Online Supplement for An integer programming approach for fault-tolerant connected dominating sets

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1 Submitted to INFORMS Journal on Computing manuscript (Please, provide the mansucript number!) Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Online Supplement for An integer programming approach for fault-tolerant connected dominating sets Austin Buchanan Industrial & Systems Engineering, Texas A&M University, College Station, TX , USA, buchanan@tamu.edu, Je Sang Sung Industrial & Systems Engineering, Texas A&M University, College Station, TX , USA, je.sung@tamu.edu, Sergiy Butenko Industrial & Systems Engineering, Texas A&M University, College Station, TX , USA, butenko@tamu.edu, Eduardo L. Pasiliao Air Force Research Laboratory, Munitions Directorate, Eglin, FL 32542, USA, eduardo.pasiliao@eglin.af.mil, In this online supplement, we provide detailed computational results for solving the minimum k-d-cds problem using a lazy-constraint approach. We also list all facets of the 1-1-CDS polytope of the Grötzsch graph. Key words : connected dominating set; k-connected m-dominating set; integer programming; fault-tolerant History : In this online supplement, we provide detailed computational results for solving the minimum k-d-cds problem using a lazy-constraint approach. We also list all facets of the 1-1-CDS polytope of the Grötzsch graph. For information not discussed here, please see Section 4.1 of the main paper. All of our computational experiments were conducted on a Dell Precision WorkStation T7500 R machine with two Intel Xeon R E GHz quad-core processors and 12 GB RAM. The solver used was Gurobi Optimizer version 5.5 with its lazy-constraint callback (Gurobi Optimization, Inc 2013). In Table 1, we compare the runtime of the lazy-constraint approach (referred to in the table as Lazy ) with other approaches for the MCDS problem. The first six approaches 1

2 2 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) were proposed by Gendron et al. (2014), and include stand-alone (SA) and iterative-probing (IP) versions of Bender s decomposition (BE), branch-and-cut (BC), and a hybrid (HY) of BE and BC. The next three approaches are as follows: p-sabc is a branch-and-cut approach from Simonetti et al. (2011); DGR is a branch-and-cut approach from Lucena et al. (2010); and MTZ uses Miller-Tucker-Zemlin constraints to enforce connectivity (as proposed by Fan and Watson (2012)). We note that the experiments of Gendron et al. (2014) for the first six approaches and the MTZ approach were conducted on a 2.0 GHz Intel Xeon R E5405 machine with 8 GB RAM. More details regarding the experiments for 1-1-CDS can be found in Tables 2 and 3. Table 2 includes the initial LP relaxation, the number of lazy cuts added, and the number of branch-and-bound nodes. Table 3 compares the computational effort expended when using arbitrary vertex-cuts versus minimal vertex-cuts. In Table 4, we report experimental results for the minimum 2-1-CDS problem using the lazy approach. A minimal vertex-cut C can be an a-b separator for many choices of a and b leading to numerous possible lazy cuts. It was not clear to us which or how many of them should be used. This lead us to consider two possible approaches: (Single cut) add the a-b-separator inequality for which a and b are the lexicographically smallest, and (All cuts) add all possible a-b-separator inequalities. In Table 5, we provide runtimes and solution sizes for the same instances for both the minimum k-total dominating set (ktds) problem and the minimum k-k-cds problem for k = 1, 2, 3, 4. Recall that the minimum k-total dominating set problem can be stated as (minimum ktds problem) min x i x {0,1} n x j k, v V. i V j N(v) 1. CDS facets for the Grötzsch graph The convex hull of characteristic vectors of 1-1-CDSs of the Mycielski graph (Mycielski 1955) on eleven vertices (also known as the Grötzsch graph) is not fully described by the r-robust vertex-cut inequalities. Below are all facet-defining inequalities except for the trivial 0-1 facets as given by PORTA (Christof et al. 1997). The vertex numbering is the same as in the graph file provided by Michael Trick for the DIMACS coloring challenge (Trick 2013).

