Engineering Branch-and-Cut Algorithms for the Equicut Problem

Anjos, Miguel F.; Liers, Frauke; Pardella, G.; Schmutzer, Andreas

doi:10.1007/978-3-319-00200-2_2

Cited by 4 publications

(4 citation statements)

References 26 publications

(57 reference statements)

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“…Buchheim et al (2008) make the separation problem for target cuts tractable by considering a sufficiently small projection of the polytope. Target cuts in this form have been computationally investigated for a number of applications: certain constrained quadratic 0-1 problems (Buchheim et al 2010), the maximum cut problem (Bonato et al 2014), the equicut problem (Anjos et al 2013), and robust network design (Buchheim et al 2011). Our method differs from their approach in that we turn the cut generation tractable by using relaxed decision diagrams instead of projection.…”

Section: Target Cutsmentioning

confidence: 99%

Target Cuts from Relaxed Decision Diagrams

Tjandraatmadja

Hoeve

2019

INFORMS Journal on Computing

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The most common approach to generate cuts in integer programming is to derive them from the linear programming relaxation. We study an alternative approach that extracts cuts from discrete relaxations known as relaxed decision diagrams. Through a connection between decision diagrams and polarity, the algorithm generates cuts that are facet-defining for the convex hull of a decision diagram relaxation. As proof of concept, we provide computational evidence that this algorithm generates strong cuts for the maximum independent set problem and the minimum set covering problem.

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Section: Target Cutsmentioning

confidence: 99%

Target Cuts from Relaxed Decision Diagrams

Tjandraatmadja

Hoeve

2019

INFORMS Journal on Computing

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“…The NP-complete problems implementable in this way include the number-partitioning problem, the integer knapsack problem [16], and the Coulomb glass problem, which-as shown in Anjos et al [49]-can be mapped to a class of opti-cut problems. We summarize the description of the necessary Hamiltonian parameters in Table 1.…”

Section: Np-complete Models Realizable With Available Trapped-ion Tecmentioning

confidence: 99%

Probing entanglement in adiabatic quantum optimization with trapped ions

et al. 2015

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Adiabatic quantum optimization has been proposed as a route to solve NP-complete problems, with a possible quantum speedup compared to classical algorithms. However, the precise role of quantum effects, such as entanglement, in these optimization protocols is still unclear. We propose a setup of cold trapped ions that allows one to quantitatively characterize, in a controlled experiment, the interplay of entanglement, decoherence, and non-adiabaticity in adiabatic quantum optimization. We show that, in this way, a broad class of NP-complete problems becomes accessible for quantum simulations, including the knapsack problem, number partitioning, and instances of the max-cut problem. Moreover, a general theoretical study reveals correlations of the success probability with entanglement at the end of the protocol. From exact numerical simulations for small systems and linear ramps, however, we find no substantial correlations with the entanglement during the optimization. For the final state, we derive analytically a universal upper bound for the success probability as a function of entanglement, which can be measured in experiment. The proposed trapped-ion setups and the presented study of entanglement address pertinent questions of adiabatic quantum optimization, which may be of general interest across experimental platforms.

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“…Alternatively, this latter constraint added to the max-cut problem gives the equicut problem which can be motivated by an application to Coulomb glasses in theoretical physics. Motivated by this application, Anjos et al [5] recently proposed an enhanced branch-and-cut algorithm for equicut based on an approach proposed by Brunetta et al [13].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, Anjos et al [5] compared basic LP and SDP relaxations for the equicut problem, and found that linear bounds can be competitive with the semidefinite ones and can be computed much faster. While their results appear to contradict the above observations, it is important to note that they focus on dense instances coming from the physics application, and that their specialized relaxation includes constraints that are not valid for the minimum bisection polytope in general.…”

Section: Introductionmentioning

confidence: 99%

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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Engineering Branch-and-Cut Algorithms for the Equicut Problem

Cited by 4 publications

References 26 publications

Target Cuts from Relaxed Decision Diagrams

Target Cuts from Relaxed Decision Diagrams

Probing entanglement in adiabatic quantum optimization with trapped ions

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

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