# Computational Study on a PTAS for Planar Dominating Set Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. PTAS for Planar Dominating Set Problem

#### 3.1. Baker’s Framework for Minimization Problem

- Let G be a plane graph with m levels of vertices for an outer face and $k\ge 2$ be an integer. Compute the vertex sets ${V}_{1},\dots ,{V}_{m}$.
- for $s=2,\dots ,k+1$
- (a)
- Compute subgraphs $G\left[U\right(i,s\left)\right]$ for $i=0,1,\dots ,r$.
- (b)
- For every subgraph $G\left[U\right(i,s\left)\right]$, find an optimal solution $S(i,s)$ by an exact algorithm.
- (c)
- Let ${S}_{s}={\mathrm{\cup}}_{i=0}^{r}S(i,s)$.

- Let S be a set of ${S}_{2},\dots ,{S}_{k+1}$ with the minimum cardinality.

#### 3.2. Modified Framework for Planar Dominating Set Problem

**Figure 2.**(

**a**) Subgraph $G\left[U\right(0,2\left)\right]$ and (

**b**) Subgraph $G\left[U\right(1,2\left)\right]$ of G.

**Figure 3.**(

**a**) Subgraph $G\left[U\right(0,3\left)\right]$ and (

**b**) Subgraph $G\left[U\right(1,3\left)\right]$ of G.

- Let G be a plane graph with m levels of vertices for an outer face and $k\ge 2$ be an integer. Compute the vertex sets ${V}_{1},\dots ,{V}_{m}$.
- for $s=2,\dots ,k+1$
- (a)
- Compute subgraphs $G\left[W\right(i,s\left)\right]$ for $i=0,1,\dots ,r$.
- (b)
- For subgraph $G\left[W\right(0,s\left)\right]$, find a minimum subset $S(0,s)$ of $W(0,s)$ that dominates every vertex of ${\mathrm{\cup}}_{j=1}^{s-1}{V}_{j}$ (every interior vertex).For every subgraph $G\left[W\right(i,s\left)\right]$, $i=1,\dots ,r-1$, find a minimum subset $S(i,s)$ of $W(i,s)$ that dominates every vertex of ${\mathrm{\cup}}_{j=(i-1)\times k+s}^{i\times k+s-1}{V}_{j}$ (every interior vertex).For subgraph $G\left[W\right(r,s\left)\right]$, find a minimum subset $S(r,s)$ of $W(r,s)$ that dominates every vertex of ${\mathrm{\cup}}_{j=(r-1)\times k+s}^{m}{V}_{j}$ (every interior vertex).
- (c)
- Let ${S}_{s}={\mathrm{\cup}}_{i=0}^{r}S(i,s)$.

- Let S be a set of ${S}_{2},\dots ,{S}_{k+1}$ with the minimum cardinality.

**Theorem 3.1**The modified application of Baker’s framework gives an $O\left({2}^{\left(6{log}_{4}3\right)(k+2))}kn\right)$ time $(1+2/k)$-approximation algorithm for the planar dominating set problem.

**Proof:**We first show the approximation ratio of the framework. Notice that ${S}_{s}={\mathrm{\cup}}_{i=0}^{r}S(i,s)$ dominates every vertex of

## 4. Computational Study of PTAS

- Compute a linear size kernel H of the subgraph using the $O\left({n}^{3}\right)$ time kernelization algorithm by Alber et al. [12].
- Find an optimal solution for H by dynamic programming based on the branch-decomposition of H and compute an optimal solution for the subgraph from the optimal solution for H.

Graph G | |E(G)| | bw | Exact Alg. | k = 3 | k = 4 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

