STIC AmSud ProjectParGO Team

Manoel Campelo, Ricardo C. Correa

ParGO Research Group, Federal University of Ceara, Brazil

March 29, 2013

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Research Group Members

Comput. Dept.

◮ Claudia L. Sales

◮ Ricardo C. Correa

◮ Rudini M. Sampaio

◮ Victor A. Campos

Math. Dept.

◮ Ana Shirley F. Silva

Stat. and Appl. Math. Dept

◮ Carlos Diego Rodrigues

◮ Manoel B. Campelo Neto

PhD Students

◮ Fabio C. S. Dias

◮ Pablo M. S. Farias

◮ Wladimir A. Tavares

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Here we are

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Vertex coloring and correlated NP-Hard problems

Graph Theory

◮ Ana Shirley F. Silva

◮ Claudia L. Sales

◮ Rudini M. Sampaio

◮ Victor A. Campos

Combinatorial Optimization

◮ Carlos Diego Rodrigues

◮ Manoel B. CampeloNeto

◮ Ricardo C. Correa

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Graph Theoretic Approach

Stable Set Polytope of G = (V,E): subset of vertices pairwisenon-adjacent

vertex u ∈ W

vertex v /∈ W

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Graph Classes

NP-Hard variation of thevertex coloring problem

+ Graphs with special struc-tural properties

⇓

Polynomial time algorithms

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

A Typical Case: b-Coloring

◮ Given a coloring c, v is a b-vertex of color i if c(v) = i andv has at least one neighbor in every color class j,j 6= i.

◮ A b-coloring is a coloring such that each color class has ab-vertex.

TheoremThe Tight b-Chromatic Problem is NP-complete for

connected bipartite graphs.

TheoremThe b-chromatic number of a split graph can be determined in

polynomial time.

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Graph Decomposition: a Powerful Tool

An example: Modular Decomposition (figure from Wikipedia)

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Selected Publications and Problems

◮ J. Araujo, C. L. Sales, “On the Grundy number of graphs withfew P4’s”. Disc. Appl. Math., 2012.

◮ M. Aste, F. Havet, C. L. Sales, “Grundy number and products ofgraphs”. Disc. Math., 2010.

◮ V. Campos, A. Gyarfas, F. Havet, C. L. Sales, F. Maffray, “Newbounds on the Grundy number of products of graphs”. J. ofGraph Theory, 2012.

◮ V. Campos, S. Klein, R. Sampaio, A. Silva, “TwoFixed-Parameter Algorithms for the Cocoloring Problem”,ISAAC, 2011.

◮ V. Campos, C. L. Sales, F. Maffray, A. Silva, “b-chromaticnumber of cacti”. LAGOS, 2009.

◮ V. Campos, A. K. Maia, N. Martins, C. L. Sales, R. Sampaio,“Restricted Coloring Problems on graphs with few P4’s”.LAGOS, 2011.

◮ F. Havet, C. L. Sales and L. Sampaio, “b-coloring of m-tightgraphs”. Disc. Appl. Math., 2012.

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Combinatorial Optimization Approach

xW [u] = 1vertex u ∈ W

xW [v] = 0vertex v /∈ W

Characteristic vector xW of W ⊆ V

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Combinatorial Optimization Approach

xW [u] = 1vertex u ∈ W

xW [v] = 0vertex v /∈ W

STAB(G): Convex hull of characteristic vectors

For every x ∈ STAB(G),

x[u] + x[v] ≤ 1, uv ∈ E

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Maximum Stable Set (MSS) Problem

Linear program

α(G) = max{∑

u∈V

x[u] | x ∈ STAB(G)}

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Maximum Stable Set (MSS) Problem

Linear program

α(G) = max{∑

u∈V

x[u] | x ∈ STAB(G)}

Integer program

α(G) =max∑

u∈V

x[u]

s.t. x[u] + x[v] ≤ 1, uv ∈ E

x[u] ∈ {0, 1}, u ∈ V

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Disjoint Stable Set Problems

Finding a family W of disjoint stable sets under certainconstraints

Some examples

◮ MSS: maximize | ∪ W| under |W| = 1

◮ Maximum k-partite induced subgraph: maximize | ∪ W|under |W| ≤ k

◮ Vertex coloring: minimize |W| when, for all v ∈ V , thereexists S ∈ W such that v ∈ S

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Covering Formulation for the Vertex Coloring Problem

Smax - family of maximal ss of G

χ(G) =min∑

S∈Smax

x[S]

s.t.∑

S∈Smax:u∈S

x[S] ≥ 1, u ∈ V

x[S] ∈ {0, 1}, S ∈ Smax

Exponential number of variables[Mehrotra, Trick 1996] - column generation implementation[Hansen, Labbe, Schindl 2009] - polyhedral study;∑

S∈Smaxx[S] ≥ χ(G) if G is critical and G is connected

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Representatives

Let us choose a representative for each nonempty stable set ofG. One simple criterion is to choose the smallest vertex in thestable set according to a given order of the vertices [C, Campos,

C 2008].

