A performance study of some approximation algorithms for minimum dominating set in a graph

We implement and test the performances of several approximation algorithms for computing the minimum dominating set of a graph. These algorithms are the standard greedy algorithm, the recent LP rounding algorithms and a hybrid algorithm that we design by combining the greedy and LP rounding algorithms. All algorithms perform better than anticipated in their theoretical analysis, and have small performance ratios, measured as the size of output divided by the LP objective lower-bound. However, each may have advantages over the others. For instance, LP rounding algorithm normally outperforms the other algorithms on sparse real-world graphs. On a graph with 400,000+ vertices, LP rounding took less than 15 seconds of CPU time to generate a solution with performance ratio 1.011, while the greedy and hybrid algorithms generated solutions of performance ratio 1.12 in similar time. For synthetic graphs, the hybrid algorithm normally outperforms the others, whereas for hypercubes and k-Queens graphs, greedy outperforms the rest. Another advantage of the hybrid algorithm is to solve very large problems where LP solvers crash, as demonstrated on a real-world graph with 7.7 million+ vertices.

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1 Introduction and Summary

Domination theory has its roots in the -Queens problem in 18th century. Later in 1957, Berge [4] formally introduced the domination number of a graph. The problem of computing the domination number of a graph has extensive applications including the design of telecommunication networks, facility location, and social networks. We refer the reader to the book by Haynes, Hedetniemi, and Slater [22] as a general reference in domination theory.

We assume that the reader is familiar with general concepts of graph theory as in [12], the theory of algorithms as in [11], and linear and integer programming concepts as in [14], respectively. Throughout this paper denotes an undirected graph on vertex set and edge set with and . Two vertices where are adjacent (or they are neighbors) if . For any , degree of , denoted by is the number of vertices adjacent to in . For any , let denote the set of all vertices in that are adjacent to . Let denote . Arboricity of , denoted by is the minimum number of spanning acyclic subgraphs of that can be partitioned into. By a theorem of Nash Williams, , where and are the number of vertices and edges, respectively, of the induced subgraph on the vertex set [30]. Consequently , and thus measures how dense is. It is known that can be computed in polynomial time [19].
Let . is a dominating set if for every there exists such that . The domination number of , denoted by , is the cardinality of a minimum (smallest) dominating set of . Computing is known to be an NP-Hard problem even for unit disc graphs and grids [13].

1.1 Greedy approximation algorithm

A simple greedy algorithm attributed to Chvatal [9] and Lovas [25] (for approximating the set cover problem) is known to approximate within a multiplicative factor of from its optimal value, where is maximum degree of and is the th harmonic number.111Note that . The algorithm initially labels all vertices uncovered. At iteration one, the algorithm selects a vertex of maximum degree in , places in a set , and labels all vertices adjacent to it as covered. In general, at iteration , the algorithm selects a vertex with the largest number of uncovered vertices adjacent to it , adds to , and labels all of its uncovered adjacent vertices as covered. The algorithm stops when becomes a dominating set. it is easy to implement the algorithm in time. It is known that approximating within a factor from the optimal is NP-hard [17]. Hence, no algorithm for approximation can improve the asymptotic worse case performance ratio achieved by the greedy algorithm. Different variations of the greedy algorithm to approximate are developed and some are tested in practice; See work of Chalupa [9] Campan et. al. [8], Eubank et. al [18], Parekh [26], Sanchis [27], and Siebertz [28].

Below are two examples of worst-case graphs (one sparse and one dense) for greedy algorithm which are derived from an instance of set cover problem provided in [6]. For both instances, the solutions provided by the greedy algorithm are actually times the optimal.

Example 1.1.

Let be an integer and for , let be a star on vertices. Consider a graph on vertices whose vertices are the disjoint union of the vertices of the ’s () plus two additional vertices and . Now, place edges from and to the first half of the vertices in each (including the root), and the second half of the vertices in each , respectively. Note that the root of each has degree and the degree of both and is . Initially, greedy chooses the root of which can cover vertices (including itself). Generally, at iteration , there is a tie between the root of and since each can cover uncovered vertices. If tie breaking does not result in selecting , there will be a tie in every iteration until the algorithm returns the set of ’s (). This dominating set has cardinality , but , since is a minimum dominating set. Note that is a planar graph.

