Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem

Srivastav, Anand; Werth, Sören

doi:10.1007/978-3-540-77050-3_41

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…They have also defined a canonical super-additive boundary functional that well-approximates dbMST [29,Lemmas 3 and 4]. This, together with Proposition 2.1 proves that dbMST is near-additive.…”

Section: Proposition 71 Dbmst Is a Smooth Sub-additive And Near-addmentioning

confidence: 83%

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Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

2012

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Euclidean optimization problems such as TSP and minimum-length matching admit fast partitioning algorithms that compute near-optimal solutions on typical instances.In order to explain this performance, we develop a general framework for the application of smoothed analysis to partitioning algorithms for Euclidean optimization problems. Our framework can be used to analyze both the running-time and the approximation ratio of such algorithms. We apply our framework to obtain smoothed analyses of Dyer and Frieze's partitioning algorithm for Euclidean matching, Karp's partitioning scheme for the TSP, a heuristic for Steiner trees, and a heuristic for degree-bounded minimum-length spanning trees.

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Section: Proposition 71 Dbmst Is a Smooth Sub-additive And Near-addmentioning

confidence: 83%

“…Proof The smoothness and sub-additivity properties have been proved by Srivastav and Werth [29]. They have also defined a canonical super-additive boundary functional that well-approximates dbMST [29,Lemmas 3 and 4].…”

Section: Proposition 71 Dbmst Is a Smooth Sub-additive And Near-addmentioning

confidence: 92%

Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

2012

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“…We now consider the Euclidean functionals

{MST}_{k} (X)

defined as the minimum length of a spanning tree of

scriptX

whose vertices all have degree

\leq k

. It was shown in that

{MST}_{k} (X) \sim β_{{MST}_{k}}^{d} n^{\frac{d - 1}{d}}

for each k , and we prove separation of these asymptotic formulas as follows: Theorem We have that

β_{TSP}^{d} = β_{{MST}_{2}}^{d} > β_{{MST}_{3}}^{d} > \dots > β_{{MST}_{τ' (d)}}^{d} = β_{MST}^{d}

for all d .…”

Section: Introductionmentioning

confidence: 93%

Separating subadditive euclidean functionals

Frieze

Pegden

2016

Random Struct Algorithms

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Abstract. If we are given n random points in the hypercube [0, 1] d , then the minimum length of a Traveling Salesperson Tour through the points, the minimum length of a spanning tree, and the minimum length of a matching, etc., are known to be asymptotically βna.s., where β is an absolute constant in each case. We prove separation results for these constants. In particular, concerning the constantsand β d TF from the asymptotic formulas for the minimum length TSP, spanning tree, matching, and 2-factor, respectively, we prove thatWe also asymptotically separate the TSP from its linear programming relaxation in this setting. Our results have some computational relevance, showing that a certain natural class of simple algorithms cannot solve the random Euclidean TSP efficiently.

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“…(Note that τ (d) ≤ τ (d), and in particular, τ (2) = 5.) We prove: We now consider the Euclidean functionals MST k (X ) defined as the minimum length of a spanning tree of X whose vertices all have degree ≤ k. It was shown in [26] that MST k (X ) ∼ β d MST k n d−1 d for each k, and we prove separation of these asymptotic formulas as follows: Theorem 1.2. We have that…”

Section: Introductionmentioning

confidence: 93%

Separating subadditive euclidean functionals

Frieze

Pegden

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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The classical Beardwood-Halton-Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form β √ n for the minimum length of a Traveling Salesperson Tour throuh n random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such Euclidean functionals on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through n random points in [0, 1] 2 was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2-factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation.We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch-and-bound algorithms based on these natural lower bounds must take nearly exponential (e˜ (n) ) time to solve the TSP to optimality, even in average case. This is the first average-case superpolynomial lower bound for these branch-and-bound algorithms (a lower bound as strong as e˜ (n) was not even been known in worst-case analysis).

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Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem

Cited by 5 publications

References 15 publications

Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

Separating subadditive euclidean functionals

Separating subadditive euclidean functionals

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