γ (G) | β | time | D_{PTAS} | β | time | D_{PTAS} | β | time | ||||

(1) | kroB150 | 436 | 10 | 23 | 10 | 10 | 28 | 8 | 2.07 | - | - | - |

pr299 | 864 | 11 | 47 | 11 | 37 | 56 | 10 | 11.42 | - | - | - | |

tsp225 | 622 | 12 | 37 | 12 | 110 | 46 | 9 | 5.21 | - | - | - | |

a280 | 788 | 13 | 43 | 13 | 337 | 53 | 10 | 8.40 | 51 | 12 | 12.09 | |

rd400 | 1183 | 17 | - | - | - | 75 | 10 | 35.30 | 74 | 12 | 351.93 | |

pcb442 | 1286 | 17 | - | - | - | 79 | 10 | 10.46 | 78 | 10 | 10.86 | |

d657 | 1958 | 22 | - | - | - | 123 | 10 | 64.89 | 120 | 12 | 604.10 | |

pr1002 | 2972 | 21 | - | - | - | 190 | 10 | 115.65 | 182 | 12 | 1253.9 | |

(2) | tri2000 | 5977 | 8 | 321 | 7 | 198 | 361 | 7 | 175.59 | - | - | - |

tri4000 | 11969 | 9 | 653 | 7 | 1903 | 724 | 7 | 733.06 | - | - | - | |

tri6000 | 17979 | 9 | 975 | 8 | 3576 | 1136 | 8 | 1994.53 | - | - | - | |

tri8000 | 23975 | 9 | 1283 | 7 | 7750 | 1430 | 7 | 2858.63 | - | - | - | |

tri10000 | 29976 | 9 | 1606 | 7 | 16495 | 1804 | 7 | 4977.06 | - | - | - | |

tri11000 | 32972 | 14 | - | - | - | 1987 | 8 | 5910.8 | 1958 | 8 | 12341.1 | |

tri12000 | 35974 | 14 | - | - | - | 2164 | 7 | 5370.18 | 2132 | 7 | 6865.08 | |

tri14000 | 41974 | 15 | - | - | - | 2514 | 7 | 8220.49 | 2434 | 7 | 9208.72 | |

tri16000 | 47969 | 16 | - | - | - | 2920 | 7 | 10060.1 | 2885 | 7 | 12794.4 | |

(3) | rand6000 | 10293 | 11 | 1563 | 9 | 150 | 1658 | 8 | 104.85 | - | - | - |

rand10000 | 17578 | 13 | 2535 | 10 | 869 | 2850 | 8 | 535.87 | 2692 | 9 | 432.23 | |

rand15000 | 26717 | 14 | 3758 | 12 | 2769 | 4144 | 10 | 1313.14 | - | - | - | |

rand16000 | 28624 | 13 | 4002* | 13 | 5917 | 4379 | 10 | 2443.27 | 4295 | 11 | 2027.7 | |

rand20000 | 35975 | 14 | 4963* | 14 | 13993 | 5465 | 10 | 4241.65 | 5368 | 12 | 5017.02 | |

rand25000 | 40378 | 16 | - | - | - | 7101 | 8 | 6407.91 | 6632 | 12 | 9470 | |

(4) | Gab500 | 949 | 13 | 115 | 12 | 238 | 136 | 10 | 18.02 | 129 | 10 | 18.95 |

Gab600 | 1174 | 14 | 135 | 14 | 3074 | 164 | 10 | 26.05 | 156 | 10 | 22.10 | |

Gab700 | 1302 | 14 | 162 | 14 | 5710 | 187 | 10 | 22.81 | 183 | 10 | 24.30 | |

Gab800 | 1533 | 17 | - | - | - | 225 | 10 | 51.82 | 205 | 12 | 24.30 | |

Gab900 | 1758 | 17 | - | - | - | 243 | 10 | 48.39 | 231 | 12 | 344.50 | |

Gab1000 | 1901 | 18 | - | - | - | 260 | 10 | 49.69 | 259 | 12 | 781.89 | |

Gab1500 | 2870 | 21 | - | - | - | 402 | 10 | 116.37 | 385 | 12 | 960.71 |

**Table 2.**Computational results for heuristic algorithms and PTAS for the planar dominating set problem (time in seconds).

Graph G | |E(G)| | γ(G) | Greedy Alg. | Greedy-Rev Alg. | Greedy-Vote Alg. | PTAS | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

