2015
DOI: 10.48550/arxiv.1510.02197
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A characterization of linearizable instances of the quadratic minimum spanning tree problem

Abstract: We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph G = (V, E) that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in O(|E| 2 ) time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified… Show more

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Cited by 3 publications
(7 citation statements)
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“…If we expand objective value expressions in (10) and cancel out identical parts from the left and right sides, we get q 1111 +q 11kℓ +q ij11 +q ijkℓ +q 1j1ℓ +q 1jk1 +q i11ℓ +q i1k1 = q 111ℓ +q 11k1 +q ij1ℓ +q ijk1 +q 1j11 +q 1jkℓ +q i111 +q i1kℓ , from which it follows that q ijkℓ = q ijk1 + q ij1ℓ + q i1kℓ + q 1jkℓ − q ij11 − q i1k1 − q i11ℓ − q 1jk1 − q 1j1ℓ − q 11kℓ + q i111 + q 1j11 + q 11k1 + q 111ℓ − q 1111 (11) for every i, j, k, ℓ ≥ 2. However, note that (11) also holds true when some of i, j, k, ℓ are equal to 1.…”
Section: Characterization Of Linearizable Instancesmentioning
confidence: 95%
See 1 more Smart Citation
“…If we expand objective value expressions in (10) and cancel out identical parts from the left and right sides, we get q 1111 +q 11kℓ +q ij11 +q ijkℓ +q 1j1ℓ +q 1jk1 +q i11ℓ +q i1k1 = q 111ℓ +q 11k1 +q ij1ℓ +q ijk1 +q 1j11 +q 1jkℓ +q i111 +q i1kℓ , from which it follows that q ijkℓ = q ijk1 + q ij1ℓ + q i1kℓ + q 1jkℓ − q ij11 − q i1k1 − q i11ℓ − q 1jk1 − q 1j1ℓ − q 11kℓ + q i111 + q 1j11 + q 11k1 + q 111ℓ − q 1111 (11) for every i, j, k, ℓ ≥ 2. However, note that (11) also holds true when some of i, j, k, ℓ are equal to 1.…”
Section: Characterization Of Linearizable Instancesmentioning
confidence: 95%
“…Domination analysis has been successfully pursued by many researchers to provide performance guarantee of heuristics for various combinatorial optimization problems [2,4,11,[16][17][18][19][20][21][22]25,26,29,30,32,33,[35][36][37][43][44][45]. Domination analysis is also linked to exponential neighborhoods [12] and very large-scale neighborhood search [1,27].…”
Section: Domination Analysismentioning
confidence: 99%
“…Linearizable instances have been studied by various authors for the case of quadratic assignment problem [17,38,46], quadratic spanning tree problem [20] and bilinear assignment problem [22]. Here we generalize the ideas from [22] and suggest an approach for finding a characterization of linearizable instances of COPIC's.…”
Section: Linearizable Instancesmentioning
confidence: 99%
“…In this paper we also pose the problem of identifying cost structures of COPIC instances that can be reduced to an instance with no interaction costs. These instances are called linearizable instances [17,38,46,20,22]. We suggest an approach of identifying such instances for COPIC with specific feasible solution structures along with a characterization of linearizable instances.…”
Section: Introductionmentioning
confidence: 99%
“…Most of the works on QMST have been focussed on heuristic algorithms [4, 9, 13, 17-19, 24, 25, 28]. Ćustić and Punnen [5] provided a characterization QMST instances that can be solved as a minimum spanning tree problem. Exact algorithm for AQMST and QMST was studied by Pereira, Gendreau, and Cunha [2,[19][20][21].…”
Section: Subject To T ∈ Fmentioning
confidence: 99%