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# Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Matrix Parameters

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## Acknowledgements

The authors are grateful to Koblich for enlightening discussions about communication complexity.

## Cite As

Carla Groenland, Isja Mannens, Jesper Nederlof, Marta Piecyk, and Paweł Rzążewski. Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Matrix Parameters. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 77:1-77:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.77

## Abstract

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). In the graph homomorphism problem, denoted by Hom(H), the graph H is fixed and we need to determine if there exists a homomorphism from an instance graph G to H. We study the complexity of the problem parameterized by the cutwidth of G, i.e., we assume that G is given along with a linear ordering v_1,…,v_n of V(G) such that, for each i ∈ {1,…,n-1}, the number of edges with one endpoint in {v_1,…,v_i} and the other in {v_{i+1},…,v_n} is at most k. We aim, for each H, for algorithms for Hom(H) running in time c_H^k n^𝒪(1) and matching lower bounds that exclude c_H^{k⋅o(1)} n^𝒪(1) or c_H^{k(1-Ω(1))} n^𝒪(1) time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call mimsup(H). Our main contribution is strong evidence of a close connection between c_H and mimsup(H): - an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most mimsup(H)^k, - lower bounds that show that for almost all graphs H indeed we have c_H ≥ mimsup(H), assuming the (Strong) Exponential-Time Hypothesis, and - an algorithm with running time exp(𝒪(mimsup(H)⋅k log k)) n^𝒪(1). In the last result we do not need to assume that H is a fixed graph. Thus, as a consequence, we obtain that the problem of deciding whether G admits a homomorphism to H is fixed-parameter tractable, when parameterized by cutwidth of G and mimsup(H). The parameter mimsup(H) can be thought of as the p-th root of the maximum induced matching number in the graph obtained by multiplying p copies of H via a certain graph product, where p tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of H. Such parameters play a central role in, among others, algebraic complexity theory and additive combinatorics. Our results tightly link the parameterized complexity of a problem to such an asymptotic matrix parameter for the first time.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• graph homomorphism
• cutwidth
• asymptotic matrix parameters

