Author:
Adrian Dumitrescu
ad.dumitrescu@algoresearch.org
Adrian Dumitrescu
Algoresearch L.L.C., Milwaukee, WI , USA
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We revisit the algorithmic problem of finding a triangle in a graph (Triangle Detection), and examine its relation to other problems such as 3Sum, Independent Set, and Graph Coloring. We obtain several new algorithms:
(I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study a question of Pˇatraşcu (2010) regarding the triangle detection problem.
(II) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚 + 𝑛) (i.e., linear) time: (i) compute a Ω(1/√𝑛)-approximation of a maximum independent set in 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks.
(III) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚+𝑛3/2) time: (i) compute √𝑛-approximation for Graph Coloring of 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with 𝑚 = (𝑛9/8).
(IV) Results (II) and (III) above suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the dual pair concept we introduce. We provide several instances of dual-pair approximation, relating Longest Path, (1,2)-TSP, and other NP-hard problems.
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-
[1]
A. Abboud, V. V. Williams, and H. Yu. Matching triangles and basing hardness on an extremely popular conjecture. SIAM J. Comput., 47(3):1098–1122, 2018.
-
[2]
A. Adamaszek, M. Mnich, and K. Paluch. New approximation algorithms for (1, 2)-TSP. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of LIPIcs. Leibniz Int. Proc. Inform., page article no. 9. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018.
-
[3]
M. Ajtai, J. Komlós, and E. Szemerédi. A note on Ramsey numbers. J. Combin. Theory Ser. A, 29(3):354–360, 1980.
-
[4]
N. Alon, R. Yuster, and U. Zwick. Color-coding. J. Assoc. Comput. Mach., 42(4):844–856, 1995.
-
[5]
N. Alon, R. Yuster, and U. Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209–223, 1997.
-
[6]
R. Bar-Yehuda and S. Even. On approximating a vertex cover for planar graphs. Proc. 14th Annual ACM Symposium on Theory of Computing (STOC), pages 303–309, 1982.
-
[7]
I. Baran, E. D. Demaine, and M. Pˇatraşcu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584–596, 2008.
-
[8]
S. Bereg and A. Dumitrescu. The lifting model for reconfiguration. Discrete Comput. Geom., 35(4):653–669, 2006.
-
[9]
P. Berman and M. Karpinski. 8/7-approximation algorithm for (1, 2)-TSP. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 641–648. ACM, New York, 2006.
-
[10]
H. L. Bodlaender. On linear time minor tests with depth-first search. J. Algorithms, 14(1):1–23, 1993.
-
[11]
B. Bollobás. Modern Graph Theory, volume 184 of Grad. Texts in Math. Springer, New York, 1998.
-
[12]
N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1):210–223, 1985.
-
[13]
D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symbolic Comput., 9(3):251–280, 1990.
-
[14]
D. de Caen. An upper bound on the sum of squares of degrees in a graph. Discrete Math., 185(1-3):245–248, 1998.
-
[15]
R. Duan, H. Wu, and R. Zhou. Faster matrix multiplication via asymmetric hashing. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science—FOCS 2023, pages 2129–2138. IEEE Computer Soc., Los Alamitos, CA, 2023.
-
[16]
L. Engebretsen and J. Holmerin. Clique is hard to approximate within 𝑛1−𝑜(1). In Automata, languages and programming (Geneva, 2000), volume 1853 of Lecture Notes in Comput. Sci., pages 2–12. Springer, Berlin, 2000.
-
[17]
D. Eppstein. Paired approximation problems and incompatible inapproximabilities. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1076–1086. SIAM, Philadelphia, PA, 2010.
-
[18]
U. Feige. Approximating maximum clique by removing subgraphs. SIAM J. Discrete Math., 18(2):219–225, 2004.
-
[19]
H. N. Gabow. Finding paths and cycles of superpolylogarithmic length. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, STOC’04, page 407–416. ACM, jun 2004.
