Metric dimension (graph theory)

In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.

Detailed definition

For an ordered subset $W=\{w_{1},w_{2},\dots ,w_{k}\}$ of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple $r(v|W)=(d(v,w_{1}),d(v,w_{2}),\dots ,d(v,w_{k}))$ , where d(x,y) represents the distance between the vertices x and y. The set W is a resolving set (or locating set) for G if every two vertices of G have distinct representations. The metric dimension of G is the minimum cardinality of a resolving set for G. A resolving set containing a minimum number of vertices is called a basis (or reference set) for G. Resolving sets for graphs were introduced independently by Slater (1975) and Harary & Melter (1976), while the concept of a resolving set and that of metric dimension were defined much earlier in the more general context of metric spaces by Blumenthal in his monograph Theory and Applications of Distance Geometry. Graphs are special examples of metric spaces with their intrinsic path metric.

Trees

If a tree is a path, its metric dimension is one. Otherwise, let L denote the set of leaves, degree-one vertices in the tree. Let K be the set of vertices that have degree greater than two, and that are connected by paths of degree-two vertices to one or more leaves. Then the metric dimension is |L| − |K|. A basis of this cardinality may be formed by removing from L one of the leaves associated with each vertex in K.^[1] The same algorithm is valid for the line graph of the tree, and thus any tree and its line graph have the same metric dimension.^[2]

Properties

In Chartrand et al. (2000), it is proved that:

The metric dimension of a graph $G$ is 1 if and only if $G$ is a path.
The metric dimension of an $n$ -vertex graph is $n - 1$ if and only if it is a complete graph.
The metric dimension of an $n$ -vertex graph is $n - 2$ if and only if the graph is a complete bipartite graph $K s, t$ , a split graph $K_{s}+{\overline {K_{t}}}(s\geq 1,t\geq 2)$ , or $K_{s}+(K_{1}\cup K_{t})(s,t\geq 1)$ .

Relations between the order, the metric dimension and the diameter

Khuller, Raghavachari & Rosenfeld (1996) prove the inequality $n\leq D^{\beta }+\beta$ for any $n$ -vertex graph with diameter $D$ and metric dimension $\beta$ . This bounds follows from the fact that each vertex that is not in the resolving set is uniquely determined by a distance vector of length $\beta$ with each entry being an integer between 1 and $D$ (there are precisely $D^{\beta }$ such vectors). However, the bound is only achieved for $D\leq 3$ or $\beta =1$ ; the more precise bound $n\leq \left(\lfloor 2D/3\rfloor +1\right)^{\beta }+\beta \sum _{i=1}^{\lceil D/3\rceil }(2i-1)^{\beta -1}$ is proved by Hernando et al. (2010).

For specific graph classes, smaller bounds can hold. For example, Beaudou et al. (2018) proved that $n\leq (\beta D+4)(D+2)/8$ for trees (the bound being tight for even values of $D$ ), and a bound of the form $n=O(D^{2}\beta )$ for outerplanar graphs. The same authors proved that $n\leq (D\beta +1)^{t-1}$ for graphs with no complete graph of order $t$ as a minor and also gave bounds for chordal graphs and graphs of bounded treewidth. The authors Foucaud et al. (2017a) proved bounds of the form $n=O(D\beta ^{2})$ for interval graphs and permutation graphs, and bounds of the form $n=O(D\beta )$ for unit interval graphs, bipartite permutation graphs and cographs.

Computational complexity

Decision complexity

Deciding whether the metric dimension of a graph is at most a given integer is NP-complete.^[3] It remains NP-complete for bounded-degree planar graphs,^[4] split graphs, bipartite graphs and their complements, line graphs of bipartite graphs,^[5] unit disk graphs,^[6] interval graphs of diameter 2 and permutation graphs of diameter 2,^[7] and graphs of bounded treewidth.^[8]

For any fixed constant k, the graphs of metric dimension at most k can be recognized in polynomial time, by testing all possible k-tuples of vertices, but this algorithm is not fixed-parameter tractable (for the natural parameter k, the solution size). Answering a question posed by Lokshtanov (2010), Hartung & Nichterlein (2013) show that the metric dimension decision problem is complete for the parameterized complexity class W[2], implying that a time bound of the form n^O(k) as achieved by this naive algorithm is likely optimal and that a fixed-parameter tractable algorithm (for the parameterization by k) is unlikely to exist. Nevertheless, the problem becomes fixed-parameter tractable when restricted to interval graphs,^[7] and more generally to graphs of bounded tree-length,^[9] such as chordal graphs, permutation graphs or asteroidal-triple-free graphs.