Example 1.2.

Let be an integer, and let be a graph with vertices ,where and . Now make a clique and an independent set of vertices, respectively. Next, consider a linear ordering on : for , the set of neighbors of in , denoted by , has cardinality and is disjoint from , for any . Finally, for place edges between and the first half of the vertices in each , and place edges between and the second half of the vertices in each . Now note that the greedy algorithm will be forced to pick the vertices , in that order but the minimum dominating set in is and .

1.2 Linear programing rounding approximation algorithms

One can formulate the computation of as an integer programming problem stated below. However, since integer programming problems are known to be NP-hard [23], the direct applications of the integer programming method would not be computationally fruitful.

[colback=white, arc=0mm] IP1:
Minimize  
Subject to  

Now observe that by relaxing the integer program IP1 one obtains the following linear program. [colback=white,arc=0mm]

LP1:
Minimize  
Subject to  
  

Note that , where and are the values of and at optimality. Since the class of linear programming problems are solvable in polynomial time [24], LP1 can be solved in polynomial time. Very recently, Bansal and Umboh [3] and Dvořák [16] have shown that an appropriate rounding of fractional solutions of LP1 gives integer solutions to IP1 whose values are at most and , respectively, in polynomial time. Hence, for sparse graphs (graphs with bounded arboricity), one can get a better approximation ratio than which is achieved by the greedy algorithm. To our knowledge, and in contrast to the greedy algorithm, the performances of the LP rounding approaches have not been tested in practice.

1.3 Other approximation algorithms

There are other approximation algorithms for very specific classes of graphs including planar graphs which have better than constant performance ratio in the worst case but are more complex than algorithms described here. See [28] for a brief reference to some related papers.

1.4 Our work

Greedy is simple and fast, since it can be implemented in linear time. Its performance ratio in the worst case scenario is logarithmic. Linear programming works in polynomial time but is more time consuming than greedy. For sparse graphs, recent linear programming rounding methods in [3, 16] have a constant performance ratio, but there have not been any experimental study of their performances.

In this paper, we implement three types of algorithms and compare and contrast their performances in practice. These algorithms are the greedy algorithm, the LP rounding algorithms, and a hybrid algorithm that combines the greedy and LP approach. The hybrid algorithm first solves the problem using the greedy algorithm and finds a dominating set . It then takes a portion of vertices in , forces their weights to be in linear program LP1, solves the resulting (partial) linear program, and then properly rounds the solution to the partial LP. Finally, it returns the rounded solution plus the portion of the greedy solution that was forced to LP1.

1.5 Environment, implementation and datasets

We used a laptop with modest computational power - 8th generation Intel i5 (1.6GHz) and 8GB RAM - to perform the experiments. We implemented the time version of the greedy algorithm in C++. We used IBM Decision Optimization CPLEX Modeling (DOCPLEX) for Python to solve the LP relaxation of the problem. Python and DOCPLEX were used to implement the LP rounding and hybrid algorithms.

The graph generator at was used to create the planar graphs, trees, k-planar graphs (graphs embedded in the plane with at most crossings per edge) , and k-trees (graphs with tree width with largest number of edges) up to 20,000 vertices. The k-Queens graphs, hypercubes (up to 12 dimensions) and graph implementations of the cases described in 1.1 and 1.2 were created ourselves. We also used publicly available Google+ and Pokec social-network graphs, as well as real-world DIMACS Graphs with up to more than 7,700,000 vertices.

