CS774. Markov Random Field : Theory and Application

Lecture 17

Kyomin JungKAIST

Nov 05 2009

Remind: MRF forMaximum Weight Independent Set (MWIS)

Given a graph G=(V,E), a subset I of V is called an Indepen-dent Set, if for all , the two end points of e does not belong to I simultaneously.

When the vertices are weighted, an independent set I is called MWIS if the sum of the weights of is maximum.

Finding a MWIS is equivalent to finding a MAP in the following MRF on

where , and

Evuvu

Vvvv xxxWxXP

),(

),,(exp][

vW

||}1,0{ VX

),( 21 xx 0

1

121 xxif

otherwise

is the weight at node v.

Ee

Iv

MRF for Combinatorial Optimization

Example: Maximum Independent Set (MIS).

Input: A graph.Feasible solution: A set S of vertices.Value of solution: |S|.Objective: Maximize.

Similar example: Max-cut, vertex coloring problem

NP-hardness

Finding the optimal solution is NP-hard. Practical implication: no polynomial time

algorithm always finds optimum solution.

Approximation algorithms: polynomial time, guaranteed to find “near op-timal” solutions for every input.

Heuristics: useful algorithmic ideas that of-ten work, but fail on some inputs. (ex, Be-lief Propagation)

Approximation Ratio

For maximization problems (ex maxcut):

Approximation Algorithms

Many approximation algorithms are de-signed.

A large variation between approximation ratios of different problems. FPTAS (Fully Polynomial Time Approxi-

mation Scheme)PTAS (Polynomial Time Approximation

Scheme)Constant ratio, super-constant…

K-Cliques

A K-clique is a set of K nodes with all K(K-1)/2 possible edges between them

This graph contains a 4-clique

Given: (G, k)Question: Does G contain a k-clique?

K-Cliques

Two problems are essentially the same

Clique / Independent Set

Complement of G

Given a graph G, let G*, the complement of G, be the graph such that two nodes are connected in G* if and only if the corresponding nodes are not connected in G

G G*

Vertex Coloring Problem

The assignment of labels or colors to the ver-tices of a graph.

Each edge has different color at its end nodes.

This problem can be expressed by MRF so that # of total coloring is the same as the partition function.

The smallest number of colors needed to color a graph G is called its chromatic num-ber

χ(G).

Hardness of MIS and Coloring Problem

For any constant ε > 0 there is no polyno-mial-time n1−ε -approximation algorithm for the maximum independent set problem un-less P = NP.

For any constant ε > 0 there is no polyno-mial-time n1−ε -approximation algorithm for computing the chromatic number of G un-less P = NP.

Hence we consider some practical and re-stricted class of graphs, like planar graph and unit disk graph.

Definition – Planar Graph

Grid Minors for Planar Graphs

r r grid:r2 verticesTreewidth = r

r r grid is the canonical planar graph of treewidth Θ(r) :every planar graph of treewidth w has

an Ω(w) Ω(w) grid minor [Robert-son, Seymour, Thomas, 1994]

r

r

PTAS for Independent set on planar graphs

Method originated by Baker (1994)

Extended by several authors to more general/other classes of graphs

e.g. SODA 2005: Demaine & Hajigitani – more problems and minor closed classes of graphs

k-outerplanar graphs

Label vertices of a plane graph by level.

All vertices on exterior face level 1.

All vertices on exterior face when vertices of levels 1 … i are removed, are on level i+1.

Graph is k-outerplanar when at most k levels.

Theorem: k-outerplanar graphs have treewidth at most 3k – 1.

3-outerplanar

Independent set on k-outerplanar graphs

For fixed k, finding a maximum inde-pendent set in a k-outerplanar graph can be solved in linear time (approxim-ately 8kn time). By dynamic programming using tree-

decomposition

Baker’s scheme

For each i in {1,2, …, k} doRemove all vertices in levels i, i+k,

i+2k, i+3k, …Each connected component of the

remaining graph is (k-1)-outerplanar.Solve independent set exactly on the

remaining graph.

Output the best of the k obtained inde-pendent sets.

Approximation Ratio

Look at a maximum independent set S.

Each of the k runs deletes a different subset of S.

So, there is a run that deletes at most |S|/k vertices from S one of the runs gives an answer that is at

least (k-1)/k times the size of the optimum.

This gives a PTAS.

Unit Disk Graph

A unit disk graph is the intersection graph of a set of unit disks in the Eu-clidean plane.

Two disks have edge when they intersect.

There exists a PTAS for the MIS (select-ing disjoint disks).

The idea is similar to that of Baker.

Independent Set of Disk Graph

Independent set (Greedy)

0 1 2 3 4 5 6 7 8

Independent set (Greedy)

0 1 2 3 4 5 6 7 8

Independent set (Greedy)

0 1 2 3 4 5 6 7 8

Independent set

0 1 2 3 4 5 6 7 8

K=2

Independent set

0 1 2 3 4 5 6 7 8