Deciding whether the metric dimension of a tree is at most a given integer can be done in linear time^[10] Other linear-time algorithms exist for cographs,^[5] chain graphs,^[11] and cactus block graphs^[12] (a class including both cactus graphs and block graphs). The problem may be solved in polynomial time on outerplanar graphs.^[4] It may also be solved in polynomial time for graphs of bounded cyclomatic number,^[5] but this algorithm is again not fixed-parameter tractable (for the parameter "cyclomatic number") because the exponent in the polynomial depends on the cyclomatic number. There exist fixed-parameter tractable algorithms to solve the metric dimension problem for the parameters "vertex cover",^[13] "max leaf number",^[14] and "modular width".^[9] Graphs with bounded cyclomatic number, vertex cover number or max leaf number all have bounded treewidth, however it is an open problem to determine the complexity of the metric dimension problem even on graphs of treewidth 2, that is, series–parallel graphs.^[9]

Approximation complexity

The metric dimension of an arbitrary n-vertex graph may be approximated in polynomial time to within an approximation ratio of $2\log n$ by expressing it as a set cover problem, a problem of covering all of a given collection of elements by as few sets as possible in a given family of sets. ^[15] In the set cover problem formed from a metric dimension problem, the elements to be covered are the ${\tbinom {n}{2}}$ pairs of vertices to be distinguished, and the sets that can cover them are the sets of pairs that can be distinguished by a single chosen vertex. The approximation bound then follows by applying standard approximation algorithms for set cover. An alternative greedy algorithm that chooses vertices according to the difference in entropy between the equivalence classes of distance vectors before and after the choice achieves an even better approximation ratio, $\log n+\log \log _{2}n+1$ .^[16] This approximation ratio is close to best possible, as under standard complexity-theoretic assumptions a ratio of $(1-\epsilon )\log n$ cannot be achieved in polynomial time for any $\epsilon >0$ .^[16] The latter hardness of approximation still holds for instances restricted to subcubic graphs,^[13] and even to bipartite subcubic graphs.^[17]

References

Notes

^ Slater 1975; Harary & Melter 1976; Khuller, Raghavachari & Rosenfeld 1996. Note Slater's nonstandard definition of the leaves of a tree.
^ Feng, Xu & Wang 2013.
^ Garey & Johnson 1979.
^ ^a ^b Díaz et al. 2012.
^ ^a ^b ^c Epstein, Levin & Woeginger 2012.
^ Hoffmann & Wanke 2013.
^ ^a ^b Foucaud et al. 2017b.
^ Li & Pilipczuk 2022.
^ ^a ^b ^c Belmonte et al. 2015.
^ Slater 1975; Harary & Melter 1976.
^ Fernau et al. 2015.
^ Hoffmann, Elterman & Wanke 2016.
^ ^a ^b Hartung & Nichterlein 2013.
^ Eppstein 2015.
^ Khuller, Raghavachari & Rosenfeld 1996.
^ ^a ^b Hauptmann, Schmied & Viehmann 2012.
^ Hartung 2014.

