Multilinear Algebraic Branching Programs and the Min-Partition Rank Method

It is a long-standing open problem in algebraic complexity to prove lower bounds against multilinear algebraic branching programs (mlABPs), however the best lower bounds are still quadratic (Alon, Kumar and Volk (Combinatorica 2020)). At the same time, it remains a possibility that the “min-partition rank” method introduced by Raz (Theory Comput. 2006), which is used to prove all known multilinear lower bounds, can also be used to prove superpolynomial lower bounds on the size of mlABPs. In this paper, we analyze the potential of the min-partition rank method to prove lower bounds on the size of mlABPs, and show the following results: 1. We relate this method to a purely combinatorial question regarding the minimum size of set systems whose chains satisfy a discrepancy condition. In the case of set-multilinear ABPs, this combinatorial measure characterizes the best lower bound that can be achieved via the min-partition rank method. 2. We prove a non-trivial upper bound on the size of a set system satisfying this combinatorial property. Together with our construction of full-rank mlABPs from set systems, this recovers a superpolynomial separation between mlABPs and multilinear formulas (Dvir, Malod, Perifel and Yehudayoff (STOC 2012)) via a conceptually different proof. 3. The property we study extends combinatorial notions of “balancing sets” considered in previous works, for which near-tight bounds are known via intervals families. We show that any intervals set system is very far from satisfying our property. This showcases how our methods capture combinatorial structures that evade previous techniques, and also allows us to improve and generalize known lower bounds for sum of ordered set-multilinear ABPs (Chatterjee, Kush, Saraf, Shpilka (CCC 2024)). These results build a bridge between algebraic complexity theory and the behavior of random walks. Our upper bound uses the fact that, with noticeable probability, a random walk of length n on the integers returns to its starting point at least once every n/log n steps (Csáki, Erdős, and Révész (PTRF 1985)), while, for our lower bound, we prove that two independent random walks are “far” from each other in discrete Fréchet distance.