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A Faster Algorithm for Determining the Linear Feasibility of Systems of BTVPI Constraints

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SOFSEM 2023: Theory and Practice of Computer Science (SOFSEM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13878))

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Abstract

In this paper, we discuss an approach for determining the feasibility of a polyhedron defined by a system of Binary Two Variable Per Inequality (BTVPI) constraints. A constraint of the form: \(a_i\cdot x_i+a_j \cdot x_j \ge b_{ij}\) is called a BTVPI constraint, if \(a_i,a_j \in \{0,1,-1,2,-2\}\) and \(b_{ij} \in \mathbb {Z}\). These constraints find applications in a number of domains, including scheduling and abstract interpretation. Our algorithm is based on a rewrite version of the well-known Fourier-Motzkin elimination procedure for linear programs. We show that our algorithm converges in polynomial time and is faster than all known algorithms for this class of problems.

This research was made possible by the NASA Established Program to Stimulate Competitive Research, Grant # 80NSSC22M0027.

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory. Prentice-Hall, Algorithms and Applications (1993)

    MATH  Google Scholar 

  2. Àlvarez, C., Greenlaw, R.: A compendium of problems complete for symmetric logarithmic space. Electronic Colloquium on Computational Complexity (ECCC), vol. 3, no. 39 (1996). https://doi.org/10.1007/PL00001603

  3. Aspvall, B., Shiloach, Y.: A fast algorithm for solving systems of linear equations with two variables per equation. Linear Algebra Appl. 34, 117–124 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekaran, R., Subramani, K.: A combinatorial algorithm for Horn programs. Discret. Optim. 10, 85–101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chandru, V., Hooker, J.N.: Optimization methods for logical inference. Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc. (1999)

    Google Scholar 

  6. Cohen, E., Meggido, N.: Improved algorithms for linear inequalities with two variables per inequality. SIAM J. Comput. 23(6), 1313–1347 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge, MA (2009)

    MATH  Google Scholar 

  8. CPLEX Manual. IBM ILOG CPLEX Optimization Studio V12.8.0 Documentation (2017)

    Google Scholar 

  9. Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton, NJ (1963)

    Book  MATH  Google Scholar 

  10. Dantzig, G.B., Eaves, B.C.: Fourier-motzkin elimination and its dual. J. Comb. Theor. (A) 14, 288–297 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fiduccia, C.M., Mattheyses, R.M.: A linear time heuristics for improving network partitions. In: Proceedings of the 19th Design Automation Conference, pp. 175–181 (1982)

    Google Scholar 

  12. Gallo, G., Pallottino, S.: Shortest path algorithms. Ann. Oper. Res. 13, 1–79 (1988)

    Article  MathSciNet  Google Scholar 

  13. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hochbaum, D.S., (Seffi) Naor, J.: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput. 23(6), 1179–1192 (1994)

    Google Scholar 

  15. Karmarkar, N.K.: A new polynomial-time algorithm for linear programming. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, pp. 302–311 (1984)

    Google Scholar 

  16. Lahiri, S.K., Musuvathi, M.: An efficient decision procedure for UTVPI constraints. In: Gramlich, B. (ed.) FroCoS 2005. LNCS (LNAI), vol. 3717, pp. 168–183. Springer, Heidelberg (2005). https://doi.org/10.1007/11559306_9

    Chapter  Google Scholar 

  17. Luyo, L.E.F., Agra, A., Figueiredo, R., Anaya, E.O.: Mixed integer formulations for a routing problem with information collection in wireless networks. Eur. J. Oper. Res. 280(2), 621–638 (2020)

    Google Scholar 

  18. Miné, A.: The octagon abstract domain. Higher-Order Symbolic Comput. 19(1), 31–100 (2006)

    Article  MATH  Google Scholar 

  19. Nelson, C.G.: An \(n^o( \log n)\) algorithm for the two-variable-per-constraint linear program satisfiability problem. Technical Report Technical Note STA, Stanford University, Computer Science Department, pp. 78–689 (1978)

    Google Scholar 

  20. Gurobi Optimization. Gurobi optimizer reference manual4. http://www.gurobi.com (2014)

  21. Revesz, P.Z.: Tightened transitive closure of integer addition constraints. In: Symposium on Abstraction, Reformulation, and Approximation (SARA), pp. 136–142 (2009)

    Google Scholar 

  22. Singh, G., Gehr, T., Püschel, M., Vechev, M.T.: An abstract domain for certifying neural networks. PACMPL 3(POPL), 1–30 (2019)

    Google Scholar 

  23. Subramani, K., Wojciechowski, P.J.: A combinatorial certifying algorithm for linear feasibility in UTVPI constraints. Algorithmica 78(1), 166–208 (2017)

    Google Scholar 

  24. Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34(2), 250–256 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Veinott, A.F., Dantzig, G.B.: Integral extreme points. SIAM Rev. 10, 371–372 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  26. Veinott, A.F., LiCalzi, M.: Subextremal functions and lattice programming, unpublished manuscript (1992)

    Google Scholar 

  27. Vollmer, H.: Introduction to Circuit Complexity. Springer (1999). https://doi.org/10.1007/978-3-662-03927-4

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Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    Piotr Wojciechowski & K. Subramani

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  1. Piotr Wojciechowski
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  2. K. Subramani
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Correspondence to K. Subramani .

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  1. University of Liverpool, Liverpool, UK

    Leszek Gąsieniec

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Wojciechowski, P., Subramani, K. (2023). A Faster Algorithm for Determining the Linear Feasibility of Systems of BTVPI Constraints. In: Gąsieniec, L. (eds) SOFSEM 2023: Theory and Practice of Computer Science. SOFSEM 2023. Lecture Notes in Computer Science, vol 13878. Springer, Cham. https://doi.org/10.1007/978-3-031-23101-8_21

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  • DOI: https://doi.org/10.1007/978-3-031-23101-8_21

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  • Online ISBN: 978-3-031-23101-8

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