Polyhedral results for the Equitable Coloring Problem

Polyhedral results for the Equitable Coloring Problem

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Abstract

In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence of the effectiveness of including these inequalities as cuts in a Branch & Cut algorithm.

\NoHyper

a]I. Méndez-Diaz\thanksrefGRANT b]G. Nasini\thanksrefGRANT b]D. Severín\thanksrefGRANT

FCEyN, Universidad de Buenos Aires, Argentina, imendez@dc.uba.ar

FCEIA, Universidad Nacional de Rosario, Argentina, {nasini, daniel}@fceia.unr.edu.ar


  Keywords:  equitable graph coloring, integer programming, branch & cut 1991 MSC: 90C27, 05C15  \endNoHyper11footnotetext: Partially supported by grants UBACyT X143 (2008-2010), PID-CONICET 204 (2010-2012) and PICT 2006-1600.

1 Introduction and preliminary results

The Equitable Coloring Problem (ECP), originally presented in [2], is a variation of the widely studied Graph Coloring Problem (GCP) with additional constraints imposing that any pair of color classes has to differ in size by at most one. Further references and applications can be seen in [1].

A -coloring of a graph is a partition of in stable sets, , with . The stable set is the class of color . An equitable -coloring (or just -eqcol) of is a -coloring satisfying the equity constraints, i.e. for each , where .

Unlike GCP, a graph admiting a -eqcol may not admit a -eqcol. This leads us to define as the set of such that does not admit any -eqcol. For instance, .

The equitable chromatic number of , , is the minimum for which has a -eqcol. Computing for arbitrary graphs is an -hard problem [1].

Although many integer programming formulations are known for GCP, as far as we know, just two of these models were adapted for ECP. One case is the model in [3], adapted in [5]. Preliminary results concerning a Branch & Cut algorithm based on one of the models in [4] were presented in [6]. The algorithm turns out to be competitive compared to the one presented in [5]. This encouraged us to delve into a polyhedral study with the aim of finding strong inequalities that allow us to improve the performance of our algorithm.

2 The polytope

From now on, we assume that is a graph with vertices such that and . Other cases are trivial.

In [4], colorings of are identified with binary vectors where and , satisfying the following constraints:

(assign a unique color to each vertex)
(adjacent vertices do not share the same color)
(eliminate some symmetric colorings)

where if color is assigned to vertex and if color is used, i.e. . The coloring polytope is defined as the convex hull of colorings of . In this work, equitable colorings are identified with binary vectors defining colorings which also satisfy

\hb@xt@.01(1)
\hb@xt@.01(2)

where is a dummy variable set to 0, constraints (1) ensure that isolated vertices use enabled colors and (2) are the equity constraints. The Equitable Coloring Polytope is the convex hull of the equitable colorings of .

Next we state the main results related to the polyhedral structure of .

Proposition 2.1

The dimension of is .

In [4], clique inequalities and block inequalities are proven to be facet-defining inequalities of . In our case, we have:

Proposition 2.2

(i) Let and be maximal clique of such that . Then, the clique inequality defines a facet of .
(ii) Let and . Then, the block inequality is valid for and defines a facet of if .

By lifting rank inequalities and neighborhood inequalities, also studied in [4], we obtain new families of valid inequalities which often define facets.

Proposition 2.3

Let , with and . Then, the -2-rank inequality defined as

is valid for . Let us assume that and no connected component of the complement graph of is bipartite. The inequality defines a facet of if one of the following conditions holds:

  • for all , is not a clique,

  • is odd, and for all such that , there exists a stable set of size 3 such that and , and the complement of has a perfect matching,

  • is even, and for all such that , there exist two disjoint stable sets of size 3, and , such that and , and the complement of has a perfect matching.

If or , the -2-rank inequality is respectively dominated by the inequalities

which also usually define facets of .

Proposition 2.4

Given , and with , the -subneighborhood inequality defined as

where , is a valid inequality for . If or , the inequality defines a facet of when the following conditions hold:

  • for all , there exists a -eqcol such that ,

  • for all , there exists an equitable coloring such that and .

Finally, we obtain three new families of valid inequalities for , which were not derived from any of the valid inequalities given in [4].