Bibliography

Beaudou, Laurent; Dankelmann, Peter; Foucaud, Florent; Henning, Michael A.; Mary, Arnaud; Parreau, Aline (2018), "Bounding the order of a graph using its diameter and metric dimension: a study through tree decompositions and VC dimension", SIAM Journal on Discrete Mathematics, 32 (2): 902–918, arXiv:1610.01475, doi:10.1137/16M1097833, S2CID 51882750
Belmonte, R.; Fomin, F. V.; Golovach, P. A.; Ramanujan, M. S. (2015), "Metric dimension of bounded width graphs", in Italiano, G. F.; Pighizzini, G.; Sannella, D. T. (eds.), Mathematical Foundations of Computer Science 2015 – MFCS 2015: 40th International Symposium, Milan, Italy, August 24-28, 2015, Proceedings, Lecture Notes in Computer Science, vol. 9235, Springer, pp. 115–126, doi:10.1007/978-3-662-48054-0_10, ISBN 978-3-662-48053-3.
Blumenthal, L.M. (1953), Theory and Applications of Distance Geometry, Clarendon, Oxford.
Bonnet, E.; Purohit, N. (2019), "Metric Dimension Parameterized by Treewidth", in Jansen, B. M. P.; Telle, J. A. (eds.), Parameterized and Exact Computation 2019 – IPEC 2019: 14th International Symposium, Proceedings, Leibniz International Proceedings in Informatics (LIPIcs), vol. 148, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, pp. 5:1–5:15, arXiv:1907.08093, doi:10.4230/LIPIcs.IPEC.2019.5.
Buczkowski, P.; Chartrand, G.; Poisson, C.; Zhang, P. (2003), "On $k$ -dimensional graphs and their bases", Periodica Mathematica Hungarica, 46 (1): 9–15, doi:10.1023/A:1025745406160, MR 1975342, S2CID 33390310.
Chartrand, G.; Eroh, L.; Johnson, M. A.; Oellermann, O. R. (2000), "Resolvability in graphs and the metric dimension of a graph", Discrete Applied Mathematics, 105 (1–3): 99–113, doi:10.1016/S0166-218X(00)00198-0, hdl:10338.dmlcz/127843, MR 1780464.
Díaz, J.; Pottonen, O.; Serna, M. J.; van Leeuwen, E. J. (2012), "On the complexity of metric dimension" (PDF), in Epstein, Leah; Ferragina, Paolo (eds.), Algorithms – ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10-12, 2012, Proceedings, Lecture Notes in Computer Science, vol. 7501, Springer, pp. 419–430, arXiv:1107.2256, doi:10.1007/978-3-642-33090-2_37, ISBN 978-3-642-33089-6.
Eppstein, David (2015), "Metric dimension parameterized by max leaf number", Journal of Graph Algorithms and Applications, 19 (1): 313–323, arXiv:1506.01749, doi:10.7155/jgaa.00360, S2CID 1318601.
Epstein, Leah; Levin, Asaf; Woeginger, Gerhard J. (2012), "The (weighted) metric dimension of graphs: hard and easy cases", in Golumbic, Martin Charles; Stern, Michal; Levy, Avivit; et al. (eds.), Graph-Theoretic Concepts in Computer Science: 38th International Workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7551, pp. 114–125, doi:10.1007/978-3-642-34611-8_14, ISBN 978-3-642-34610-1.
Feng, Min; Xu, Min; Wang, Kaishun (2013), "On the metric dimension of line graphs", Discrete Applied Mathematics, 161 (6): 802–805, arXiv:1107.4140, doi:10.1016/j.dam.2012.10.018, S2CID 36010185.
Fernau, Henning; Heggernes, Pinar; van 't Hof, Pim; Meister, Daniel; Saei, Reza (2015), "Computing the metric dimension for chain graphs", Information Processing Letters, 115 (9): 671–676, doi:10.1016/j.ipl.2015.04.006.
Foucaud, Florent; Mertzios, George B.; Naserasr, Reza; Parreau, Aline; Valicov, Petru (2017a), "Identification, location-domination and metric dimension on interval and permutation graphs. I. Bounds", Theoretical Computer Science, 68: 43–58, arXiv:1507.08164, doi:10.1016/j.tcs.2017.01.006, S2CID 25244200
Foucaud, Florent; Mertzios, George B.; Naserasr, Reza; Parreau, Aline; Valicov, Petru (2017b), "Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity", Algorithmica, 78 (3): 914–944, arXiv:1405.2424, doi:10.1007/s00453-016-0184-1, S2CID 1520161.
Garey, M. R.; Johnson, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, ISBN 0-7167-1045-5 A1.5: GT61, p. 204.
Harary, F.; Melter, R. A. (1976), "On the metric dimension of a graph", Ars Combinatoria, 2: 191–195, MR 0457289.
Hartung, Sepp (2014), Exploring parameter spaces in coping with computational intractability, PhD thesis, Technische Universität Berlin, retrieved 2015-09-15.
Hartung, Sepp; Nichterlein, André (2013), "On the parameterized and approximation hardness of metric dimension", 2013 IEEE Conference on Computational Complexity (CCC), Stanford, CA, USA, June 5-7, 2013, Proceedings, IEEE, pp. 266–276, arXiv:1211.1636, doi:10.1109/CCC.2013.36, ISBN 978-0-7695-4997-2, S2CID 684505.
Hauptmann, Mathias; Schmied, Richard; Viehmann, Claus (2012), "Approximation complexity of metric dimension problem", Journal of Discrete Algorithms, 14: 214–222, doi:10.1016/j.jda.2011.12.010, MR 2922072.
Hernando, Carmen; Mora, Mercè; Pelayo, Ignacio M.; Seara, Carlos; Wood, David R. (2010), "Extremal graph theory for metric dimension and diameter", Electronic Journal of Combinatorics, 17: #R30, doi:10.37236/302, hdl:2117/8261.
Hoffmann, Stefan; Elterman, Alina; Wanke, Egon (2016), "A linear time algorithm for metric dimension of cactus block graphs", Theoretical Computer Science, 630: 43–62, doi:10.1016/j.tcs.2016.03.024
Hoffmann, Stefan; Wanke, Egon (2013), "Metric Dimension for Gabriel Unit Disk Graphs Is NP-Complete", in Bar-Noy, Amotz; Halldórsson, Magnús M. (eds.), Algorithms for Sensor Systems: 8th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities, ALGOSENSORS 2012, Ljubljana, Slovenia, September 13-14, 2012, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7718, Springer, pp. 90–92, arXiv:1306.2187, doi:10.1007/978-3-642-36092-3_10, ISBN 978-3-642-36091-6, S2CID 9740623.
Khuller, S.; Raghavachari, B.; Rosenfeld, A. (1996), "Landmarks in graphs", Discrete Applied Mathematics, 70 (3): 217–229, doi:10.1016/0166-218x(95)00106-2, hdl:10338.dmlcz/140702.
Li, Shaohua; Pilipczuk, Marcin (July 2022), "Hardness of metric dimension in graphs of constant treewidth", Algorithmica, 84 (11): 3110–3155, arXiv:2102.09791, doi:10.1007/s00453-022-01005-y, S2CID 231979414
Lokshtanov, Daniel (2010), "Open problems – Parameterized complexity and approximation algorithms: Metric Dimension", in Demaine, Erik D.; Hajiaghayi, MohammadTaghi; Marx, Dániel (eds.), Parameterized Complexity and Approximation Algorithms, Dagstuhl Seminar Proceedings, Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, doi:10.4230/DagSemProc.09511.3.
Slater, P. J. (1975), "Leaves of trees", Proc. 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congressus Numerantium, vol. 14, Winnipeg: Utilitas Math., pp. 549–559, MR 0422062.
Slater, P. J. (1988), "Dominating and reference sets in a graph", Journal of Mathematical and Physical Sciences, 22 (4): 445–455, MR 0966610.

