# Resolution Complexity of Perfect Matching Principles for Sparse Graphs

## Abstract

The resolution complexity of the perfect matching principle was studied by Razborov [Raz04], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph \(G_n\) such that the resolution complexity of the perfect matching principle for \(G_n\) is \(2^{\varOmega (n)}\), where *n* is the number of vertices in \(G_n\). This lower bound is tight up to some polynomial. Our result implies the \(2^{\varOmega (n)}\) lower bounds for the complete graph \(K_{2n+1}\) and the complete bipartite graph \(K_{n, O(n)}\) that improves the lower bounds following from [Raz04]. Our results also imply the well-known exponential lower bounds on the resolution complexity of the pigeonhole principle, the functional pigeonhole principle and the pigeonhole principle over a graph.

We also prove the following corollary. For every natural number *d*, for every *n* large enough, for every function \(h:\{1, 2, \dots , n\} \rightarrow \{1, 2, \dots , d\}\), we construct a graph with *n* vertices that has the following properties. There exists a constant *D* such that the degree of the *i*-th vertex is at least *h*(*i*) and at most *D*, and it is impossible to make all degrees equal to *h*(*i*) by removing the graph’s edges. Moreover, any proof of this statement in the resolution proof system has size \(2^{\varOmega (n)}\). This result implies well-known exponential lower bounds on the Tseitin formulas as well as new results: for example, the same property of a complete graph.

## Notes

### Acknowledgements

The authors are grateful to Vsevolod Oparin for fruitful discussions, to Alexander Shen for the suggestions on the presentation of results, and to anonymous reviewers for useful comments.

### References

- [AHI05]Alekhnovich, M., Hirsch, E.A., Itsykson, D.: Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas. J. Autom. Reason.
**35**(1–3), 51–72 (2005)MATHMathSciNetGoogle Scholar - [Ale04]Alekhnovich, M.: Mutilated chessboard problem is exponentially hard for resolution. Theor. Comput. Sci.
**310**(1–3), 513–525 (2004)MATHMathSciNetCrossRefGoogle Scholar - [BSW01]Ben-Sasson, E., Wigderson, A.: Short proofs are narrow – resolution made simple. J. ACM
**48**(2), 149–169 (2001)MATHMathSciNetCrossRefGoogle Scholar - [DR01]Dantchev, S.S., Riis, S.: “Planar” tautologies hard for resolution. In: FOCS, pp. 220–229 (2001)Google Scholar
- [Hak85]Haken, A.: The intractability of resolution. Theor. Comput. Sci.
**39**, 297–308 (1985)MATHMathSciNetCrossRefGoogle Scholar - [HLW06]Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc.
**43**, 439–561 (2006)MATHMathSciNetCrossRefGoogle Scholar - [IS11]Itsykson, D., Sokolov, D.: Lower bounds for myopic DPLL algorithms with a cut heuristic. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 464–473. Springer, Heidelberg (2011) CrossRefGoogle Scholar
- [MCW02]Vadhan, S., Capalbo, M., Reingold, O., Wigderson, A.: Randomness conductors and constant-degree expansion beyond the degree/2 barrier. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 659–668 (2002)Google Scholar
- [Raz01a]Raz, R.: Resolution lower bounds for the weak pigeonhole principle. Technical report 01–021, Electronic Colloquium on Computational Complexity (2001)Google Scholar
- [Raz01b]Razborov, A.A.: Resolution lower bounds for the weak pigeonhole principle. Technical report 01–055, Electronic Colloquium on Computational Complexity (2001)Google Scholar
- [Raz03]Razborov, A.A.: Resolution lower bounds for the weak functional pigeonhole principle. Theor. Comput. Sci.
**303**(1), 233–243 (2003)MATHMathSciNetCrossRefGoogle Scholar - [Raz04]Razborov, A.A.: Resolution lower bounds for perfect matching principles. J. Comput. Syst. Sci.
**69**(1), 3–27 (2004)MATHMathSciNetCrossRefGoogle Scholar - [SB97]Pitassi, T., Buss, S.: Resolution and the weak pigeonhole principle. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 149–156. Springer, Heidelberg (1998) CrossRefGoogle Scholar
- [Urq87]Urquhart, A.: Hard examples for resolution. J. ACM
**34**(1), 209–219 (1987)MATHMathSciNetCrossRefGoogle Scholar - [Urq03]Urquhart, A.: Resolution proofs of matching principles. Ann. Math. Artif. Intell.
**37**(3), 241–250 (2003)MATHMathSciNetCrossRefGoogle Scholar