Exact Algorithms for the Vertex Coloring Problem

2 Maximal Clique ! A clique K of a graph G is a complete subgraph of G. A clique is maximal if no vertex can be added still having a clique. The cardi...

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Exact Algorithms for the Vertex Coloring Problem Branch and Bound: Algorithm DSATUR, Brélaz (Comm. ACM, 1979); Improvement of DSATUR: Sewell (2nd DIMACS Implementation Challenge, 1993). Branch-and-Price: Mehrotra, Trick (INFORMS J. on. Computing, 1996), Gualandi, Malucelli (Politecnico di Milano Technical Report, 2010), Malaguti, Monaci, T. (Discrete Optimization, 2010). Branch and Cut: Méndez-Díaz, Zabala (Disc. Appl. Math. 2006, 2008). 1

Maximal Clique n  n 

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A clique K of a graph G is a complete subgraph of G. A clique is maximal if no vertex can be added still having a clique.

The cardinality of any (maximal) clique of graph G represents a Lower Bound for the problem. A fast greedy algorithm (Johnson, J. Comp. Syst. Sci. 1974) can be used to compute a maximal clique K of G(V,E): Given an ordering of the vertices, consider the candidate vertex set W. Set W = V, K = 0 and iteratively: * Choose the vertex v of W of maximum degree and add it to the current clique K. * Remove from W vertex v and all the vertices not adjacent to the current clique K. Different orderings of the vertices generally produce different maximal cliques.

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ILP models for VCP: Model VCP-ASSIGN n 

Binary variables:

xih = { yh = {

n

1 if vertex i has color h 0 otherwise 1 if color h is used 0 otherwise

min ∑ yh

i =1,…,n h =1,…,n h =1,…,n

(1)

h =1

n

∑x

ih

=1

i = 1,.., n

(2)

h =1

xih + x jh ≤ yh xi ,h ∈ {0,1} yh ∈ {0,1}

∀i, j : (i, j ) ∈ E h = 1,..., n i = 1,..., n h = 1,..., n

(3)

h = 1,..., n ( 4) 3

Independent Sets n  n 

An Independent Set (or Stable Set) of G = (V, E) is a subset of V such that there is no edge in E connecting a pair of vertices. It is maximal if no vertex can be added still having an independent set. 1

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For VCP: all the vertices of an independent set can have the same color Feasible coloring -> partitioning of the graph into independent sets. 1

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5

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Independent Sets and Cliques n 

Given a graph G = (V, E) 1

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5 Independent Set

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Its “complementary graph” G = (V, E), with 1

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E = {(i, j): (i, j) ∈/ E}

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clique

independent set of G → clique of G (and viceversa) clique of G → independent set of G (and viceversa)

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Set Covering Formulation SC -VCP s.t.

min ∑ xs s∈S

∑x

(1)

s

≥1

∀i ∈ V

(2)

∀s ∈ S

(3)

s:i∈s

xs ∈ {0,1}

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S can be defined as the family of all the maximal

Independent Sets (or Stable Sets) of graph G. The LP Relaxation of this formulation leads to tight lower bounds, and symmetry in the solution is avoided, but the number of maximal independent sets (i.e. the number of “columns”) can be exponential in the number of vertices n. 6

Set Covering Formulation SC-VCP Branch-and-Price Algorithms SC-VCP: Master Problem n  n 

LP Relaxation of SC-VCP: exponentially many variables (columns, independent sets) . Column Generation procedure: Solve the LP Relaxation of the SC-VCP formulation by considering a subset of independent sets (columns): Restricted Master Problem (RMP); Detect possible negative reduced cost columns by solving the corresponding “Pricing Problem”, add them to the RMP and iterate. 7

Pricing Problem n 

ci is the optimal dual variable associated with the i-th

“covering constraint” in the SC-VCP formulation (weight of vertex i). The Pricing Problem requires the solution of a Maximum Weighted Independent Set Problem (MWISP) (NP-Hard). n 

yi = 1{ if vertex i is in the independent set}, 0 otherwise n

max ∑ ci yi i =1

yi + y j ≤ 1

∀i, j : (i, j ) ∈ E

yi ∈ {0,1}

i = 1,..., n

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Pricing Problem (MWISP) (2)

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ci is the optimal dual variable associated with the i-th

“covering constraint” in the SC-VCP formulation. yi = 1{if vertex i is in the independent set}, 0 otherwise n

max ∑ ci yi i =1

yi + y j ≤ 1

∀i, j : (i, j ) ∈ E

yi ∈ {0,1}

i = 1,..., n

If the optimal solution value is greater than 1, then an independent set (column) with negative reduced cost has been found.