1.6 Our results

Through experimentation, all algorithms perform better than anticipated in their theoretical analysis, particularly with respect to the performance ratios (measured with respect to the LP objective lower-bound). However, each may have advantages over the others for specific data sets. For instance, LP rounding normally outperforms the other algorithms on real-world graphs. On a graph with 400,000+ vertices, LP rounding took less than 15 seconds of CPU time to generate a solution with performance ratio 1.011, while the greedy and hybrid algorithms generated solutions of performance ratio 1.12 in similar time. For synthetic graphs (generated k-trees, k-planar) the hybrid algorithm normally outperforms the others, whereas for hypercubes and k-Queens graphs, the greedy outperforms the rest. Particularly, on the 12-dimensional hypercube, greedy finds a solution with performance ratio 1.7 in 0.01 seconds. On the other hand, the LP rounding and hybrid algorithms produce solutions with performance ratio 13 and 3.3 using 7.5 and 0.08 seconds of CPU time, respectively. It is notable that greedy gives optimal results in some cases where the domination number is known. Specifically, the greedy algorithm produces an optimal solution on hypercubes with dimensions where k=1, 2, 3, and 4.

The hybrid algorithm can solve very large problems when the size of LP1 becomes formidable in practice. For instance, the hybrid algorithm solved a real-world graph with 7.7 million+ vertices in 106 seconds of CPU time with a performance ratio of 2.0075. The LP solver crashed on this problem.

This paper is organized as follows. In section two, we formally describe LP rounding and hybrid algorithms. When the size of problem is so large that LP1 can not be solved in practice, then can not be computed, and hence the performance ratio of the hybrid algorithm can not be determined. We resolved this problem by decomposing LP1 in to two smaller linear programs so each of them has an objective value not exceeding and used the maximum objective value of the two smaller LP’s, instead of , to measure the performance ratio of the hybrid algorithm. Section 3, 4, and 5 contains results for Planar, k-Planar, and k-Tree graphs, hypercubes and k-Queen graphs, and real-world graphs respectively.

2 Linear Programming and hybrid approach

The following algorithm is due to Bansal and Umboh [3].

[colback=white,arc=0mm] Algorithm ([3])

Solve LP1, and let be the set of all vertices that have weight at least , where is the arboricity of graph . Let be the set of all vertices not adjacent to any vertex in and returns . Dvořák[15, 16] studied -domination problem, that is, when a vertex dominates all vertices at distance at most from it and its combinatorial dual, or a -independent set [1]. In [16] he employed the LP rounding approach of Bansal and Umboh, as a part of his frame work and consequently, for , he improved the approximation ratio of Algorithm by showing that the algorithm given below provides a approximation.

[colback=white,arc=0mm] Algorithm ([16])

Solve LP1, and let be the set of all vertices that have weight at least , where is the arboricity of graph . Let be the set of all vertices that are not adjacent to any vertex of and return .

Remark 2.1.

Graph in example 1.1 is planar, so . Thus, algorithms and have a worst-case performance ratio of nine and seven respectively, whereas greedy exhibits a worst-case performance ratio. Throughout our experiments, rounding algorithms returned an optimal solution of size two for both examples, whereas greedy returned a set of size three for Example 1.1. Furthermore, in Example 1.2, it can be verified that for graph and hence in theory the worse case performance ratios of the rounding algorithms are not constant either. Interestingly enough, in our experiments, was always two for graphs of type Example 1.2, and LP rounding algorithms also always found a solution of size two which is the optimal value. Thus the performance ratio was always one and much smaller than the predicted worst case.

Next, we provide a description of the decomposition approach for approximating LP1 and our hybrid algorithm. Recall that a separation in is a partition of so that no vertex of is adjacent to any vertex of . In this case is called a vertex separator in . Let be a feasible solution to LP1, and let . Then denotes .

Lemma 2.1.

Let be a separation in and consider the following linear programs: [colback=white,arc=0mm] LP2:
Minimize  
Subject to
and [colback=white,arc=0mm] LP3:
Minimize  
Subject to

Then .

Proof. Let be an optimal solution to LP1. Note that the restrictions of to and give feasible solutions for LP3 and LP2 of values and , and hence the claim for the lower bound on follows. .

Note that in LP2, LP3 the constraints are not written for all variables, and rounding method in [3] may not directly be applied.

Theorem 2.1.

Let , let , let and let . Let be an optimal solution for LP3, and let denote the sum of the weights assigned to all vertices in . Then there is dominating set in of size at most .