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## References

1. Noga Alon and Eyal Lubetzky. The shannon capacity of a graph and the independence numbers of its powers. IEEE Trans. Inf. Theory, 52(5):2172-2176, 2006. URL: https://doi.org/10.1109/TIT.2006.872856.
2. Srinivasan Arunachalam, Péter Vrana, and Jeroen Zuiddam. The asymptotic induced matching number of hypergraphs: Balanced binary strings. The Electronic Journal of Combinatorics, 27(3), 2020. URL: https://doi.org/10.37236/9019.
3. Andreas Blatter, Jan Draisma, and Filip Rupniewski. Countably many asymptotic tensor ranks. arXiv, 2022. URL: https://arxiv.org/abs/2212.12219.
4. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
5. Glencora Borradaile and Hung Le. Optimal dynamic program for r-domination problems over tree decompositions. In Jiong Guo and Danny Hermelin, editors, IPEC 2016, volume 63 of LIPIcs, pages 8:1-8:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.8.
6. Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam. Discreteness of asymptotic tensor ranks. arXiv, 2023. URL: https://arxiv.org/abs/2306.01718.
7. Andrei A. Bulatov and Amirhossein Kazeminia. Complexity classification of counting graph homomorphisms modulo a prime number. In Stefano Leonardi and Anupam Gupta, editors, STOC 2022, pages 1024-1037. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520075.
8. Peter Bürgisser, Michael Clausen, and Amin Shokrollahi. Algebraic complexity theory. In Grundlehren der mathematischen Wissenschaften, 1997.
9. Jin-Yi Cai and Ashwin Maran. The complexity of counting planar graph homomorphisms of domain size 3. In Barna Saha and Rocco A. Servedio, editors, STOC 2023, pages 1285-1297. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585173.
10. Airlie Chapman and Mehran Mesbahi. On strong structural controllability of networked systems: A constrained matching approach. In 2013 American Control Conference, pages 6126-6131, 2013. URL: https://doi.org/10.1109/ACC.2013.6580798.
11. Prasad Chaugule, Nutan Limaye, and Aditya Varre. Variants of homomorphism polynomials complete for algebraic complexity classes. ACM Trans. Comput. Theory, 13(4):21:1-21:26, 2021. URL: https://doi.org/10.1145/3470858.
12. Rajesh Chitnis, László Egri, and Dániel Marx. List h-coloring a graph by removing few vertices. Algorithmica, 78(1):110-146, 2017. URL: https://doi.org/10.1007/s00453-016-0139-6.
13. Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
14. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, STOC 2017, pages 210-223. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055502.
15. Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A tight lower bound for counting Hamiltonian cycles via matrix rank. In Artur Czumaj, editor, SODA 2018, pages 1080-1099. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.70.
16. Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Robert Krauthgamer, editor, SODA 2016, pages 1650-1669. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch113.
17. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. J. ACM, 65(3):12:1-12:46, 2018. URL: https://doi.org/10.1145/3148227.
18. Ronald de Wolf. Nondeterministic quantum query and communication complexities. SIAM Journal on Computing, 32(3):681-699, 2003. URL: https://doi.org/10.1137/S0097539702407345.
19. MARTIN E. DYER and CATHERINE S. GREENHILL. The complexity of counting graph homomorphisms. RANDOM STRUCT. ALGORITHMS, 17(3-4):260-289, 2000.
20. László Egri, Andrei A. Krokhin, Benoît Larose, and Pascal Tesson. The complexity of the list homomorphism problem for graphs. In Jean-Yves Marion and Thomas Schwentick, editors, STACS 2010, volume 5 of LIPIcs, pages 335-346. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. URL: https://doi.org/10.4230/LIPIcs.STACS.2010.2467.
21. László Egri, Dániel Marx, and Paweł Rzążewski. Finding list homomorphisms from bounded-treewidth graphs to reflexive graphs: a complete complexity characterization. In Rolf Niedermeier and Brigitte Vallée, editors, STACS 2018, volume 96 of LIPIcs, pages 27:1-27:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.27.
22. Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio mathematica, 2:463-470, 1935.
23. Baris Can Esmer, Jacob Focke, Dániel Marx, and Paweł Rzążewski. List homomorphisms by deleting edges and vertices: tight complexity bounds for bounded-treewidth graphs. CoRR, abs/2210.10677, 2022. URL: https://doi.org/10.48550/arXiv.2210.10677.
24. M. Fekete. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift, 17(1):228-249, December 1923. URL: https://doi.org/10.1007/bf01504345.
25. Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, and Philip Wellnitz. Tight complexity bounds for counting generalized dominating sets in bounded-treewidth graphs. In Nikhil Bansal and Viswanath Nagarajan, editors, SODA 2023, pages 3664-3683. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch140.
26. Jacob Focke, Dániel Marx, and Paweł Rzążewski. Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds. In Joseph (Seffi) Naor and Niv Buchbinder, editors, SODA 2022, pages 431-458. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.22.
27. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. URL: https://doi.org/10.1145/2886094.
28. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Representative families of product families. ACM Trans. Algorithms, 13(3):36:1-36:29, 2017. URL: https://doi.org/10.1145/3039243.
29. Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The fine-grained complexity of graph homomorphism parameterized by clique-width. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, ICALP 2022, volume 229 of LIPIcs, pages 66:1-66:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.66.
