Quasi-bipartite graph

In the mathematical field of graph theory, an instance of the Steiner tree problem (consisting of an undirected graph G and a set R of terminal vertices that must be connected to each other) is said to be quasi-bipartite if the non-terminal vertices in G form an independent set, i.e. if every edge is incident on at least one terminal. This generalizes the concept of a bipartite graph: if G is bipartite, and R is the set of vertices on one side of the bipartition, the set R is automatically independent.

The first graph has non-terminal points which are directly connected, and is therefore non-quasi-bipartite. The second has no non-terminal points which are directly connected, and is therefore quasi-bipartite.

This concept was introduced by Rajagopalan and Vazirani [1] who used it to provide a (3/2 + ε) approximation algorithm for the Steiner tree problem on such instances. Subsequently, the ε factor was removed by Rizzi[2] and a 4/3 approximation algorithm was obtained by Chakrabarty et al.[3] The same concept has been used by subsequent authors on the Steiner tree problem, e.g.[4] Robins and Zelikovsky[5] proposed an approximation algorithm for Steiner tree problem which on quasi-bipartite graphs has approximation ratio 1.28. The complexity of Robins and Zelikovsky's algorithm is O(m n2), where m and n are the numbers of terminals and non-terminals in the graph, respectively. In 2012, Goemans et al.[6] gave a 73/60 ≈ 1.217-approximation algorithm for the Steiner tree problem on quasi-bipartite graphs; an algorithm achieving the same approximation factor was previously known for the special case of quasi-bipartite graphs with unit cost edges.[7]

References

  1. ^ Rajagopalan, Sridhar; Vazirani, Vijay V. (1999), "On the bidirected cut relaxation for the metric Steiner tree problem", Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 742–751.
  2. ^ Rizzi, Romeo (2003), "On Rajagopalan and Vazirani's 3/2-approximation bound for the Iterated 1-Steiner heuristic", Inf. Process. Lett., 86 (6): 335–338, doi:10.1016/S0020-0190(03)00210-2.
  3. ^ Chakrabarty, Deeparnab; Devanur, Nikhil R.; Vazirani, Vijay V. (2008), "New Geometry-Inspired Relaxations and Algorithms for the Metric Steiner Tree Problem", Proc. 13th IPCO, Lecture Notes in Computer Science, vol. 5035, pp. 344–358, doi:10.1007/978-3-540-68891-4_24, ISBN 978-3-540-68886-0.
  4. ^ Gröpl, Clemens; Hougardy, Stefan; Nierhoff, Till; Prömel, Hans Jürgen (2001), "Lower Bounds for Approximation Algorithms for the Steiner Tree Problem", Graph-Theoretic Concepts in Computer Science : 27th International Workshop, WG 2001, Lecture Notes in Computer Science, vol. 2204, Springer-Verlag, Lecture Notes in Computer Science 2204, pp. 217–228, doi:10.1007/3-540-45477-2_20, ISBN 978-3-540-42707-0.
  5. ^ Robins, Gabriel; Zelikovsky, Alexander (2000), "Improved Steiner tree approximation in graphs", Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 770–779.
  6. ^ Goemans, Michel; Olver, Neil; Rothvoss, Thomas; Zenklusen, Rico (2012). "Matroids and integrality gaps for hypergraphic steiner tree relaxations". Proceedings of the forty-fourth annual ACM symposium on Theory of computing. p. 1161-1176. doi:10.1145/2213977.2214081. ISBN 9781450312455. S2CID 13424446.
  7. ^ Gröpl, Clemens; Hougardy, Stefan; Nierhoff, Till; Prömel, Hans Jürgen (2002), "Steiner trees in uniformly quasi-bipartite graphs", Information Processing Letters, 83 (4): 195–200, doi:10.1016/S0020-0190(01)00335-0.