Directed Graph Minors and Serial-Parallel Width
Graph minors are a primary tool in understanding the structure of undirected graphs, with many conceptual and algorithmic implications. We propose new variants of directed graph minors and directed graph embeddings, by modifying familiar definitions. For the class of 2-terminal directed acyclic graphs (TDAGs) our two definitions coincide, and the class is closed under both operations. The usefulness of our directed minor operations is demonstrated by characterizing all TDAGs with serial-parallel width at most k; a class of networks known to guarantee bounded negative externality in nonatomic routing games. Our characterization implies that a TDAG has serial-parallel width of 1 if and only if it is a directed series-parallel graph. We also study the computational complexity of finding a directed minor and computing the serial-parallel width.
directed minors
pathwidth
Theory of computation~Graph algorithms analysis
Mathematics of computing~Graph theory
44:1-44:14
Regular Paper
https://tinyurl.com/y9hcukyz
Argyrios
Deligkas
Argyrios Deligkas
Leverhulme Research Centre, University of Liverpool, UK
Reshef
Meir
Reshef Meir
Faculty of Industrial Engineering and Management, Technion, Israel
10.4230/LIPIcs.MFCS.2018.44
I. Ashlagi, D. Monderer, and M. Tennenholtz. Two-terminal routing games with unknown active players. Artificial Intelligence, 173(15):1441-1455, 2009.
M. Babaioff, R. Kleinberg, and C. Papadimitriou. Congestion games with malicious players. In EC, pages 103-112. ACM, 2007.
A. Blum and M. Furst. Fast planning through planning graph analysis. Artificial intelligence, 90(1-2):281-300, 1997.
C. Chang and J. Slagle. An admissible and optimal algorithm for searching AND/OR graphs. Artificial Intelligence, 2(2):117-128, 1971.
B. Codenotti and M. Leoncini. Parallel Complexity of Linear System Solution. World Scientific, 1991.
D. Cohen, M. Cooper, P. Jeavons, and S. Zivny. Tractable classes of binary csps defined by excluded topological minors. In IJCAI, pages 1945-1951, 2015.
G. Cooper. The computational complexity of probabilistic inference using bayesian belief networks. Artificial intelligence, 42(2-3):393-405, 1990.
E. Demaine, M. Hajiaghayi, and K. Kawarabayashi. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In FOCS, pages 637-646. IEEE, 2005.
R.J Duffin. Topology of series-parallel networks. Journal of Mathematical Analysis and Applications, 10(2):303-318, 1965.
D. Eppstein. Parallel recognition of series-parallel graphs. Inf. and Comp., 98(1):41-55, 1992.
A. Epstein, M. Feldman, and Y. Mansour. Efficient graph topologies in network routing games. Games and Economic Behavior, 66(1):115-125, 2009.
Fortune, Hopcroft, and Wyllie. The directed subgraph homeomorphism problem. TCS: Theoretical Computer Science, 10, 1980.
V. Gogate and R. Dechter. A complete anytime algorithm for treewidth. In UAI, pages 201-208, 2004.
R. Holzman and N. Law-Yone. Network structure and strong equilibrium in route selection games. Mathematical Social Sciences, 46(2):193-205, 2003.
E. Horvitz, J. Breese, and M. Henrion. Decision theory in expert systems and artificial intelligence. International journal of approximate reasoning, 2(3):247-302, 1988.
A. Jakoby, M. Liśkiewicz, and R. Reischuk. Space efficient algorithms for directed series-parallel graphs. Journal of Algorithms, 60(2):85-114, 2006.
T. Johnson, N. Robertson, P. Seymour, and R. Thomas. Directed tree-width. Journal of Combinatorial Theory, Series B, 82(1):138-154, 2001.
T. Johnson, N. Robertson, P. Seymour, and R. Thomas. Excluding a grid minor in planar digraphs. arXiv:1510.00473, 2015.
K. Kawarabayashi and S. Kreutzer. Towards the graph minor theorems for directed graphs. In ICALP, pages 3-10. Springer, 2015.
S. Kintali and Q. Zhang. Forbidden directed minors and kelly-width. arXiv:1308.5170, 2013.
J. Kleinberg and S. Oren. Time-inconsistent planning: a computational problem in behavioral economics. In EC, pages 547-564. ACM, 2014.
S. Kreutzer. Nowhere crownful classes of directed graphs. In Encyclopedia of Algorithms, pages 1416-1419. Springer, 2016.
Casimir Kuratowski. Sur le probleme des courbes gauches en topologie. Fundamenta mathematicae, 15(1):271-283, 1930.
Michael Lampis, Georgia Kaouri, and Valia Mitsou. On the algorithmic effectiveness of digraph decompositions and complexity measures. Discrete Optimization, 8(1):129-138, 2011. Parameterized Complexity of Discrete Optimization. URL: http://dx.doi.org/10.1016/j.disopt.2010.03.010.
http://dx.doi.org/10.1016/j.disopt.2010.03.010
László Lovász. Graph minor theory. Bulletin of the American Mathematical Society, 43(1):75-86, 2006.
A. Mackworth. Consistency in networks of relations. In Readings in AI, pages 69-78. Tioga Publ. Col., 1981.
K. Meer. An extended tree-width notion for directed graphs related to the computation of permanents. Computer Science-Theory and Applications, pages 247-260, 2011.
R. Meir and D. Parkes. Playing the wrong game: Bounding negative externalities in diverse populations of agents. In AAMAS'18, 2018. To appear.
I. Milchtaich. Network topology and the efficiency of equilibrium. GEB, 57:321-346, 2006.
E. Nikolova and N. Stier-Moses. The burden of risk aversion in mean-risk selfish routing. In EC, pages 489-506, 2015.
Ira Pohl. Heuristic search viewed as path finding in a graph. Artificial intelligence, 1(3-4):193-204, 1970.
N. Robertson and P. Seymour. Graph minors. II. algorithmic aspects of tree-width. Journal of algorithms, 7(3):309-322, 1986.
N. Robertson and P. Seymour. Graph minors. xx. wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004.
M. Rowland, A. Pacchiano, and A. Weller. Conditions beyond treewidth for tightness of higher-order lp relaxations. In AI and Statistics, pages 10-18, 2017.
M. Safari. D-width: A more natural measure for directed tree width. In MFCS, pages 745-756. Springer, 2005.
C. Shannon. The synthesis of two-terminal switching circuits. Bell Labs Technical Journal, 28(1):59-98, 1949.
K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. JACM, 29(3):623-641, 1982.
P. Tang, Y. Teng, Z. Wang, S. Xiao, and Y. Xu. Computational issues in time-inconsistent planning. In AAAI, pages 3665-3671, 2017.
J. Vygen. NP-completeness of some edge-disjoint paths problems. Discrete Applied Mathematics, 61(1):83-90, 1995.
K. Wagner. Über eine eigenschaft der ebenen komplexe. Mathematische Annalen, 114(1):570-590, 1937.
Z. Wang, J. Zhang, J. Feng, and Z. Chen. Knowledge graph embedding by translating on hyperplanes. In AAAI, pages 1112-1119, 2014.
Argyrios Deligkas and Reshef Meir
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode