# Total dual integrality of the linear complementarity problem

## Abstract

In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

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1. 1.

A decision problem P is said to be coNP-complete if P is in class coNP and P is coNP-hard. The class coNP is the set of decision problems for which there exists a polynomial-time algorithm that verifies “no” instances when given a certificate. A decision problem P is called coNP-hard if for each problem $$P'$$ in coNP, there exists a polynomial-time algorithm that transforms any instance $$L'$$ of $$P'$$ into some instance L of P such that the answer of $$L'$$ is “no” if and only if the answer of L is “no.” See also the book of Garey and Johnson (1990) for precise definitions.

## References

1. Bouchet, A. (1992). A characterization of unimodular orientations of simple graphs. Journal of Combinatorial Theory, Series B, 56(1), 45–54.

2. Chandrasekaran, R. (1984). Integer programming problems for which a simple rounding type algorithm works. Progress in Combinatorial Optimization, 8, 101–106.

3. Chandrasekaran, R., Kabadi, S. N., & Sridhar, R. (1998). Integer solution for linear complementarity problem. Mathematics of Operations Research, 23(2), 390–402.

4. Chung, S. J. (1989). NP-completeness of the linear complementarity problem. Journal of Optimization Theory and Applications, 60(3), 393–399.

5. Cook, W., Lovász, L., & Schrijver, A. (1984). A polynomial-time test for total dual integrality in fixed dimension (pp. 64–69). Berlin Heidelberg, Berlin, Heidelberg: Springer.

6. Cottle, R. W. (1968). The principal pivoting method of quadratic programming. In Mathematics of decision sciences, part 1 (pp. 142–162). Providence R.I.: American Mathematical Society.

7. Cottle, R. W., & Dantzig, G. B. (1968). Complementary pivot theory of mathematical programming. Linear Algebra and its Applications, 1(1), 103–125.

8. Cottle, R. W., Pang, J. S., & Stone, R. E. (1992). The linear complementarity problem. Boston: Academic Press.

9. Cottle, R. W., Pang, J. S., & Venkateswaran, V. (1989). Sufficient matrices and the linear complementarity problem. Linear Algebra and its Applications, 114–115, 231–249 (special Issue Dedicated to Alan J. Hoffman).

10. Cunningham, W. H., & Geelen, J. F. (1998). Integral solutions of linear complementarity problems. Mathematics of Operations Research, 23(1), 61–68.

11. Ding, G., Feng, L., & Zang, W. (2008). The complexity of recognizing linear systems with certain integrality properties. Mathematical Programming, 114(2), 321–334.

12. Edmonds, J., & Giles, R. (1977). A min-max relation for submodular functions on graphs. In P. L. Hammer, E. L. Johnson, B. H. Korte & G. L. Nemhauser (Eds.), Studies in integer programming, Annals of discrete mathematics (Vol. 1, pp. 185–204). Amsterdam: Elsevier.

13. Fukuda, K., & Terlaky, T. (1992). Linear complementarity and oriented matroids. Journal of the Operations Research Society of Japan, 35(1), 45–61.

14. Gabriel, S. A., Conejo, A. J., Ruiz, C., & Siddiqui, S. (2013). Solving discretely constrained, mixed linear complementarity problems with applications in energy. Computers & Operations Research, 40(5), 1339–1350.

15. Garey, M. R., & Johnson, D. S. (1990). Computers and intractability; A guide to the theory of NP-completeness. New York: W. H. Freeman & Co.

16. Harville, D. A. (1997). Matrix algebra from a statistician’s perspective. New York: Springer.

17. Hoffman, A. J., & Kruskal, J. B. (1956). Integral boundary points of convex polyhedra. In H. Kuhn & A. Tucker (Eds.), Linear inequalities and related systems (pp. 223–246). Princeton: Princeton University Press.

18. Howson, J. T., Jr. (1972). Equilibria of polymatrix games. Management Science, 18(5-part-1), 312–318

19. Kronecker, L. (1884). Näherungsweise ganzzahlige auflösung linearer gleichungen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1179–1193, 1271–1299.

20. Lemke, C. E. (1965). Bimatrix equilibrium points and mathematical programming. Management Science, 11(7), 681–689.

21. Murty, K. G. (1997). Linear complementarity, linear and nonlinear programming. Internet Edition, http://www-personal.umich.edu/~murty/books/linear_complementarity_webbook/

22. Pap, J. (2011). Recognizing conic TDI systems is hard. Mathematical Programming, 128(1), 43–48.

23. Pardalos, P. M., & Nagurney, A. (1990). The integer linear complementarity problem. International Journal of Computer Mathematics, 31(3–4), 205–214.

24. Ruiz, C., Conejo, A. J., & Gabriel, S. A. (2012). Pricing non-convexities in an electricity pool. IEEE Transactions on Power Systems, 27(3), 1334–1342.

25. Schrijver, A. (1986). Theory of linear and integer programming. Toronto: Wiley

26. Takayama, T., & Judge, G. G. (1971). Spatial and temporal price allocation models. Amsterdam: North-Holland.

## Acknowledgements

The authors thank the referees for their valuable comments on this manuscript. The first author is supported by JST ERATO Grant Number JPMJER1201, Japan, and JSPS KAKENHI Grant Numbers JP14J10346 and JP17K12646. The second author is supported by JSPS KAKENHI Grant Numbers JP25730001, JP24106002, and JP17K00028. The third author is supported by JSPS KAKENHI Grant Numbers JP24106002, JP25280004, JP26280001, and JST CREST Grant Number JPMJCR1402, Japan.

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Correspondence to Hanna Sumita.

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