Branch-and-cut-and-price algorithms for the Degree Constrained Minimum Spanning Tree Problem

Bicalho, Luis Henrique; Cunha, Alexandre Salles da; Lucena, Abílio

doi:10.1007/s10589-015-9788-7

Cited by 13 publications

(5 citation statements)

References 25 publications

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“…Instead of selecting all the violated cuts found, we only add to Γ ψ the most violated cut (if any) and the ones that are sufficiently orthogonal to it. As verified in several works (see, e.g., [37,38,39]), this strategy is able to balance the strength and diversity of the cuts separated, while limiting the model size. Here, this strategy is applied to select both GCCs and CCs, but separately (lines 7 and 8, Figure 7).…”

mentioning

confidence: 68%

“…Consider a (not necessarily minimal) cover C ⊆ P for the knapsack constraint (37) and the corresponding cover inequality (39). Moreover, let ω ∈ Z |P| be the coefficient vector of the y variables in (39), such that ω i = 1 for all i ∈ C, and ω i = 0 for all i ∈ P\C. Accordingly, (39) can be alternatively stated as…”

Section: Lcismentioning

confidence: 99%

“…For simplicity, we described the process of down-lifting by considering a classical cover inequality of type (39). However, this kind of lifting is usually applied only after some or all the variables identified in P\C have already been up-lifted.…”

Section: Lcismentioning

confidence: 99%

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A cutting-plane algorithm for the Steiner team orienteering problem

Assunção

Mateus

2019

Computers & Industrial Engineering

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The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximizing the total sum of rewards collected within the time limit, but, now, every mandatory location must be visited. In this work, we propose a new commoditybased formulation for STOP and use it within a cutting-plane scheme. The algorithm benefits from the compactness and strength of the proposed formulation and works by separating three families of valid inequalities, which consist of some general connectivity constraints, classical lifted cover inequalities based on dual bounds and a class of conflict cuts. To our knowledge, the last class of inequalities is also introduced in this work. A state-of-the-art branch-and-cut algorithm from the literature of TOP is adapted to STOP and used as baseline to evaluate the performance of the cutting-plane. Extensive computational experiments show the competitiveness of the new algorithm while solving several STOP and TOP instances. In particular, it is able to solve, in total, 14 more TOP instances than any other previous exact algorithm and finds eight new optimality certificates. With respect to the new STOP instances introduced in this work, our algorithm solves 30 more instances than the baseline.

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confidence: 68%

Section: Lcismentioning

confidence: 99%

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A cutting-plane algorithm for the Steiner team orienteering problem

Assunção

Mateus

2019

Computers & Industrial Engineering

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“…Instead of selecting all the violated CCCs found, we only add to the model the most violated cut (if any) and the ones that are sufficiently orthogonal to it. Such strategy is able to balance the strength and diversity of the cuts separated, while limiting the model size (see, e.g., Wesselmann and Suhl (2012); Samer and Urrutia (2015); Bicalho et al (2016)). Naturally, this filtering procedure does not apply to LCIs, since at most a single LCI is separated per iteration.…”

Section: Reinforcing the Original Formulation Through Cutting Planesmentioning

confidence: 99%

Coupling Feasibility Pump and Large Neighborhood Search to solve the Steiner team orienteering problem

Assunção,

Mateus

2021

Preprint

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The Steiner Team Orienteering Problem (STOP) is defined on a digraph in which arcs are associated with traverse times, and whose vertices are labeled as either mandatory or profitable, being the latter provided with rewards (profits). Given a homogeneous fleet of vehicles M , the goal is to find up to m = |M | disjoint routes (from an origin vertex to a destination one) that maximize the total sum of rewards collected while satisfying a given limit on the route's duration. Naturally, all mandatory vertices must be visited. In this work, we show that solely finding a feasible solution for STOP is NP-hard and propose a Large Neighborhood Search (LNS) heuristic for the problem. The algorithm is provided with initial solutions obtained by means of the matheuristic framework known as Feasibility Pump (FP). In our implementation, FP uses as backbone a commodity-based formulation reinforced by three classes of valid inequalities. To our knowledge, two of them are also introduced in this work. The LNS heuristic itself combines classical local searches from the literature of routing problems with a long-term memory component based on Path Relinking. We use the primal bounds provided by a state-of-the-art cutting-plane algorithm from the literature to evaluate the quality of the solutions obtained by the heuristic. Computational experiments show the efficiency and effectiveness of the proposed heuristic in solving a benchmark of 387 instances. Overall, the heuristic solutions imply an average percentage gap of only 0.54% when compared to the bounds of the cutting-plane baseline. In particular, the heuristic reaches the best previously known bounds on 382 of the 387 instances. Additionally, in 21 of these cases, our heuristic is even able to improve over the best known bounds.

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“…The problem models a number of applications, including the branching of cables in offshore wind farms (Klein et al 2015). It has been addressed with several exact and heuristic techniques, such as variable neighborhood search (de Souza and Martins 2008), branch-and-cut (Martinez and da Cunha 2014) and branch-and-cut-and-price (Bicalho et al 2016). An analysis of problem variants as well as new MILP formulations have been recently presented by Dias et al (2017).…”

Section: Brief Literature Reviewmentioning

confidence: 99%

Solution of minimum spanning forest problems with reliability constraints

Ahani¹,

Salari²,

Hosseini³

et al. 2019

Preprint

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We propose the reliability constrained k-rooted minimum spanning forest, a relevant optimization problem whose aim is to find a k-rooted minimum cost forest that connects given customers to a number of supply vertices, in such a way that a minimum required reliability on each path between a customer and a supply vertex is satisfied and the cost is a minimum. The reliability of an edge is the probability that no failure occurs on that edge, whereas the reliability of a path is the product of the reliabilities of the edges in such path. The problem has relevant applications in the design of networks, in fields such as telecommunications, electricity and transports. For its solution, we propose a mixed integer linear programming model, and an adaptive large neighborhood search metaheuristic which invokes several shaking and local search operators. Extensive computational tests prove that the metaheuristic can provide good quality solutions in very short computing times.

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Branch-and-cut-and-price algorithms for the Degree Constrained Minimum Spanning Tree Problem

Cited by 13 publications

References 25 publications

A cutting-plane algorithm for the Steiner team orienteering problem

A cutting-plane algorithm for the Steiner team orienteering problem

Coupling Feasibility Pump and Large Neighborhood Search to solve the Steiner team orienteering problem

Solution of minimum spanning forest problems with reliability constraints

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