polynomial-time approximation schemes for geometric intersection graphs

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Polynomial-Time Approximation Schemes for Geometric Intersection Graphs. Authors: T. Erlebach, L. Jansen, and E. Seidel Presented by: Ping Luo 10/ 1 7/2005. Agenda. Preliminaries Disk partition and plane subdivision Dynamic programming. Basic Concepts. - PowerPoint PPT Presentation

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• Polynomial-Time Approximation Schemes for Geometric Intersection GraphsAuthors: T. Erlebach, L. Jansen, and E. Seidel

Presented by: Ping Luo10/17/2005

• AgendaPreliminariesDisk partition and plane subdivisionDynamic programming

• Basic ConceptsIntersection graph: for a set V of geometric objects, the corresponding intersection graph is the undirected graph with vertex set V and an edge between two vertices if the corresponding objects intersect.Disk graph: a disk graph is the intersection graph of a set of disksDisk graph recognition is NP-hard

• Disk graph an example

• Problem definitionsMaximum Weight Independent Set (MWIS) For a given set of geographic objects, find a subset of disjoint objects such that the total weight is maximum.This paper deals with disks and is the topic of this talk.Minimum Weight Vertex Cover (MWVC)Find a subset of a given objects with minimum total weight such that, for any two intersection objects, at least one is in the subset.

• Basic idea of the PTAS algorithm for MWISThe plane is partitioned into squares on each level. Some disks in each level are removed so that different squares on the same level yield independent subproblems with respect to all disks that are on this level or smaller levels.Each square has at most a constant number of disks so that enumeration is possible

• AgendaPreliminariesDisk partition and plane subdivisionDynamic programming

• Partition disks into levelsZoom in or out to make the largest diameter to be 1. The smallest is denoted as dmin

Compute disk levels

Determine disk level j of each disk with diameter d:

• Partition disks an exampleLevel 0Level 1Level 2

• Plane subdivisionAn example: k = 3, l = 2

• Definitions I(r, s) active Lines (0r,s
• Observation about j-squares in different levels For any j, , every (j+1)-square is completely contained in some j-square.Every j-square is the union of (j+1)-squares.

• Definition II - D(r,s)Delete disks that hit (r,s) active lines. The rest of the disks forms a new disk graph called D(r,s).There are many different D(r,s) since 0
• Definitions IIIRelevant square: a j-squarethat contains at least one disk in D(r,s) of level jForest structure: a forestthat stores the child-parentrelationship between relevant squaresChild square: a relevantj-square S that is containedin a relevant j-square Sand no relevant square S of levels j such that j
• Useful upper boundsThe number of disks (level
• AgendaPreliminariesDisk partition and plane subdivisionDynamic programming

• TablesTwo types of tables are used for each relevant j-square S to store the intermediate results. Main table . It is indexed by I, a set of independent disks of level =j) contained in S, and X U I= Auxiliary tables and . J is a set of independent disks of level = j and contained in S, X U I U J=

• Outline of the algorithmFor each pair of r and s (0
• Base auxiliary table construction pseudocode

• Auxiliary table base construction an example

• Auxiliary table combining pseudocode(continue on next slide)

• Auxiliary table combining pseudocode

• Caveat of the combining algorithmNot independent!

• A correctionInstead of enumerating disks that intersect the boundary of R1 or R2, we enumerate disks that intersect private boundary of R1, or private boundary of R2, or shared boundary of R1 and R2. The time complexity wont change due to this correctionshared boundary

• Deal with missing child squaresSome squares contain no disks and hence are not relevant. No table will be constructed for irrelevant squares.If a j-square S contains some irrelevant (j+1)-squares S, all the child squares of S contained in S will be considered.

• Summary of the algorithmConstruct D(r,s) for 0
• Time complexity D(r,s) need to be considered. The relevant squares and their forest structure can be constructed in time polynomial in n: For each relevant square S, missing child squares can be handled in time . Then a dynamic approach is used to construct table Ts for the square S, which takesTotal time complexity:

• Questions?HomeworkWhy does the paper assumes the input of its algorithms is the set D of disks, not only the corresponding intersection graph? In the MWIS presented in this paper, if each disk has weight 1, then MWIS can be reduced to a MIS problem. Describe in your own words what the difference is between the MIS in this paper and the MIS you learned in the class. List at least two aspects which you can think of.

• Reference"Polynomial-time approximation schemes for geometric intersection graphs", T. Erlebach, K. Jansen, and E. Seidel, SIAM Journal on Computing 34, pp. 1302-1323 (2005).