Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order

Jäger, Gerold; Molitor, Paul

doi:10.1007/978-3-540-85097-7_20

Cited by 25 publications

(40 citation statements)

References 20 publications

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“…If a cut of value less than one is found, we separate constraints (10) instead of constraints (4). Furthermore, we built separators for inequalities (8) and (9) and for inequalities (10) with sets S ⊂ V, |S| = 2, which are based on complete enumeration. We carried out all experiments on a PC with an Intel R Core TM i7 CPU 920 with 2.67 GHz and 12 GB RAM.…”

Section: Experimental Studymentioning

confidence: 99%

“…Several heuristics for approximately solving QTSPs associated with PM models and PVLM models of order two exist [8,9]. However, these heuristics yield quite different solutions with quite different likelihoods, and there are no guarantees about the deviations of the suboptimal solutions from the optimal solution.…”

Section: Introductionmentioning

confidence: 99%

“…There are several exact algorithms for solving QTSPs [8][9][10], which provide optimal solutions in acceptable running times at least for small instances. Specifically, it has been shown that a branch-and-cut algorithm tailored to the QHPP and the QTSP can solve these problems for random instances up to size 30 in a few hours [9].…”

Section: Introductionmentioning

confidence: 99%

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Computational Recognition of RNA Splice Sites by Exact Algorithms for the Quadratic Traveling Salesman Problem

Fischer

Jäger

et al. 2015

Computation

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One fundamental problem of bioinformatics is the computational recognition of DNA and RNA binding sites. Given a set of short DNA or RNA sequences of equal length such as transcription factor binding sites or RNA splice sites, the task is to learn a pattern from this set that allows the recognition of similar sites in another set of DNA or RNA sequences. Permuted Markov (PM) models and permuted variable length Markov (PVLM) models are two powerful models for this task, but the problem of finding an optimal PM model or PVLM model is NP-hard. While the problem of finding an optimal PM model or PVLM model of order one is equivalent to the traveling salesman problem (TSP), the problem of finding an optimal PM model or PVLM model of order two is equivalent to the quadratic TSP (QTSP). Several exact algorithms exist for solving the QTSP, but it is unclear Computation 2015, 3 286 if these algorithms are capable of solving QTSP instances resulting from RNA splice sites of at least 150 base pairs in a reasonable time frame. Here, we investigate the performance of three exact algorithms for solving the QTSP for ten datasets of splice acceptor sites and splice donor sites of five different species and find that one of these algorithms is capable of solving QTSP instances of up to 200 base pairs with a running time of less than two days.

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Section: Experimental Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computational Recognition of RNA Splice Sites by Exact Algorithms for the Quadratic Traveling Salesman Problem

Fischer

Jäger

et al. 2015

Computation

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“…The QCCP is closely related to the quadratic traveling salesman problem (QTSP) which is introduced in [24]. The QTSP is the problem of finding a Hamiltonian cycle in a graph minimizing a quadratic cost function.…”

Section: Introductionmentioning

confidence: 99%

“…When we remove the subtour elimination constraints, the QTSP boils down to the QCCP. Therefore, the QCCP is often used to provide lower bounds for the QTSP [20,24,36]. For this reason, the quadratic cycle cover problem is an interesting optimization problem that has received more attention in the past few years.…”

Section: Introductionmentioning

confidence: 99%

The quadratic cycle cover problem: special cases and efficient bounds

Meijer

Sotirov

2020

J Comb Optim

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The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds.In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property.The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account.

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