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.

The research is partially supported by the RFBR grant 14-01-00545, by the President’s grant MK-2813.2014.1 and by the Government of the Russia (grant 14.Z50.31.0030).