D_{Gr} | time | D_{Rev} | time | D_{Vote} | time | D_{PTAS} | time | ||||

(1) | kroB150 | 436 | 23 | 27 | 0.002 | 31 | 0.01 | 31 | 0.002 | 28 | 2.08 |

pr299 | 864 | 47 | 54 | 0.003 | 63 | 0.032 | 62 | 0.005 | 56 | 11.42 | |

tsp225 | 622 | 37 | 49 | 0.153 | 54 | 0.02 | 50 | 0.003 | 46 | 5.21 | |

a280 | 788 | 43 | 51 | 0.004 | 62 | 0.025 | 62 | 0.006 | 51 | 12.09 | |

rd400 | 1183 | - | 78 | 0.007 | 92 | 0.032 | 90 | 0.009 | 74 | 351.93 | |

pcb442 | 1286 | - | 76 | 0.908 | 90 | 0.063 | 87 | 0.01 | 78 | 10.86 | |

d657 | 1958 | - | 126 | 0.016 | 146 | 0.128 | 143 | 0.021 | 120 | 604.10 | |

pr1002 | 2972 | - | 190 | 0.032 | 236 | 0.328 | 194 | 0.04 | 182 | 1253.9 | |

(2) | tri2000 | 5977 | 321 | 365 | 0.116 | 379 | 1.119 | 464 | 0.168 | 361 | 175.59 |

tri4000 | 11969 | 653 | 729 | 0.183 | 765 | 1.792 | 787 | 0.544 | 724 | 733.06 | |

tri6000 | 17979 | 975 | 1118 | 0.418 | 1166 | 4.14 | 1306 | 0.541 | 1136 | 1994.53 | |

tri8000 | 23975 | 1283 | 1449 | 0.715 | 1522 | 7.003 | 1653 | 0.918 | 1430 | 2858.63 | |

tri10000 | 29976 | 1606 | 1819 | 1.117 | 1906 | 11.524 | 2302 | 1.572 | 1804 | 4977.06 | |

tri11000 | 32972 | - | 2040 | 1.375 | 2116 | 14.092 | 3431 | 2.561 | 1958 | 12341.1 | |

tri12000 | 35974 | - | 2186 | 1.607 | 2278 | 16.538 | 2741 | 2.243 | 2132 | 6865.08 | |

tri14000 | 41974 | 2576 | 2.462 | 2664 | 22.976 | 3317 | 3.163 | 2434 | 9208.72 | ||

tri16000 | 47969 | - | 2917 | 2.839 | 3033 | 30.694 | 3684 | 4.005 | 2885 | 12794.4 | |

(3) | rand6000 | 10293 | 1563 | 1932 | 0.748 | 2166 | 4.517 | 2908 | 1.206 | 1658 | 104.85 |

rand10000 | 17578 | 2535 | 3197 | 2.06 | 3618 | 13.33 | 4164 | 2.878 | 2692 | 432.23 | |

rand15000 | 26717 | 3758 | 4698 | 4.861 | 5402 | 29.487 | 7277 | 7.641 | 4144 | 1313.14 | |

rand16000 | 28624 | 4002* | 5039 | 5.176 | 5744 | 35.589 | 7552 | 10.327 | 4295 | 2027.7 | |

rand20000 | 35975 | 4963* | 6273 | 8.053 | 7168 | 55.948 | 8571 | 11.903 | 5398 | 5017.02 | |

rand25000 | 45327 | - | 7772 | 12.467 | 8942 | 91.039 | 11865 | 20.615 | 6632 | 9470 | |

(4) | Gab500 | 949 | 115 | 146 | 0.006 | 173 | 0.039 | 160 | 0.007 | 129 | 18.95 |

Gab600 | 1174 | 135 | 168 | 0.007 | 199 | 0.051 | 171 | 0.009 | 156 | 22.10 | |

Gab700 | 1302 | 162 | 200 | 0.01 | 242 | 0.072 | 238 | 0.012 | 183 | 24.30 | |

Gab800 | 1533 | - | 227 | 0.012 | 270 | 0.097 | 307 | 0.019 | 205 | 24.30 | |

Gab900 | 1758 | - | 254 | 0.016 | 303 | 0.103 | 323 | 0.022 | 231 | 344.50 | |

Gab1000 | 1901 | - | 280 | 0.019 | 344 | 0.146 | 423 | 0.03 | 259 | 781.89 | |

Gab1500 | 2870 | - | 426 | 0.042 | 507 | 0.335 | 496 | 0.051 | 385 | 960.71 |

**Table 3.**Computational results for Exact, Greedy and PTAS algorithms for small instances (time in seconds).

Graph G | |E(G)| | Exact Alg. | Greedy Alg. | PTAS | ||||
---|---|---|---|---|---|---|---|---|

γ(G) | time | D_{G} | time | D_{PTAS} | time | |||

(1) | kroB150 | 436 | 23 | 10 | 27 | 0.002 | 28 | 2.08 |

pr299 | 864 | 47 | 37 | 54 | 0.032 | 56 | 11.42 | |

tsp225 | 622 | 37 | 110 | 49 | 0.153 | 46 | 5.21 | |

a280 | 788 | 43 | 337 | 51 | 0.004 | 51 | 12.09 | |

(2) | tri2000 | 5977 | 321 | 198 | 365 | 0.116 | 361 | 175.59 |

tri4000 | 11969 | 653 | 1903 | 729 | 0.183 | 724 | 733.06 | |

tri6000 | 17979 | 975 | 3576 | 1118 | 0.418 | 1136 | 1994.53 | |

tri8000 | 23975 | 1283 | 7750 | 1449 | 0.715 | 1430 | 2858.63 | |

tri10000 | 29976 | 1606 | 16495 | 1819 | 1.117 | 1804 | 4977.06 | |

(3) | rand6000 | 10293 | 1563 | 150 | 1932 | 0.748 | 1658 | 104.85 |

rand10000 | 17578 | 2535 | 869 | 3197 | 2.06 | 2692 | 432.23 | |

rand15000 | 26727 | 3758 | 2769 | 4698 | 4.861 | 4144 | 1313.14 | |

rand16000 | 28624 | 4002* | 5917 | 5039 | 5.176 | 4295 | 2027.7 | |

rand20000 | 35975 | 4963* | 13993 | 6273 | 8.053 | 5398 | 5017.02 | |

(4) | Gab500 | 949 | 115 | 238 | 146 | 0.006 | 129 | 18.95 |

Gab600 | 1174 | 135* | 3074 | 168 | 0.007 | 156 | 22.10 | |

Gab700 | 1302 | 162* | 5710 | 200 | 0.01 | 183 | 24.30 |

Graph G | |E(G)| | Greedy Alg. | PTAS | |||
---|---|---|---|---|---|---|

D_{Gr} | time | D_{PTAS} | time | |||

(1) | rd400 | 1183 | 78 | 0.007 | 74 | 351.93 |

pcb442 | 1286 | 76 | 0.908 | 78 | 10.86 | |

d657 | 1958 | 126 | 0.016 | 120 | 604.10 | |

pr1002 | 2972 | 190 | 0.032 | 182 | 1253.9 | |

(2) | tri11000 | 32972 | 2040 | 1.375 | 1958 | 12341.1 |

tri12000 | 35974 | 2186 | 1.607 | 2132 | 6865.08 | |

tri14000 | 41974 | 2576 | 2.462 | 2434 | 9208.72 | |

tri16000 | 47969 | 2917 | 2.839 | 2885 | 12794.4 | |

(3) | rand25000 | 45327 | 7772 | 12.467 | 6632 | 9470 |

(4) | Gab800 | 1533 | 227 | 0.012 | 205 | 24.30 |

Gab900 | 1758 | 254 | 0.016 | 231 | 344.50 | |

Gab1000 | 1901 | 280 | 0.019 | 259 | 781.89 | |

Gab1500 | 2870 | 426 | 0.042 | 385 | 960.71 |

## 5. Concluding Remarks

## Acknowledgements

## References

- Berge, C. Graphs and Hypergraphs; American Elsevier: New York, NY, USA, 1973. [Google Scholar]
- Liu, C. Introduction to Combinatorial Mathematics; McGraw-Hill: New York, NY, USA, 1963. [Google Scholar]
- Norman, R.; Harary, F.; Cartwright, D. Structural Models: An Introduction to the Theory of Directed Graphs; Wiley: Weinheim, Germany, 1966. [Google Scholar]
- Haynes, T.W.; Hedetniemi, S.T.; Slater, P.J. Domination in Graphs. In Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Haynes, T.W.; Hedetniemi, S.T.; Slater, P.J. Fundamentals of Domination in Graphs. In Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Garey, M.R.; Johnson, D.S. Computers and Intractability, a Guide to the Theory of NP-Completeness; Freeman: New York, NY, USA, 1979. [Google Scholar]
- Johnson, D.S. Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci
**1974**, 9, 256–278. [Google Scholar] [CrossRef] - Fiege, U. A threshold of ln n for approximating set cover. J. ACM
**1998**, 45, 634–652. [Google Scholar] [CrossRef] - Baker, B.S. Approximation algorithms for NP-complete problems on planar graphs. J. ACM
**1994**, 41, 153–180. [Google Scholar] [CrossRef] - Downey, R.G.; Fellows, M.R. Parameterized Complexity. In Monographs in Computer Science; Springer-Verlag: Berlin/Heidelberg, Germany, 1999. [Google Scholar]
- Fomin, F.V.; Grandoni, F.; Kratch, D. Some new techniques in design and analysis of exact (exponential) algorithms. Bull. EATCS