30. Chris Godsil. Problems in algebraic combinatorics. Electr. J. Comb., 2, January 1995. URL: https://doi.org/10.37236/1224.
31. Carla Groenland, Isja Mannens, Jesper Nederlof, Marta Piecyk, and Pawel Rzazewski. Towards tight bounds for the graph homomorphism problem parameterized by cutwidth via asymptotic rank parameters. CoRR, abs/2312.03859, 2023. URL: https://doi.org/10.48550/arXiv.2312.03859.
32. Carla Groenland, Isja Mannens, Jesper Nederlof, and Krisztina Szilágyi. Tight bounds for counting colorings and connected edge sets parameterized by cutwidth. In Petra Berenbrink and Benjamin Monmege, editors, STACS 2022, volume 219 of LIPIcs, pages 36:1-36:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.36.
33. Pavol Hell and Jaroslav Nešetřil. The core of a graph. Discrete Mathematics, 109(1-3):117-126, 1992. URL: https://doi.org/10.1016/0012-365X(92)90282-K.
34. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. J. Comb. Theory, Ser. B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
35. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. Discret. Appl. Math., 327:33-46, 2023. URL: https://doi.org/10.1016/j.dam.2022.11.011.
36. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. Theor. Comput. Sci., 795:520-539, 2019. URL: https://doi.org/10.1016/j.tcs.2019.08.006.
37. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters, tight bounds, and approximation for (k, r)-center. Discret. Appl. Math., 264:90-117, 2019. URL: https://doi.org/10.1016/j.dam.2018.11.002.
38. Tomasz Kociumaka and Marcin Pilipczuk. Deleting vertices to graphs of bounded genus. Algorithmica, 81(9):3655-3691, 2019. URL: https://doi.org/10.1007/s00453-019-00592-7.
39. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
40. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM J. Discret. Math., 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
41. Benoit Larose. Families of strongly projective graphs. Discussiones Mathematicae Graph Theory, 22:271-292, 2002.
42. Benoit Larose and Claude Tardif. Strongly rigid graphs and projectivity. Multiple-Valued Logic, 7:339-361, 2001.
43. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the Exponential Time Hypothesis. Bulletin of the EATCS, 105:41-72, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/92.
44. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
45. Tomasz Łuczak and Jaroslav Nešetřil. Note on projective graphs. Journal of Graph Theory, 47(2):81-86, 2004.
46. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and gaps: Tight complexity results of general factor problems parameterized by treewidth and cutwidth. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, ICALP 2021, volume 198 of LIPIcs, pages 95:1-95:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.95.
47. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard). In Holger Dell and Jesper Nederlof, editors, IPEC 2022, volume 249 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:23, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2022.22.
48. Burkhard Monien. The complexity of determining paths of length k. In Manfred Nagl and Jürgen Perl, editors, WG '83, pages 241-251. Universitätsverlag Rudolf Trauner, Linz, 1983.
49. Jesper Nederlof. Algorithms for np-hard problems via rank-related parameters of matrices. In Fedor V. Fomin, Stefan Kratsch, and Erik Jan van Leeuwen, editors, Treewidth, Kernels, and Algorithms - Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday, volume 12160 of Lecture Notes in Computer Science, pages 145-164. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_11.
50. Jesper Nederlof. Bipartite TSP in O(1.9999ⁿ) time, assuming quadratic time matrix multiplication. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, STOC 2020, pages 40-53. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384264.
51. Jaroslav Nešetřil and Aleš Pultr. A Dushnik - Miller type dimension of graphs and its complexity. In Marek Karpiński, editor, Fundamentals of Computation Theory, pages 482-493, Berlin, Heidelberg, 1977. Springer Berlin Heidelberg.
52. Jaroslav Nešetřil and Vojtéch Rödl. A simple proof of the Galvin-Ramsey property of the class of all finite graphs and a dimension of a graph. Discrete Mathematics, 23(1):49-55, 1978. URL: https://doi.org/10.1016/0012-365X(78)90186-3.
53. Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, ESA 2020, volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.74.
54. Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs. SIAM J. Comput., 50(2):487-508, 2021. URL: https://doi.org/10.1137/20M1320146.
55. D.D. Olesky, Michael Tsatsomeros, and P. van den Driessche. Qualitative controllability and uncontrollability by a single entry. Linear Algebra and its Applications, 187:183-194, 1993. URL: https://doi.org/10.1016/0024-3795(93)90134-A.
56. Marta Piecyk and Paweł Rzążewski. Fine-grained complexity of the list homomorphism problem: Feedback vertex set and cutwidth. In Markus Bläser and Benjamin Monmege, editors, STACS 2021, volume 187 of LIPIcs, pages 56:1-56:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.56.
57. Michał Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In MFCS 2011, volume 6907, pages 520-531. Springer, 2011.
58. Marc Roth and Philip Wellnitz. Counting and finding homomorphisms is universal for parameterized complexity theory. In Shuchi Chawla, editor, SODA 2020, pages 2161-2180. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.133.
59. Maguy Trefois and Jean-Charles Delvenne. Zero forcing number, constrained matchings and strong structural controllability. Linear Algebra and its Applications, 484:199-218, 2015.
60. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Amos Fiat and Peter Sanders, editors, ESA 2009, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_51.