Proposition 2.5

Let . The -color inequality defined as

where and , is a valid inequality for . In addition, if , contains all the colors greater than and the complement of has a matching of size , then the -color inequality defines a facet of .

Proposition 2.6

Given a non universal vertex of and such that , the -outside-neighborhood inequality defined as

where , is valid for and defines a facet of if the following conditions hold:

  • there exists such that ,

  • if is odd, the complement of has a perfect matching,

  • for all , there exists a -eqcol such that ,

  • for all such that and , there exists a -eqcol such that , and a -eqcol such that and ,

  • for all , there exists a -eqcol lying on the face defined by the inequality.

Proposition 2.7

Given , be a clique of such that and such that and . The -clique-neighborhood inequality defined as

where , is a valid inequality for . If there exists such that , the inequality defines a facet of when the following conditions hold:

  • for all , there exists an -eqcol lying on the face defined by the inequality,

  • for all , there exist two -eqcols lying on the face defined by the inequality, with in the first one and where the second one is obtained from the first by only changing the color of , i.e. ,

  • for all , if , there exist two -eqcols lying on the face defined by the inequality such that in one of them and in the other.

Although the sufficient conditions in the previous results are strong, we find several cases where they hold. Moreover, even when the inequalities do not define facets, the dimension of the faces defined by them is quite high. For example, if , it can be proved that the dimension of the face defined by the -clique-neighborhood inequality is at least .

3 Computational performance of valid inequalities

In this section, we report on the computational performance of the families of valid inequalities studied in the previous section, embedded as cuts in a B&C algorithm for solving ECP.

In order to strengthen the formulation and avoid considering classes of symmetric colorings, constraints are considered within the initial relaxation, and are handled as cuts during the optimization.

The cutting process consists in looking for violated clique and -2-rank inequalities with a greedy algorithm. During the separation of clique inequalities, it attempts to find violated -clique-neighborhood inequalities by scanning vertices not adjacent to a given clique . Whenever not enough cuts were generated, it tries to add block, -subneighborhood and -outside-neighborhood inequalities, handled by enumeration, and -color inequalities with a greedy algorithm. Separation routines for clique and block inequalities are exposed in [4]. The B&C algorithm also includes an initial heuristic, a primal heuristic and a custom branching rule.

Experiments were carried out over random instances of 70 vertices with different density percentages and 2 hours time limit. We compare our B&C algorithm with (BC) and without (BC) our new inequalities against the general purpose IP-solver CPLEX 12.1 and results reported in [5].

% % solved inst. Nodes (average) Time in sec. (average)
dens. BC BC CPX [5] BC BC CPX [5] BC BC CPX [5]

10
100 100 100 100 3.4 4 13.3 57 0.3 0.3 4 109
30 90 90 0 0 2135 3949 276 224
50 70 70 0 0 7932 21595 1354 2145
70 80 80 10 100 525 2970 214 678 128 446 4380 273
90 100 100 100 100 5.1 14.5 30 9.4 2.6 2.8 29 11

As one may appreciate from the table, the addition of our cutting planes has shown to be particularly useful in substantially decreasing the number of Branch-and-Bound nodes and the CPU time was significantly reduced on medium and high density instances.

References

  • [1] Kubale, M. et al. “Graph Colorings”, AMS, Providence, Rhode Island, 2004.
  • [2] Meyer, W. Equitable Coloring, Amer. Math. Monthly, 80 (1973), 920–922.
  • [3] Campêlo, M., R. Corrêa and V. Campos, On the asymmetric representatives formulation for the vertex coloring problem, Discrete Appl. Math., 156 (2008), 1097–1111.
  • [4] Méndez-Díaz, I. and P. Zabala, A cutting plane algorithm for graph coloring, Discrete Appl. Math., 156 (2008), 159–179.
  • [5] Bahiense, L., Y. Frota, N. Maculan, T. Noronha and C. Ribeiro, A branch-and-cut algorithm for equitable coloring based on a formulation by representatives, Electr. Notes Discrete Math., 35 (2009), 347–352.
  • [6] Méndez-Díaz, I., G. Nasini and D. Severin, A branch-and-cut algorithm for the equitable graph coloring problem, ALIO-INFORMS Joint International Meeting, Buenos Aires, Argentina, 2010.
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