**2005**, 87, 47–77. [Google Scholar] - Alber, J.; Fellows, M.R.; Niedermeier, R. Polynomial time data reduction for dominating set. J. ACM
**2004**, 51, 363–384. [Google Scholar] [CrossRef] - Alber, J.; Bodlaender, H.L.; Fernau, H.; Kloks, T.; Niedermeier, R. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica
**2002**, 33, 461–493. [Google Scholar] [CrossRef] - Fomin, F.V.; Thilikos, D.M. Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM J. Comput.
**2006**, 36, 281–309. [Google Scholar] [CrossRef] - Kanj, I.A.; Perkovic, L. Improved Parameterized Algorithms for Planar Dominating Set. In Proceedings of the 27th Mathematical Foundations of Computer Science. LNCS 2420, Warsaw, Poland, Augaust, 2002; pp. 399–410.
- Robertson, N.; Seymour, P.D. Graph minors X. Obstructions to tree decomposition. J. Comb. Theory Ser. B
**1991**, 52, 153–190. [Google Scholar] [CrossRef] - Gu, Q.; Tamaki, H. Optimal branch-decomposition of planar graphs in O(n
^{3}) time. ACM Trans. Algorithm**2008**, 4, 30:1–30:13. [Google Scholar] [CrossRef] - Seymour, P.D.; Thomas, R. Call routing and the ratcatcher. Combinatorica
**1994**, 14, 217–241. [Google Scholar] [CrossRef] - Fomin, F.V.; Thilikos, D.M. New upper bounds on the decomposability of planar graphs. J. Graph Theory
**2006**, 51, 53–81. [Google Scholar] [CrossRef] - Dorn, F. Dynamic Programming and Fast Matrix Multiplication. In Proceedings of the 14th Annual European Symposium on Algorithms (ESA2006) LNCS 4168, Zurich, Switerland, September, 2006; pp. 280–291.
- Marzban, M.; Gu, Q.; Jia, X. Computational study on planar dominating set problem. Theor. Comput. Sci.
**2009**, 410, 5455–5466. [Google Scholar] [CrossRef] - Sanchis, L.A. Experimental analysis of heuristic algorithms for the dominating set problem. Algorithmica
**2002**, 33, 3–18. [Google Scholar] [CrossRef] - Hopcroft, J.; Tarjan, R. Efficient planarity testing. J. ACM
**1974**, 21, 549–568. [Google Scholar] [CrossRef] - Tamaki, H. A linear Time Heuristic for the Branch-decomposition of Planar Graphs. In Proceedings of the 11th Annual European Symposium, Budapest, Hungary, 16–19 September 2003; pp. 765–775.
- Reinelt, G. TSPLIB-A traveling salesman library. ORSA J. Comput.
**1991**, 3, 376–384. [Google Scholar] [CrossRef] - Library of Efficient Data Types and Algorithms, Version 5.2. Available online: http://www.algorithmic-solutions.com/leda/index.html (accessed on 1 July 2007).

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Marzban, M.; Gu, Q.-P.
Computational Study on a PTAS for Planar Dominating Set Problem. *Algorithms* **2013**, *6*, 43-59.
https://doi.org/10.3390/a6010043

**AMA Style**

Marzban M, Gu Q-P.
Computational Study on a PTAS for Planar Dominating Set Problem. *Algorithms*. 2013; 6(1):43-59.
https://doi.org/10.3390/a6010043

**Chicago/Turabian Style**

Marzban, Marjan, and Qian-Ping Gu.
2013. "Computational Study on a PTAS for Planar Dominating Set Problem" *Algorithms* 6, no. 1: 43-59.
https://doi.org/10.3390/a6010043