Perfect Graphs Lecture 23: Apr 17

Hard Optimization Problems Independent set Clique Colouring Clique cover Hard to approximate within a factor of coding scheduling

Graph Product Given G1=(V1,E1) and G2=(V2,E2), their product G1xG2 is the graph whose vertex set is V1xV2 and the edge set is {((u1,v1),(u2,v2)) : u1=u2 and (v1,v2) in E2 or v1=v2 and (u1,u2) in E1 or (u1,u2) in E1 and (v1,v2) in E2. Claim: There is a clique of size >= k in G if and only if there is a clique of size >= k 2 in GxG.

Constant Factor Hardness Assuming clique is hard to approximate within a factor of (1+є), then it is also hard to approximate within any constant factor. Idea: take graph product.

Perfect Graph Graphs in which these hard problems are “nice”. Independent set Clique Colouring Clique cover Easy equalities: Easy inequalities:

Perfect Graph In what graphs does the equalitiy holds? A graph is perfect if for every induced subgraph H of G, A graph is co-perfect if for every induced subgraph H of G, What graphs are perfect? Bipartite graphs What graphs are co-perfect? Bipartite graphs What graphs are not perfect? Odd cycles What graphs are not co-perfect? Odd cycles

Line Graph of Bipartite Graph Vertices correspond to edges of a bipartite graph, and two vertices have an edge if and only if the corresponding edges share an endpoint. Line graph of bipartite graphs are perfect. Line graph of bipartite graphs are co-perfect.

Interval Graph Vertices correspond to intervals, and two vertices have an edge if and only if the corresponding intervals overlap. Interval graphs are perfect. Interval graphs are co-perfect. Many applications.

Chordal Graph Also known as triangulated graphs. A graph is chordal if every cycle of length >=4 has a chord. A graph is chordal if it is the intersection graph of subtrees of a tree. A graph is chordal if it has an ordering such that for each vertex the neighbours in front form a clique. Chordal graphs are perfect. Chordal graphs are co-perfect.

Conjectures Perfect graph conjecture: A graph is perfect if and only if it is co-perfect. Strong perfect graph conjecture [Berge 1960]: A graph is perfect if and only if it does not contain odd cycles and the complement of odd cycles as induced subgraphs. Lovasz 1970 Chudnovsky, Robinson, Seymour, Thomas 2003

LP-Perfect In which graphs do the LP always have integral solutions for independent set? This LP is integral only if the input graph is bipartite.

LP-Perfect In which graphs do the LP always have integral solutions for independent set? for each clique C No known polynomial time separation oracle. A graph is LP-perfect if and only if the above linear program is integral.

Perfect Graph Theorem The following are equivalent 1.A graph is perfect. 2.A graph is LP-perfect. 3.A graph is co-perfect.

Duplication Lemma Let G be a graph and v be a vertex. Let G+ be the graph obtained from G by adding a new vertex v’ and connecting it to v and the neighbours of v. Lemma. If G is perfect, then G+ is perfect.

Proof of Duplication Lemma It is enough to prove If v is in a maximum clique, then both sides plus 1. Consider an optimal colouring of G. Let v be coloured red. Consider G-R+v, with maximum clique at most w(G)-1. Colour G-R+v using w(G)-1 colours. So colour G using w(G) colours as R-v+v’ is an independent set.

Perfect => LP-perfect for each clique C Compute an optimal solution of the LP. Let qx be integral. Duplicate each vertex by qx(v) times to obtain a graph G’. G’ is perfect and has clique size exactly q and total cost at most qLP. Decompose G’ into q independent set, one must have cost at most LP.

LP-perfect => co-Perfect for each clique Cfor each vertex v Consider a clique C with positive value in an optimal dual solution. This clique C must intersect every maximum independent set.

co-Perfect => Perfect Take the complement of G and apply perfect => co-perfect!

Strong Perfect Graph Theorem A polynomial time algorithm to recognize perfect graphs.

What’s Next Shannon coding Lovasz Theta function Solve the clique LP using SDP Colouring 3-colourable graphs