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Representatives

Let us choose a representative for each nonempty stable set ofG. One simple criterion is to choose the smallest vertex in thestable set according to a given order of the vertices [C, Campos,

C 2008].

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2

3

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6

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Representatives

Let us choose a representative for each nonempty stable set ofG. One simple criterion is to choose the smallest vertex in thestable set according to a given order of the vertices [C, Campos,

C 2008].

1

2

3

4

5

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10

9

8

6

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Representatives

Let us choose a representative for each nonempty stable set ofG. One simple criterion is to choose the smallest vertex in thestable set according to a given order of the vertices [C, Campos,

C 2008].

1

2

3

4

5

7

10

9

8

6

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Notation for the Representatives

G+(u)

u1

4

5

7

10

9

8

6

Anti-neighborhood

N+(u) = {v > u | uv 6∈ E}

Subgraph induced byanti-neighbors

G+(u) = G[N+[u] = N+(u)∪{u}]

Stable sets represented by u

Stable sets of G+(u)containing u itself

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Variables

G+(u)

xu[v1] = 1

xu[v2] = 0

v1

v2

xu[u] = 1u

xu[u] ∈ {0, 1} : u is a representative

xu[v] ∈ {0, 1} : u represents v ∈ N+[u]

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Formulations by Representatives

χ(G) = min{∑

v∈Vxv[v] | x ∈ X

}

, where X is given by

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Formulations by Representatives

χ(G) = min{∑

v∈Vxv[v] | x ∈ X

}

, where X is given by

Integer program

xu[v] + xu[w] ≤ xu[u],u ∈ V, v, w ∈ N+(u), vw ∈ E∑

u∈N−[v]

xu[v] ≥ 1, v ∈ V

xu[v] ∈ {0, 1}, u ∈ V, v ∈ N+[u]

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Formulations by Representatives

χ(G) = min{∑

v∈Vxv[v] | x ∈ X

}

, where X is given by

Integer program

xu[v] + xu[w] ≤ xu[u],u ∈ V, v, w ∈ N+(u), vw ∈ E∑

u∈N−[v]

xu[v] ≥ 1, v ∈ V

xu[v] ∈ {0, 1}, u ∈ V, v ∈ N+[u]

“Linear” program

xv ∈ STABxv [v](G+(v)),

∑

u∈N−[v]

xu[v] ≥ 1, xv[v] ∈ {0, 1}, v ∈ V

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

General Disjoint Stable Set Problem

max f(V ) =∑

v∈V

∑

w∈N+(v) fv[w]xv[w] + e

av1x[v1] . . . avnx[vn]

xV [V ] . . .

−xv1 [v1] xv1 [N+(v1)]

. . .. . .

−xvn [vn] xvn [N+(vn)]

cv1xv1 [v1] . . . cvnxvn [vn] cv1xv1 [N+(v1)]. . . cvnxvn [N+(vn)]

≤

b

x[V ]

0

. . .

0

d

Selected Publications and Problems

◮ M. C., V. Campos, R. C. C., “On the asymmetric representativesformulation for the vertex coloring problem,” DAM, 2008.

◮ M. C., V. A. Campos, R. C. C., “Um Algoritmo dePlanos-de-Corte para o Numero Cromatico Fracionario de umGrafo”. Pesquisa Operacional, 2009.

◮ M. C., R. C. C., “A Combined Parallel LagrangianDecomposition and Cutting-Plane Generation for MaximumStable Set Problems”. ISCO, 2010.

◮ M. C., R. C. C., P. F. S. Moura, M. C. Santos, “On optimalk-fold colorings of webs and antiwebs”. DAM, 2013.

◮ A. Basu, M. C., M. Conforti, G. Cornuejols, G. Zambelli,“Unique Lifting of Integer Variables in Minimal Inequalities”.Math. Program., 2012.

◮ M. C., G. Cornuejols, “Stable sets, corner polyhedra and theChvatal closure”. Op. Res. Let., 2009.

◮ M. C., G. Cornuejols, “The Chvatal closure of generalized stablesets in bidirected graphs”. LAGOS, 2009.

Manoel Campelo, Ricardo C. Correa STIC AmSud Project

Thank you