Proof. Let be the set of all vertices in with , and let . Now apply the method in [3] to to obtain a rounded solution, or a dominating set , of at most vertices in . Finally, note that is a dominating set in with cardinality at most .

[colback=white,arc=0mm]

Algorithm (Hybrid Algorithm)

Apply the greedy algorithm to to obtain a dominating set , and let be the first vertices in . Now solve the following linear program on the induced subgraph of with the vertex set .

(1)
(3)

Next, let and , and apply the rounding scheme in algorithms or to , and let and be corresponding sets, and output the set .

Remark 2.2.

Note that by Theorem 2.1 Algorithm can be implemented in polynomial time. Furthermore, , and thus Algorithm has a bounded performance ratio.

3 Performance on Planar Graphs, k-Planar Graphs, and k-Trees

In this section, we compare the performance ratios of Greedy, , , Hybrid, and Hybrid on planar graphs, k-planar graphs k-trees. In Tables 2 and 3, we present the performance of the algorithms on k-trees where and k-planar graphs where , respectively. These graphs are dense. We also present the algorithms’ performance on sparse k-trees and sparse k-planar graphs in tables 4 and 5. The planar graphs k-trees, and k-planar graphs were all made using described in section 1.5.

In most cases, the and variants of the hybrid algorithm outperformed the others, producing the lowest performance ratio to the LP lower bound . Greedy performs close to hybrid and outperforms it for the larger dense k-trees and a few of the k-planar graphs. The LP-rounding algorithms performed the worst across the board. All algorithms were able to compute dominating sets in less than 2 seconds across the different types of graphs and their range of sizes.

The arboricity of each of the planar graphs is at most 3. For k-trees, we use for arboricity. For k-planar graphs, we use the upper bound of on arboricity.

Greedy/ / Hybrid/ / Hybrid/
2000, 5980 316.93 1.12 1.40 1.11 1.39 1.11
4000, 11972 620.72 1.16 1.35 1.14 1.34 1.14
6000, 17978 942.59 1.13 1.29 1.13 1.29 1.13
8000, 23974 1239.16 1.14 1.41 1.13 1.40 1.13
10000, 29972 1579.06 1.13 1.27 1.13 1.27 1.13
12000, 35973 1874.66 1.13 1.36 1.12 1.35 1.12
14000, 41974 2185.35 1.14 1.33 1.14 1.32 1.14
16000, 47975 2514.62 1.14 1.33 1.13 1.33 1.13
18000, 53971 2811.98 1.15 1.35 1.14 1.35 1.14
20000, 59971 3127.20 1.14 1.32 1.13 1.31 1.13
Table 1: Results for Planar Graphs
Greedy/ / Hybrid/ / Hybrid/
2000, 13972 15.00 1.07 1.20 1.00 1.20 1.00
4000, 31964 10.00 1.00 1.00 1.00 1.00 1.00
6000, 53955 11.00 1.00 1.00 1.00 1.00 1.00
8000, 71955 13.00 1.00 1.00 1.00 1.00 1.00
10000, 99945 11.19 1.07 2.23 1.07 2.23 1.07
12000, 119945 12.00 1.00 1.00 1.00 1.00 1.00
14000, 139945 18.50 1.08 1.89 1.14 1.89 1.14
16000, 175934 11.25 1.16 1.60 1.33 1.60 1.33
18000, 197934 11.00 1.18 2.00 1.18 2.00 1.18
20000, 219934 10.50 1.14 1.43 1.43 1.43 1.43
Table 2: Results for k-Trees where
Greedy/ / Hybrid/ / Hybrid/
2000, 12986 151.97 1.26 2.16 1.24 2.11 1.24
4000, 27254 289.69 1.27 2.65 1.29 2.64 1.29
6000, 40885 431.77 1.26 2.50 1.26 2.50 1.26
8000, 54568 568.01 1.24 2.57 1.25 2.57 1.25
10000, 71414 684.20 1.27 2.57 1.28 2.56 1.28
12000, 85580 821.65 1.26 2.62 1.27 2.62 1.27
14000, 100241 957.77 1.25 2.47 1.26 2.46 1.26
16000, 114270 1098.18 1.27 2.21 1.27 2.21 1.27
18000, 128725 1238.09 1.27 2.23 1.27 2.22 1.27
20000, 142891 1368.44 1.26 2.24 1.25 2.23 1.25
Table 3: Results for k-Planar Graphs where
Greedy/ / Hybrid/ / Hybrid/
2000, 9985 39.00 1.05 1.08 1.05 1.08 1.05
4000, 19985 70.50 1.04 1.06 1.04 1.06 1.04
6000, 29985 90.83 1.03 1.17 1.03 1.17 1.03
8000, 39985 132.25 1.03 1.07 1.03 1.07 1.03
10000, 49985 158.00 1.03 1.03 1.03 1.03 1.03
12000, 59985 209.67 1.02 1.08 1.02 1.08 1.02
14000, 69985 225.58 1.04 1.09 1.04 1.09 1.04
16000, 79985 270.25 1.02 1.09 1.02 1.09 1.02
18000, 89985 291.83 1.02 1.06 1.02 1.06 1.02
20000, 99985 339.58 1.04 1.08 1.04 1.08 1.04
Table 4: Results for k-Trees where
Greedy/ / Hybrid/ / Hybrid/
2000, 11465 171.42 1.19 1.65 1.20 1.65 1.20
4000, 23033 336.57 1.21 1.63 1.22 1.63 1.22
6000, 34577 510.02 1.24 2.20 1.25 2.19 1.25
8000, 46130 680.88 1.25 1.91 1.25 1.91 1.25
10000, 57786 840.92 1.23 2.12 1.24 2.10 1.24
12000, 69220 1019.54 1.23 2.02 1.22 2.02 1.22
14000, 80680 1181.05 1.22 1.90 1.22 1.90 1.22
16000, 92300 1355.13 1.23 2.03 1.23 2.03 1.23
18000, 103862 1516.14 1.24 1.99 1.24 1.99 1.24
20000, 115354 1689.35 1.22 2.08 1.21 2.08 1.21
Table 5: Results for k-Planar Graphs where