[1] Slater 1975; Harary & Melter 1976; Khuller, Raghavachari & Rosenfeld 1996. Note Slater's nonstandard definition of the leaves of a tree.

[FOOTNOTEFengXuWang2013-2] Feng, Xu & Wang 2013.

[FOOTNOTEGareyJohnson1979-3] Garey & Johnson 1979.

[FOOTNOTEDíazPottonenSernavan_Leeuwen2012-4] Díaz et al. 2012.

[FOOTNOTEEpsteinLevinWoeginger2012-5] Epstein, Levin & Woeginger 2012.

[FOOTNOTEHoffmannWanke2013-6] Hoffmann & Wanke 2013.

[FOOTNOTEFoucaudMertziosNaserasrParreau2017b-7] Foucaud et al. 2017b.

[FOOTNOTELiPilipczuk2022-8] Li & Pilipczuk 2022.

[FOOTNOTEBelmonteFominGolovachRamanujan2015-9] Belmonte et al. 2015.

[10] Slater 1975; Harary & Melter 1976.

[FOOTNOTEFernauHeggernesvan_'t_HofMeister2015-11] Fernau et al. 2015.

[FOOTNOTEHoffmannEltermanWanke2016-12] Hoffmann, Elterman & Wanke 2016.

[FOOTNOTEHartungNichterlein2013-13] Hartung & Nichterlein 2013.

[FOOTNOTEEppstein2015-14] Eppstein 2015.

[FOOTNOTEKhullerRaghavachariRosenfeld1996-15] Khuller, Raghavachari & Rosenfeld 1996.

[FOOTNOTEHauptmannSchmiedViehmann2012-16] Hauptmann, Schmied & Viehmann 2012.

[FOOTNOTEHartung2014-17] Hartung 2014.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]