3 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 3 x1+ x2+ x3+ x4+ x5+ x6+ x7+ x8+ x9+ x10+ x11 >= 4 x1+ x2+ x3+ x4+ x5+ x6 + x8 +2x11 >= 3 x1+ x2+ x3+ x4+ x5+ x6 + x10+2x11 >= 3 x1+ x2+ x3+ x4+ x5 + x7 + x9 +2x11 >= 3 x1+ x2+ x3+ x4+ x5 + x7 + x10+2x11 >= 3 x1+ x2+ x3+ x4+ x5 + x8+ x9 +2x11 >= 3 2x1 + x3 + x5+2x6+ x7+ x8+ x9+ x10+ x11 >= 3 x1+ x2 +2x5+ x6+ x7+ x8+ x9+2x10+ x11 >= 3 x1 +2x3+ x4 + x6+ x7+2x8+ x9+ x10+ x11 >= 3 2x2 + x4+ x5+ x6+2x7+ x8+ x9+ x10+ x11 >= 3 x2+ x3+2x4 + x6+ x7+ x8+2x9+ x10+ x11 >= 3 x1+ x2+ x3+ x4+ x5 +3x11 >= 3 x1+ x2+ x3+ x4 + x8+ x9 + x11 >= 2 x1+ x2+ x3 + x5+ x6 + x10+ x11 >= 2 x1+ x2 + x4+ x5 + x7 + x10+ x11 >= 2 x1 + x3+ x4+ x5+ x6 + x8 + x11 >= 2 x2+ x3+ x4+ x5 + x7 + x9 + x11 >= 2 x1+ x2 + x5+ x6+ x7+ x8+ x9+2x10 >= 2 x1 + x3+ x4 + x6+ x7+2x8+ x9+ x10 >= 2 x1 + x3 + x5+2x6+ x7+ x8+ x9+ x10 >= 2 x2+ x3+ x4 + x6+ x7+ x8+2x9+ x10 >= 2 x2 + x4+ x5+ x6+2x7+ x8+ x9+ x10 >= 2 x1 + x3 + x11 >= 1 x1 + x5 + x11 >= 1 x2 + x4 + x11 >= 1 x2 + x5 + x11 >= 1 x3+ x4 + x11 >= 1 x1 + x3 + x6 + x8 >= 1 x1 + x5+ x6 + x10 >= 1 x2 + x4 + x7 + x9 >= 1 x2 + x5 + x7 + x10 >= 1 x3+ x4 + x8+ x9 >= 1 x6+ x7+ x8+ x9+ x10 >= 1 References Christof, T., A. Löbel, M. Stoer PORTA-polyhedron representation transformation algorithm. URL Fan, N., J.P. Watson Solving the connected dominating set problem and power dominating set problem by integer programming. G. Lin, ed., Combinatorial Optimization and Applications, Lecture Notes in Computer Science, vol Springer,

4 4 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Gendron, B., A. Lucena, A.S. da Cunha, L. Simonetti Benders decomposition, branch-and-cut, and hybrid algorithms for the minimum connected dominating set problem. INFORMS Journal on Computing To Appear. Gurobi Optimization, Inc Gurobi Optimizer Reference Manual. URL Lucena, A., N. Maculan, L. Simonetti Reformulations and solution algorithms for the maximum leaf spanning tree problem. Computational Management Science Mycielski, J Sur le coloriage des graphes. Colloq. Math Simonetti, L., A.S. da Cunha, A. Lucena The minimum connected dominating set problem: formulation, valid inequalities and a branch-and-cut algorithm. Network Optimization Trick, M Graph coloring instances. URL

5 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 5 Table 1 A comparison of runtimes, in seconds, to solve the MCDS problem (i.e., minimum 1-1-CDS). Aside from the Lazy approach, all times are from Gendron et al. (2014). A dash indicates unsolved in time limit. Instance SABE IPBE SABC IPBC SAHY IPHY p-sabc DGR MTZ Lazy v30 d v30 d v30 d v30 d v30 d v50 d v50 d v50 d v50 d v50 d v50 d v70 d v70 d v70 d v70 d v70 d v70 d v100 d v100 d v100 d v100 d v100 d v100 d v120 d v120 d v120 d v120 d v120 d v120 d v150 d v150 d v150 d v150 d v150 d v150 d v200 d v200 d v200 d v200 d v200 d v200 d IEEE-14-Bus IEEE-30-Bus IEEE-57-Bus RTS IEEE-118-Bus IEEE-300-Bus # in 1 sec # in 10 sec # in 100 sec # in 1000 sec # solved # fastest