-
[20]
A. Gajentaan and M. H. Overmars. On a class of 𝑂(𝑛2) problems in computational geometry. Comput. Geom., 45(4):140–152, 2012.
-
[21]
M. R Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., New York, 1979.
-
[22]
J. Gimbel and C. Thomassen Coloring triangle-free graphs with fixed size. Discrete Math., 219(1-3):275–277, 2000.
-
[23]
O. Gold and M. Sharir. Improved bounds for 3SUM, 𝑘-SUM, and linear degeneracy. In 25th European Symposium on Algorithms, volume 87 of LIPIcs. Leibniz Int. Proc. Inform., page article no. 42. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017.
-
[24]
A. Grønlund and S. Pettie. Threesomes, degenerates, and love triangles. J. ACM, 65(4): article no. 22, 2018.
-
[25]
M. M. Halldórsson. A still better performance guarantee for approximate graph coloring. Inform. Process. Lett., 45(1):19–23, 1993.
-
[26]
D. S. Hochbaum. Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems, pages 94–143. PWS Publishing Co., 1997. (D. S. Hochbaum, editor).
-
[27]
A. Itai and M. Rodeh. Finding a minimum circuit in a graph. SIAM J. Comput., 7(4):413–423, 1978.
-
[28]
M. Karpinski, M. Lampis, and R. Schmied. New inapproximability bounds for TSP. J. Comput. System Sci., 81(8):1665–1677, 2015.
-
[29]
M. Karpinski and R. Schmied. On approximation lower bounds for tsp with bounded metrics. Electronic Colloquium on Computational Complexity, 2012. TR12-008.
-
[30]
J. H. Kim. The Ramsey number 𝑅(3, 𝑡) has order of magnitude 𝑡2/ log 𝑡. Random Structures Algorithms, 7(3):173–207, 1995.
-
[31]
T. Kloks, D. Kratsch, and H. Müller. Finding and counting small induced subgraphs efficiently. Inform. Process. Lett., 74(3-4):115–121, 2000.
-
[32]
T. Kopelowitz, S. Pettie, and E. Porat. Higher lower bounds from the 3SUM conjecture. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1272–1287. ACM, New York, 2016.
-
[33]
C. Lund and M. Yannakakis. The approximation of maximum subgraph problems. In Proc. 20th International Colloquium on Automata, Languages, and Programming, volume 700 of LNCS, pages 40–51, 1993. ICALP.
-
[34]
J. Matoušek. Thirty-three Miniatures, volume 53 of Stud. Math. Libr. American Mathematical Society, Providence, RI, 2010. Mathematical and algorithmic applications of linear algebra.
-
[35]
M. Mitzenmacher and E. Upfal. Probability and computing, Randomization and probabilistic techniques in algorithms and data analysis. Cambridge University Press, Cambridge, second edition, 2017.
-
[36]
C. H. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two. Math. Oper. Res., 18(1):1–11, 1993.
-
[37]
M. Pˇatraşcu. Towards polynomial lower bounds for dynamic problems. In STOC’10— Proceedings of the 2010 ACM International Symposium on Theory of Computing, pages 603–609. ACM, New York, 2010.
-
[38]
S. Poljak. A note on stable sets and colorings of graphs. Comment. Math. Univ. Carolinae, 15:307–309, 1974.
-
[39]
P. Turán. Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok, 48:436–452, 1941.
-
[40]
V. Vassilevska, R. Williams, and Shan L. M. Woo. Confronting hardness using a hybrid approach. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1–10. ACM, New York, 2006.
-
[41]
D. B. West. Combinatorial Mathematics. Cambridge University Press, Cambridge, 2021.
-
[42]
V. V. Williams, Y. Xu, Z. Xu, and R. Zhou. New bounds for matrix multiplication: from alpha to omega. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3792–3835. SIAM, Philadelphia, PA, 2024.
-
[43]
D. P. Williamson and D. B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, Cambridge, 2011.
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