users by a polynomial in the number of steps, provided a widely accepted complexity-theoretical assumption holds.

A probabilistic approach to problems parameterized above or below tight bounds

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds or below tight upper bounds. To illustrate the approach we consider two problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar [M. Mahajan, V. Raman, S. Sikdar, Parameterizing above or below guaranteed values, J. Comput. System Sci. 75 (2) (2009) 137-153]. We show that a generalization of one of the problems and three non-trivial special cases of the other problem admit kernels of quadratic size. As a byproduct we obtain a new probabilistic inequality that could be of independent interest. Our new inequality is dual to the Hypercontractive Inequality.