Chapter 4 Partition (3) Double Partition Ding-Zhu Du

Partition a big thing (hard to deal) into small ones (easy to deal). It is a natural idea.

Partition is also an important technique in design of approximation algorithms. Example: To find a dominating set, we may find a dominating set in each small area.

Weighted Dominating Set in unit disk graphs Given a unit disk graph G=(D,E) with node weight c:DR, find a dominating set with minimum total weight. + <1<1

Backgroud 72-approximation (Ambuhl, et al. 2006).

(6+ε)-approximation (Gao, et al. 2008).

Partition into big cells General Case

Partition No node lies on a cut-line.

Problem A(i,j) 1 Dominating area

Lemma Problem A(i,j) has 2-approximation. Theorem

16 ? 14 !

2-approximation for A(I,j) Case 1 Minimum weight of node in D ij

2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ? ALAM AR CL CR BL BM BR

A problem on strip: outside disks cover inside points p1p1 p2p2 pipi Ti(D,D) : minimum weight set with D, D, dominating p 1, …, p i such that (1) D (lowest intersection point on L) among disks above the strip (2) D(highest intersection point on L) among disks below the strip L

Dynamic Programming

p1p1 p2p2 p i-1 D 1 (lowest intersection point on L) among disks above the strip, in T i (D,D) D 2 (highest intersection point on L) among disks below the strip, in T i (D,D) L

pjpj pipi p i-1 D D1

2-approximation for A(I,j) Case 2. nodes in N(D ij ) dominate nodes in D ij ?

LM Lemma If p is dominated by u in LM area, then every point in is dominated by u.

p u v p u v

p p Lemma If p and p can be dominated by nodes in BM but not nodes in CL and CR, then every node in can be dominated in nodes in A and B. A B CLCR

is the leftmost one for p dominated by a node in BM, but not any node in CL and CR is the rightmost one for p dominated by a node in LM, but not any node in CL and CR contains all nodes dominated by nodes in BM but not nodes in CL and CR. pp Consider OPT

is the leftmost one for p dominated by a node in UM, but not any node in CL and CR is the rightmost one for p dominated by a node in UM, but not any node in CL and CR contains all nodes dominated by nodes in UM but not nodes in CL and CR. qq Consider OPT

L R U R

How do we find p, p, q, q? Try all possibilities. How many possibilities?

Idea: Combine cells into a strip Each strip contains m cells.

6-approximation for a special case: For every subset C of cells, 1.every cell e in C is in case 1; 2.every cell e not in C is in case 2.

6!

Partition into big cells General Case

(6+ε)-approximation in general case Shafting to minimize # of disks on boundaries

(9.875+ε)-approximation for minimum weight connected dominating set in unit disk graph. Connecting a dominating set into a cds needs to add at most opt nodes. (Zou et al, 2008) (improved 17opt)