6 6 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Table 2 Extended computational results for solving the minimum 1-1-CDS problem using the lazy approach. Instance IP Opt Root LP # Lazy cuts # B&B nodes IP Time v30 d v30 d v30 d v30 d v30 d v50 d , v50 d v50 d v50 d v50 d v50 d v70 d , v70 d v70 d v70 d v70 d v70 d v100 d , v100 d v100 d v100 d , v100 d v100 d v120 d v120 d v120 d , v120 d , v120 d v120 d v150 d , v150 d , v150 d , v150 d , v150 d v150 d v200 d , v200 d ,182, v200 d , v200 d , v200 d , v200 d IEEE-14-Bus IEEE-30-Bus IEEE-57-Bus , RTS , IEEE-118-Bus IEEE-300-Bus , ,

7 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 7 Table 3 The effect of Algorithm 1 on reducing the computational effort for the minimum 1-1-CDS problem. Without Algorithm 1 With Algorithm 1 Instance # Lazy cuts # B&B nodes Time # Lazy cuts # B&B nodes Time v30 d10 54,750 81, v30 d v30 d v30 d v30 d v50 d5 >110,425 >125,657 > , v50 d v50 d v50 d v50 d v50 d v70 d5 >88,233 >97,749 > , v70 d v70 d v70 d v70 d v70 d v100 d5 10,266 14, , v100 d v100 d v100 d30 0 1, , v100 d v100 d v120 d , v120 d v120 d20 0 2, , v120 d30 0 2, , v120 d v120 d v150 d5 1,125 15, , v150 d , , v150 d , , v150 d30 1 5, , v150 d v150 d v200 d , , v200 d10 6 2,220, ,182, v200 d , , v200 d , , v200 d50 0 1, , v200 d IEEE-14-Bus IEEE-30-Bus IEEE-57-Bus >102,288 >116,896 > , RTS96 >78,608 >77,532 > , IEEE-118-Bus >67,771 >43,083 > IEEE-300-Bus >39,576 >86,494 > , ,

8 8 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Table 4 Comparing two approaches for solving minimum 2-1-CDS: adding a single lazy cut versus adding all lazy cuts. IP Opt Single cut All cuts # Lazy Cuts # B&B nodes Time # Lazy Cuts # B&B nodes Time v30 d v30 d v30 d v30 d v30 d v50 d v50 d v50 d v50 d v50 d v50 d v70 d v70 d v70 d v70 d v70 d v70 d v100 d v100 d v100 d , , v100 d , , v100 d v100 d v120 d v120 d v120 d , , v120 d , , v120 d v120 d v150 d , , v150 d , , v150 d , , v150 d , , v150 d v150 d v200 d , , v200 d , ,328, v200 d ,296, , v200 d , , v200 d v200 d IEEE IEEE IEEE RTS , IEEE IEEE

9 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 9 Table 5 A comparison of running times for minimum k-total dominating set and minimum k-k-cds. Blank entries denote that the instance is infeasible. 1TDS 1-1-CDS 2TDS 2-2-CDS 3TDS 3-3-CDS 4TDS 4-4-CDS Graph Opt Time Opt Time Opt Time Opt Time Opt Time Opt Time Opt Time Opt Time v30 d v30 d v30 d v30 d v30 d v50 d v50 d v50 d v50 d v50 d v50 d v70 d v70 d v70 d v70 d v70 d v70 d v100 d v100 d v100 d v100 d v100 d v100 d v120 d v120 d v120 d v120 d v120 d v120 d v150 d v150 d v150 d v150 d v150 d v150 d v200 d v200 d , , , , , , v200 d , , , , v200 d , , , , v200 d , , v200 d