The symmetric quadratic traveling salesman problem

Fischer, Anja; Helmberg, Christoph

doi:10.1007/s10107-012-0568-1

Cited by 29 publications

(31 citation statements)

References 19 publications

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“…∑ {i, j,k}⊆V, j =i≤k = j y i jk = 2x jk for all { j, k} ∈ E, for all j ∈ V Each of these equations induces one variable more than originally demanded, namely y k jk as the linearized substitute for the square term x jk x jk . Thus we may safely subtract y k jk from the left and x jk from the right hand side and obtain ∑ {i, j,k}⊆V, j =i<k = j y i jk = x jk for all { j, k} ∈ E, for all j ∈ V which are exactly the linearization constraints as presented by Fischer and Helmberg (2013).…”

Section: Symmetric Quadratic Traveling Salesman Problemmentioning

confidence: 99%

Compact linearization for binary quadratic problems subject to assignment constraints

Mallach

2017

4OR-Q J Oper Res

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We prove new necessary and sufficient conditions to carry out a compact linearization approach for a general class of binary quadratic problems subject to assignment constraints as it has been proposed by Liberti in 2007. The new conditions resolve inconsistencies that can occur when the original method is used. We also present a mixed-integer linear program to compute a minimally-sized linearization. When all the assignment constraints have non-overlapping variable support, this program is shown to have a totally unimodular constraint matrix. Finally, we give a polynomial-time combinatorial algorithm that is exact in this case and can still be used as a heuristic otherwise.

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Section: Symmetric Quadratic Traveling Salesman Problemmentioning

confidence: 99%

Compact linearization for binary quadratic problems subject to assignment constraints

Mallach

2017

4OR-Q J Oper Res

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“…Indeed, inequalities (37) are a particular case of the RLT inequalities (9), defined for W = {u, v, w}. They also appear in the context of the SQSTP, and were proven to be facet defining for the convex hull of the SQSTP feasible solutions in [5].…”

Section: Valid Inequalitiesmentioning

confidence: 99%

“…The AQMSTP is also related to the symmetric quadratic traveling salesman problem [5] (SQTSP). Differently from the widely known TSP, in the SQTSP case, the cost of a cycle does not depend exclusively on pairs of successive nodes traversed by the salesman.…”

Section: Introductionmentioning

confidence: 99%

Polyhedral results, branch‐and‐cut and Lagrangian relaxation algorithms for the adjacent only quadratic minimum spanning tree problem

Pereira¹,

Cunha

2017

Networks

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Given a complete and undirected graph G, the adjacent only quadratic minimum spanning tree problem (AQM-STP) consists of finding a spanning tree that minimizes a quadratic function of its adjacent edges. The strongest AQMSTP linear integer programming formulation in the literature works on an extended variable space, using exponentially many decision variables assigned to the stars of G. In this article, we characterize three families of facet defining inequalities by investigating the projection of that formulation onto the space of the canonical linearization variables. On the algorithmic side, we introduce four new branch-and-bound algorithms. Three of them are branch-and-cut algorithms based on the inequalities characterized by projection. The fourth is based on a Lagrangian relaxation scheme, also devised for the star reformulation. Two of the branch-and-cut algorithms provide very good results, almost always dominating the previously best algorithm for the problem. The Lagrangian relaxation based branch-and-bound algorithm provides even better results. It manages to solve all previously solved AQMSTP instances in the literature in about one tenth of the time needed by its competitors.

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“…These include, for example, maximum cut (or binary quadratic programming (QP)) and its variants [39][40][41], clustering [42], non-convex QP with binary variables [43], quadratically constrained QP [44], the quadratic traveling salesman problem (TSP) [45], TSP with neighborhoods [46], and polynomial optimization [47].…”

Section: Applicationsmentioning

confidence: 99%