CS774. Markov Random Field : Theory and Application Lecture 17 Kyomin Jung KAIST Nov

Remind: MRF for Maximum Weight Independent Set (MWIS) Given a graph G=(V,E), a subset I of V is called an Independent 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 if otherwise is the weight at node v.

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 optimal” solutions for every input. Heuristics: useful algorithmic ideas that often work, but fail on some inputs. (ex, Belief Propagation)

Approximation Ratio For maximization problems (ex maxcut):

Approximation Algorithms Many approximation algorithms are designed. A large variation between approximation ratios of different problems.  FPTAS (Fully Polynomial Time Approximation 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 vertices 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 number χ(G).

Hardness of MIS and Coloring Problem For any constant ε > 0 there is no polynomial-time n 1− ε -approximation algorithm for the maximum independent set problem unless P = NP. For any constant ε > 0 there is no polynomial-time n 1− ε -approximation algorithm for computing the chromatic number of G unless P = NP. Hence we consider some practical and restricted class of graphs, like planar graph and unit disk graph.

Definition – Planar Graph

Grid Minors for Planar Graphs r  r grid:  r 2 vertices  Treewidth = 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 [Robertson, 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 independent set in a k-outerplanar graph can be solved in linear time (approximately 8 k n time).  By dynamic programming using tree- decomposition

Baker’s scheme For each i in {1,2, …, k} do  Remove 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 independent 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 Euclidean plane. Two disks have edge when they intersect.  There exists a PTAS for the MIS (selecting disjoint disks).  The idea is similar to that of Baker.

Independent Set of Disk Graph

Independent set (Greedy)

Independent set (Greedy)

Independent set (Greedy)

Independent set K=2

Independent set