4 Performance on Hypercubes and k-Queen Graphs

In this section, we present the performance of Greedy, , , Hybrid, and Hybrid on hypercubes from 5-12 dimensions and k-Queens graphs.

Table 6 compares the performance ratios of the algorithms on hypercubes. We use the arboricity for hypercubes for LP rounding and hybrid [21]. For k-Queens graphs, arboricity is unknown, so we use the upper bound , where is the length of the chessboard.

For both hypercubes and k-Queens graphs, Greedy performs the best, followed by Hybrid and Hybrid. and LP rounding perform the worst by far. This is not surprising as LP Rounding approaches are known to in general perform worse on dense graphs than sparse graphs. Solutions were computed in under 8 seconds for all graphs and algorithms.

-2.5 cm-2.5 cm[!htb]Results for Hypercubes Greedy/ / Hybrid/ / Hybrid/ 5, 80 5.33 1.50 3.00 1.50 3.00 1.50 6, 192 9.14 1.75 7.00 1.75 7.00 1.75 7, 448 16.00 1.00 1.00 1.00 1.00 1.00 8, 1024 28.44 1.13 9.00 1.13 9.00 1.13 9, 2304 51.20 1.25 7.07 2.99 7.07 2.99 10, 5120 93.09 1.38 11.00 2.70 11.00 2.70 11, 11264 170.67 1.50 6.59 2.85 6.59 2.85 12, 24576 315.08 1.63 13.00 3.14 13.00 3.14

