VNS and second order heuristics for the min-degree constrained minimum spanning tree problem

Given an undirected graph with weights associated with its edges, the min-degree constrained minimum spanning tree (md$\mathit{md}$-MST) problem consists in finding a minimum spanning tree of the given graph, imposing minimum degree constraints in all nodes except the leaves. This problem was recently proposed in Almeida et al. [Min-degree constrained minimum spanning tree problem: Complexity, proprieties and formulations. Operations Research Center, University of Lisbon, Working-paper no. 6; 2006], where its theoretical complexity was characterized and showed to be NP$\mathit{NP}$-hard.     

A hybrid algorithm for the single-machine total tardiness problem

We propose a hybrid algorithm based on the ant colony optimization (ACO) meta-heuristic, in conjunction with four well-known elimination rules, to tackle the NP$\mathit{NP}$-hard single-machine scheduling problem to minimize the total job tardiness. The hybrid algorithm has the same running time as that of ACO. We conducted extensive computational experiments to test the performance of the hybrid algorithm and ACO. The computational results show that the hybrid algorithm can produce optimal or near-optimal solutions quickly, and its performance compares favourably with that of ACO for handling standard instances of the problem.     

On the connectivity preserving minimum cut problem

In this paper, we study a generalization of the classical minimum cut problem, called Connectivity Preserving Minimum Cut (CPMC) problem, which seeks a minimum cut to separate a pair (or pairs) of source and destination nodes and meanwhile ensure the connectivity between the source and its partner node(s). For this problem, we consider two variants, connectivity preserving minimum node cut (CPMNC) and connectivity preserving minimum edge cut (CPMEC). For CPMNC, we show that it cannot be approximated within α logn for some constant α   unless P=NP$P=\mathit{NP}$, and cannot be approximated within any poly(logn)$\mathit{poly}(\mathit{logn})$ unless NP has quasi-polynomial time algorithms. The hardness results hold even for graphs with unit weight and bipartite graphs. For CPMEC, we show that it is in P for planar graphs.     

Single-machine scheduling with periodic maintenance to minimize makespan

We consider a single-machine scheduling problem with periodic maintenance activities. Although the scheduling problem with maintenance has attracted researchers’ attention, most of past studies considered only one maintenance period. In this research several maintenance periods are considered where each maintenance activity is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the makespan, subject to periodic maintenance and nonresumable jobs. We first prove that the worst-case ratio of the classical LPT   algorithm is 2. Then we show that there is no polynomial time approximation algorithm with a worst-case ratio less than 2 unless P=NP$P=\text{<italic>NP</italic>}$, which implies that the LPT algorithm is the best possible.     

On a Refined Analysis of Some Problems in Interval Arithmetic Using Real Number Complexity Theory

We study some problems in interval arithmetic treated in Kreinovich et al. [13]. First, we consider the best linear approximation of a quadratic interval function. Whereas this problem (as decision problem) is known to be NP-hard in the Turing model, we analyze its complexity in the real number model and the analogous class NP . Our results substantiate that most likely it does not any longer capture the difficulty of NP in such a real number setting. More precisely, we give upper complexity bounds for the approximation problem for interval functions by locating it in (a real analogue of). This result allows several conclusions: the problem is not (any more) NP -hard under so called weak polynomial time reductions and likely not to be NP -hard under (full) polynomial time reductions; for fixed dimension the problem is polynomial time solvable; this extends the results in Koshelev et al. [12] and answers a question left open in [13].We also study several versions of interval linear systems and show similar results as for the approximation problem.Our methods combine structural complexity theory with issues from semi-infinite optimization theory.