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Branch-and-Price Algorithm 1 (Gualandi, Malucelli; Tech. Rep 2010) n 

Column Generation procedure: detection of possible negative reduced cost columns by alternatively solving:

1) Exact Algorithm CLIQUER (maximum weighted clique problem; Ostergard, Discr. Appl. Math. 2002). 2) Heuristic Algorithm QUALEX-MS (maximum weighted clique problem; Busygin, Discr. Appl. Math. 2006). 2) Constraint Programming Algorithm (weighted version of the max-clique problem algorithm proposed by Fahle, Proc. Ann. Eur. Symp. Algorithms, Springer, 2002). 10

Branch-and-Price Algorithm 2 (Malaguti, Monaci, T.; Discrete Optimization 2010) n 

Column Generation procedure: detect possible negative reduced cost columns by solving MWISP, by using:

1) a Tabu Search heuristic algorithm (TS) which produces maximum weighted independent sets; 2) an ILP Solver (CPLEX 10.2), if the algorithm TS fails in finding negative reduced cost columns. 11

Branch-and-Cut Algorithm Méndez-Díaz, Zabala (Disc. Appl. Math. 2006, 2008) n 

Improvement of the VCP-ASSIGN Formulation by adding new valid inequalities.

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Initialization Phase:

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Preprocessing procedure to reduce the number of vertices to be considered. Initial Upper Bound computed through the execution of algorithm DSATUR with a short time limit (5 seconds). Initial Lower Bound computed by finding a maximal clique through a greedy algorithm.

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New Branching Rules. 12

Computational Results for the Exact Approaches n 

Algorithm DSATUR: Brélaz (Comm. ACM, 1979), Improved by Sewell (2nd DIMACS Implem. Challenge 1993) Implemented by Mehrotra, Trick (INFORMS J. on C., 1996)

n  Branch and Cut Algorithm BC-COL (with the stronger lower bounding procedures): Méndez-Díaz, Zabala (Disc. Appl. Math. 2006, 2008) n  Branch-and-Price Algorithm G-M: Gualandi, Malucelli (Tech. Report Politecnico di Milano, 2010). n  Branch-and-Price Algorithm M-M-T: Malaguti, Monaci, T. (Discrete Optimization, 2010). n 

Comparable CPU times.

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DIMACS Benchmark Instances Johnson, Trick, 2nd DIMACS Implementation Challenge 1993 n  DIMACS

benchmark graph instances compose a variety of graph classes used for evaluating the performance of VCP algorithms: - random graphs: DSJC_n.x; - geometric random graphs: DSJR_n.x; r_n.x; - quasi-random graphs: flat_n.x; - artificial graphs: le_n.x; latin_square_10; Queen_rn.rn; myciel_k - real-world application-related graphs.

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VCP: Exact Algorithms Mendez-Diaz, Zabala (2006, 2008) BC-COL

DSATUR

GualandiMalucelli (2010)

common instances (w.r.t. M-M-T)

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proven optimal solutions only algorithm

8 1

1 0

15 2

25 (15) 13 (4)

Lower Bound LB > LB (MMT) LB = LB (MMT) LB < LB (MMT) global LB gap w.r.t. MMT

9 43 14 92

1 23 42 326

2 27 10

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Upper Bound UB < UB (MMT) UB = UB (MMT) UB > UB (MMT) global UB gap w.r.t. MMT

0 24 42 333

0 18 48 304

0 27 12

*

*

M-M-T (2010)

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