Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, <i>k</i>-MST, and Related Problems

Mitchell, Joseph S. B.

doi:10.1137/s0097539796309764

Cited by 359 publications

(238 citation statements)

References 4 publications

Supporting

Mentioning

237

Contrasting

Order By: Relevance

“…The current best values for ρ are ρ = 3 2 for general metrics [7], and ρ = 1 + ǫ (for any constant ǫ > 0) for constant dimensional Euclidean metrics [1,16]. This has been improved slightly to 1 + ρ…”

Section: Related Workmentioning

confidence: 99%

Capacitated Vehicle Routing with Nonuniform Speeds

Gørtz¹,

Molinaro

Nagarajan³

et al. 2016

Mathematics of OR

View full text Add to dashboard Cite

Abstract. The capacitated vehicle routing problem (CVRP) [17] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles having respective speeds {λi} k i=1 , the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al. [15]. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees [14]. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot.

show abstract

Section: Related Workmentioning

confidence: 99%

Capacitated Vehicle Routing with Nonuniform Speeds

Gørtz¹,

Molinaro

Nagarajan³

et al. 2016

Mathematics of OR

View full text Add to dashboard Cite

show abstract

“…Our algorithm is based on applying the m-guillotine method [20], appropriately adapted to take into account the cost function and coverage constraint. 1 We need several definitions; we largely follow the notation of [20].…”

Section: The Case C > 4: a Ptasmentioning

confidence: 99%

“…1 We need several definitions; we largely follow the notation of [20]. Let G = (V, E) be an embedding of a connected planar graph, of total Euclidean edge-length L. Let D be a set of disks centered at each vertex v of G of radius r v .…”

Section: The Case C > 4: a Ptasmentioning

confidence: 99%

See 1 more Smart Citation

Minimum-cost coverage of point sets by disks

Alt

Arkin

Brönnimann³

et al. 2006

Proceedings of the Twenty-Second Annual Symposium on Computational Geometry

Self Cite

View full text Add to dashboard Cite

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t j ) and radii (r j ) that cover a given set of demand points Y ⊂ R 2 at the smallest possible cost. We consider cost functions of the form ∑ j f (r j ), where f (r) = r α is the cost of transmission to radius r. Special cases arise for α = 1 (sum of radii) and α = 2 (total area); power consumption models in wireless network design often use an exponent α > 2. Different scenarios arise according to possible restrictions on the transmission centers t j , which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t j on a given line in order to cover demand points Y ⊂ R 2 ; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y ; (c) a proof of NP-hardness for a discrete set of transmission points in R 2 and any fixed α > 1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

show abstract

“…We would like to point out that the well-known approximation schemes of Arora [Aro98] and Mitchell [Mit99] are only applicable to the octilinear Steiner tree problem without obstacles, but it seems to be unknown whether polynomial time approximation schemes are possible. For the octilinear Steiner tree problem without obstacles heuristics have been proposed by Kahng et al [KMZ03] and Zhu et al [ZZJ + 04].…”

Section: Introductionmentioning

confidence: 99%

Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles

Müller‐Hannemann

Schulze

2006

Algorithm Theory – SWAT 2006

View full text Add to dashboard Cite

Abstract. The novel octilinear routing paradigm (X-architecture) in VLSI design requires new approaches for the construction of Steiner trees. In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and (2) a version with rectangular soft obstacles. The interior of hard obstacles has to be avoided completely by the Steiner tree. In contrast, the Steiner tree is allowed to run over soft obstacles. But if the Steiner tree intersects some soft obstacle, then no connected component of the induced subtree may be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles. For both problem types, we provide reductions to the Steiner tree problem in graphs of polynomial size with the following approximation guarantees. Our main results are (1) a 2-approximation of the octilinear Steiner tree problem in the presence of hard rectilinear or octilinear obstacles which can be computed in O(n log 2 n) time, where n denotes the number of obstacle vertices plus the number of terminals, (2) a (2 + ε)-approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles which runs in O(n 3 ) time, and (3) a polynomial time (1.55 + ε)-approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles.

show abstract

Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems

Cited by 359 publications

References 4 publications

Capacitated Vehicle Routing with Nonuniform Speeds

Capacitated Vehicle Routing with Nonuniform Speeds

Minimum-cost coverage of point sets by disks

Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles

Contact Info

Product

Resources

About