Document Open Access Logo

The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

Authors Robert Krauthgamer, Ohad Trabelsi

Thumbnail PDF


  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Author Details

Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel
Ohad Trabelsi
  • Weizmann Institute of Science, Rehovot, Israel

Cite AsGet BibTex

Robert Krauthgamer and Ohad Trabelsi. The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 45:1-45:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


In the Set Cover problem, the input is a ground set of n elements and a collection of m sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time O(mn2^n) [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed epsilon>0, no algorithm can solve Set Cover in time 2^{(1-epsilon)n} poly(m), even if set sizes are bounded by Delta=Delta(epsilon). We show strong connections between this problem and kTree, a special case of Subgraph Isomorphism where the input is an n-node graph G and a k-node tree T, and the goal is to determine whether G has a subgraph isomorphic to T. First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of size Delta=O(1/epsilon * log n), and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known 2^n poly(n)-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree, which is kTree with k=n and arbitrary directions to the edges of G and T. This would also improve the running time for Directed Hamiltonicity, for which no algorithm significantly faster than 2^n poly(n) is known despite extensive research. Second, we prove that if p-Partial Cover, a parameterized version of Set Cover that requires covering at least p elements, cannot be solved significantly faster than 2^n poly(m) (an assumption even weaker than Log-SeCoCo) then kTree cannot be computed significantly faster than 2^k poly(n), the running time of the Koutis and Williams' algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Discrete optimization
  • Theory of computation → Parameterized complexity and exact algorithms
  • Conditional lower bounds
  • Hardness in P
  • Set Cover Conjecture
  • Subgraph Isomorphism


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-Based Lower Bounds for Subset Sum and Bicriteria Path. In 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '19, pages 41-57, 2019. URL:
  2. Amir Abboud, Virginia Vassilevska-Williams, and Huacheng Yu. Matching Triangles and Basing Hardness on an Extremely Popular Conjecture. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 41-50. ACM, 2015. URL:
  3. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, July 1995. URL:
  4. Richard Bellman. Combinatorial processes and dynamic programming. In Combinatorial analysis, Proceedings of Symposia in Applied Mathematics, pages 217-249. American Mathematical Society, 1960. URL:
  5. Richard Bellman. Dynamic Programming Treatment of the Travelling Salesman Problem. J. ACM, 9(1):61-63, 1962. URL:
  6. Andreas Bjorklund. Determinant sums for undirected hamiltonicity. SIAM Journal on Computing, 43(1):280-299, 2014. URL:
  7. Andreas Björklund. Below All Subsets for Some Permutational Counting Problems . In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016), volume 53 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:11, 2016. URL:
  8. Andreas Björklund, Dell Holger, and Thore Husfeldt. The Parity of Set Systems under Random Restrictions with Applications to Exponential Time Problems. In 42nd International Colloquium on Automata, Languages and Programming (ICALP 2015), volume 9134, pages 231-242. Springer, 2015. URL:
  9. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set Partitioning via Inclusion-Exclusion. SIAM J. Comput., 39(2):546-563, July 2009. URL:
  10. Andreas Björklund, Thore Husfeldt, Kaski Ptteri, and Mikko Koivisto. Narrow Sieves for Parameterized Paths and Packings. Journal of Computer and System Sciences, 87:119-139, 2017. URL:
  11. Andreas Björklund, Petteri Kaski, and Łukasz Kowalik. Constrained Multilinear Detection and Generalized Graph Motifs. Algorithmica, 74(2):947-967, 2016. URL:
  12. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On Problems As Hard As CNF-SAT. ACM Transactions on Algorithms, 12(3):41:1-41:24, 2016. URL:
  13. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socała. Tight Bounds for Graph Homomorphism and Subgraph Isomorphism. In 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1643-1649. SIAM, 2016. URL:
  14. Anders Dessmark, Andrzej Lingas, and Andrzej Proskurowski. Faster Algorithms for Subgraph Isomorphism of k-Connected Partial k-Trees. Algorithmica, 27(3):337-347, January 2000. URL:
  15. Fedor V. Fomin, Dieter Kratsch, and Gerhard J. Woeginger. Exact (Exponential) Algorithms for the Dominating Set Problem. In 30th International Conference on Graph-Theoretic Concepts in Computer Science, WG'04, pages 245-256. Springer-Verlag, 2004. URL:
  16. Godfrey H. Hardy and Srinivasa Ramanujan. Asymptotic Formulaæ in Combinatory Analysis. Proceedings of the London Mathematical Society, s2-17(1):75-115, 1918. URL:
  17. Michael Held and Richard M. Karp. A Dynamic Programming Approach to Sequencing Problems. In Proceedings of 16th ACM National Meeting, ACM '61, pages 71.201-71.204. ACM, 1961. URL:
  18. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, March 2001. URL:
  19. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have Strongly Exponential Complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL:
  20. Yoichi Iwata and Yuichi Yoshida. On the Equivalence among Problems of Bounded Width. In 23rd Annual European Symposium on Algorithms (ESA 2015), pages 754-765. Springer, 2015. URL:
  21. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. The IBM Research Symposia Series. Springer US, 1972. URL:
  22. Mikko Koivisto. Partitioning into Sets of Bounded Cardinality. In Parameterized and Exact Computation (IWPEC 2009), volume 5917 of Lecture Notes in Computer Science, pages 258-263. Springer-Verlag, 2009. URL:
  23. Ioannis Koutis and Ryan Williams. LIMITS and applications of group algebras for parameterized problems. ACM Trans. Algorithms, 12(3):31:1-31:18, May 2016. URL:
  24. Łukasz Kowalik and Juho Lauri. On Finding Rainbow and Colorful Paths. Theoretical Computer Science, 628(C):110-114, 2016. URL:
  25. R. Krithika, Abhishek Sahu, and Prafullkumar Tale. Dynamic Parameterized Problems. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), volume 63 of Leibniz International Proceedings in Informatics (LIPIcs), pages 19:1-19:14. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. URL:
  26. Andrzej Lingas. Subgraph Isomorphism for Biconnected Outerplanar Graphs in Cubic Time. Theoretical Computer Science, 63(3):295-302, 1989. URL:
  27. László Lovász and Michael D Plummer. Matching theory, volume 367. American Mathematical Society, 2009. Google Scholar
  28. Dániel Marx and Michal Pilipczuk. Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask). In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), volume 25 of Leibniz International Proceedings in Informatics (LIPIcs), pages 542-553. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014. URL:
  29. Jiří Matoušek and Robin Thomas. On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Mathematics, 108(1):343-364, 1992. URL:
  30. Jesper Nederlof. Finding Large Set Covers Faster via the Representation Method. In 24th Annual European Symposium on Algorithms (ESA 2016), volume 57 of Leibniz International Proceedings in Informatics (LIPIcs), pages 69:1-69:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. URL:
  31. Albert Nijenhuis and Herbert S. Will. Combinatorial Algorithms: For Computers and Hard Calculators. Academic Press, 2nd edition, 1978. Google Scholar
  32. Ohad Trabelsi. Nearly Optimal Time Bounds for kPath in Hypergraphs. CoRR, 2018. URL:
  33. Gerhard J. Woeginger. Exact Algorithms for NP-hard Problems: A Survey. In Michael Jünger, Gerhard Reinelt, and Giovanni Rinaldi, editors, Combinatorial Optimization - Eureka, You Shrink!, pages 185-207. Springer-Verlag, 2003. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail