Some combinatorial-algebraic problems from complexity theory

Abstract In this paper we survey some problems which have recently appeared in the study of the complexity of computations. We also survey and prove some related results. In particular we prove a lower bound on rigidity of an explicitly defined matrix and give an upper bound for decomposition of graphs into unions of complete bipartite graphs which yields an upper bound on projective dimension of 0–1 matrices. The concepts of projective and affine dimension were introduced by A.A. Razborov (1990) and the present authors (1992) as a possible tool for finding lower bounds on formula and branching program complexity.

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