Subcubic equivalences between path, matrix, and triangle problems

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2010, Pages 645-654

  Williams, Virginia Vassilevska ^a   Williams, Ryan ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b IBM ALMADEN RESEARCH CENTER   (United States)

Author keywords

All pairs shortest paths; Equivalences; Matrix multiplication; Minimum cycle; Reductions; Replacement paths; Subcubic algorithms; Triangle detection

Indexed keywords

DIRECTED GRAPHS; EQUIVALENCE CLASSES; GRAPH ALGORITHMS; GRAPHIC METHODS; MATRIX ALGEBRA; QUANTUM THEORY; REDUCTION; STRUCTURAL OPTIMIZATION;

ALL PAIRS SHORTEST PATHS; EQUIVALENCES; MATRIX MULTIPLICATION; MINIMUM CYCLE; REPLACEMENT PATHS; SUBCUBIC ALGORITHMS; TRIANGLE DETECTIONS;

GRAPH STRUCTURES;

EID: 78751508026     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2010.67     Document Type: Conference Paper

References (41)

1. N. Alon. Personal communication. 2009.
2. N. Alon, Z. Galil, and O. Margalit. On the exponent of the all pairs shortest path problem. J. Comput. Syst. Sci., 54(2):255-262, 1997. (Pubitemid 127433373)
3. N. Alon and A. Naor. Approximating the cut-norm via Grothendieck's inequality. SIAM J. Computing, 35:787-803, 2006.
4. V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. Faradzev. On economical construction of the transitive closure of an oriented graph. Soviet Math. Dokl., 11:1209-1210, 1970.
5. N. Bansal and R. Williams. Regularity lemmas and combinatorial algorithms. In Proc. FOCS, pages 745-754, 2009.
6. A. Bernstein. A nearly optimal algorithm for approximating replacement paths and k shortest simple paths in general graphs. In Proc. SODA, pages 742-755, 2010.
7. M. Blum and S. Kannan. Designing programs that check their work. J. ACM, 42(1):269-291, 1995.
8. J. Brickell, I. Dhillon, S. Sra, and J. Tropp. The metric nearness problem. SIAM J. Matrix Anal. Appl., 30(1):375-396, 2008.
9. H. Buhrman and R. Špalek. Quantum verification of matrix products. In Proc. SODA, pages 880-889, 2006.
10. T. M. Chan. More algorithms for all-pairs shortest paths in weighted graphs. In Proc. STOC, pages 590-598, 2007.
11. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symbolic Computation, 9(3):251-280, 1990.
12. A. Czumaj and A. Lingas. Finding a heaviest triangle is not harder than matrix multiplication. In Proc. SODA, pages 986-994, 2007.
13. R. Duan and S. Pettie. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In Proc. SODA, pages 384-391, 2009.
14. D. Eppstein. Finding the k shortest paths. SIAM Journal on Computing, 28(2):652-673, 1998.
15. M. J. Fischer and A. R. Meyer. Boolean matrix multiplication and transitive closure. In Proc. FOCS, pages 129-131, 1971.
16. R. W. Floyd. Algorithm 97: shortest path. Comm. ACM, 5:345, 1962.
17. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. JACM, 34(3):596-615, 1987.
18. 2) problems in computational geometry. Computational Geometry, 5(3):165-185, 1995.
19. A. Itai and M. Rodeh. Finding a minimum circuit in a graph. SIAM J. Computing, 7(4):413-423, 1978.
20. N. Katoh, T. Ibaraki, and H. Mine. An efficient algorithm for K shortest simple paths. Networks, 12(4):411-427, 1982.
21. E. Lawler. A procedure for computing the K best solutions to discrete optimization problems and its application to the shortest path problem. Management Science, 18:401-405, 1971/72.
22. A. Lingas. A fast output-sensitive algorithm for Boolean matrix multiplication. In Proc. ESA, pages 408-419, 2009.
23. F. Magniez, M. Santha, and M. Szegedy. Quantum algorithms for the triangle problem. In Proc. SODA, pages 1109-1117, 2005.
24. n. Information Processing Letters, 38(5):277-278, 1991.
25. J. I. Munro. Efficient determination of the transitive closure of a directed graph. Inf. Process. Lett., 1(2):56-58, 1971.
26. V. Y. Pan. Strassen's algorithm is not optimal; trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations. In Proc. FOCS, pages 166-176, 1978.
27. V. Y. Pan. New fast algorithms for matrix operations. SIAM J. Comput., 9(2):321-342, 1980.
28. L. Roditty. On the K-simple shortest paths problem in weighted directed graphs. In Proc. SODA, pages 920-928, 2007.
29. L. Roditty and U. Zwick. On dynamic shortest paths problems. In ESA, pages 580-591, 2004.
30. L. Roditty and U. Zwick. Replacement paths and k simple shortest paths in unweighted directed graphs. In Proc. ICALP, pages 249-260, 2005.
31. A. Shoshan and U. Zwick. All pairs shortest paths in undirected graphs with integer weights. In Proc. FOCS, pages 605-614, 1999.
32. J. P. Spinrad. Efficient graph representations. Fields Institute Monographs, 19, 2003.
33. V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354-356, 1969.
34. V. Vassilevska. Nondecreasing paths in weighted graphs, or: how to optimally read a train schedule. In Proc. SODA, pages 465-472, 2008.
35. 3-δ time, with applications. In Proc. STOC, pages 225-231, 2006.
36. V. Vassilevska, R. Williams, and R. Yuster. Finding the smallest H-subgraph in real weighted graphs and related problems. In Proc. ICALP, pages 262-273, 2006.
37. V. Vassilevska, R. Williams, and R. Yuster. All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory of Computing, 5(1):173-189, 2009.
38. S. Warshall. A theorem on boolean matrices. J. ACM, 9(1):11-12, 1962.
39. G. J. Woeginger. Open problems around exact algorithms. Discrete Applied Mathematics, 156(3):397-405, 2008.
40. J. Y. Yen. Finding the K shortest loopless paths in a network. Management Science, 17:712-716, 1970/71.
41. 2.81 infinite-precision multiplications. Inf. Proc. Letters, 4:155-156, 1976.