-2.5 cm-2.5 cm[!htb]Results for k-Queens Graphs Greedy/ / Hybrid/ / Hybrid/ 225, 5180 4.89 2.05 38.45 6.75 36.40 6.75 256, 6320 5.19 1.93 46.98 7.70 43.90 7.12 289, 7616 5.50 1.82 45.84 8.91 44.03 8.91 324, 9078 5.80 1.90 50.34 9.83 48.27 9.83 361, 10716 6.10 1.97 52.42 9.67 50.78 9.67 400, 12540 6.41 2.03 56.81 10.14 53.06 9.68 441, 14560 6.71 1.94 59.89 11.32 56.91 11.17 484, 16786 7.02 2.00 63.86 9.55 59.29 9.12 529, 19228 7.32 1.91 65.83 10.38 62.83 10.11 576, 21896 7.62 1.97 70.82 11.93 64.00 11.67 625, 24800 7.93 2.02 74.15 10.47 69.61 10.34 676, 27950 8.23 1.94 76.27 11.78 68.50 11.30 729, 31356 8.54 1.87 80.80 11.83 74.48 11.13 784, 35028 8.84 1.92 80.07 14.82 74.64 14.25 841, 38976 9.15 1.97 85.81 12.02 78.81 11.70 900, 43210 9.45 2.01 87.18 12.91 81.26 12.38

5 Performance on Real-World Graphs

In this section, we present the performance of LP rounding, greedy, and hybrid on the real-world social network graphs from Google+ [9], Pokec [9], and DIMACS [2]. Each of these graphs are sparse, but their arboricity is unknown. Since arboricity is unknown, we experiment with the threshold applied during LP rounding and hybrid, starting with , where is a lower bound on arborictiy . We call LP Rounding with this threshold Algorithm . Similarly, Algorithm has threshold . Through experimentation, the best threshold which we found was ; the resulting Algorithm is called .

In Table 8, we compare the solution size of , , and , along with their hybrid analogs and greedy, to the LP lower bound on the Google+ graphs. Table 9 compares the same algorithms on the Pokec graphs. In Table 10, we compare the performance ratio to the LP lower bound for these algorithms on 3 social network graphs from DIMACS. In Tables 8, 9 and 10, LP Rounding performs better than the greedy and hybrid approaches, with greedy being the worst out of the algorithms tested. Out of the LP rounding approaches, performs the best.

-2.5 cm-2.5 cm[!htb] Results for Google+ Graphs Greedy Hybrid Hybrid Hybrid 500, 1006 42 42 42 42 42 42 42 42 2000, 5343 170 176 170 170 170 176 176 176 10000, 33954 860 900 864 864 864 893 893 893 20000, 81352 1715 1817 1730 1730 1716 1800 1800 1800 50000, 231583 4565 4849 4651 4607 4585 4790 4790 4790

-2.5 cm-2.5 cm[!htb] Results for Pokec Graphs Greedy Hybrid Hybrid Hybrid 500, 993 16 16 16 16 16 16 16 16 2000, 5893 75 75 75 75 75 75 75 75 10000, 44745 413 413 413 413 413 413 413 413 20000, 102826 921 928 921 921 921 923 923 923 50000, 281726 2706 2773 2712 2712 2712 2757 2757 2743

Compared to the best results from [9], which used a randomized local search algorithm that is run for up to one hour, LP Rounding approaches generally produced a smaller or as good solution using significantly less run-time at less than 0.5 seconds for each graph.

-2.5 cm-2.5 cm[!htb]Results for DIMACS Graphs Graph Greedy/ / Hybrid / Hybrid/ Hybrid coAuthorsDBLP 299067, 977676 43969.00 1.02 1.00 1.02 1.00 1.02 1.00 1.02 coPapersCiteseer 434102, 16036720 26040.92 1.12 1.01 1.12 1.01 1.12 1.01 1.12 citatinCiteseer 268495, 1156647 43318.85 1.04 1.03 1.04 1.03 1.04 1.02 1.04

Table 11 shows an example of a 7 million+ vertices graph where and cannot be run as a result of the large size. For hybrid approaches, using the first vertices from the greedy solution, where is the size of the greedy solution, resulted in the use of too much memory. We instead used the first vertices from the greedy solution. Both Hybrid and Hybrid performed better than greedy. Greedy took 14 seconds to produce a solution while hybrid took 107 seconds. is provided as a lower bound on , and therefore, .

-2.5 cm-2.5 cm[!htb] Results for the DIMACS Great Britain Street Network max{} Greedy Hybrid Hybrid 7733822, 8156517 1314133 1357189 1357189 2732935 2724608 2724608

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