Siu Man Chan’s research while affiliated with University of California, Berkeley and other places

We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of eciding whether s~pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.

We prove tight size bounds on monotone switching networks for the NP-complete problem of k-clique, and for an explicit monotone problem by analyzing a pyramid structure of height h for the P-complete problem of generation. This gives alternative proofs of the separations of m-NC from m-P and of m-NCⁱ from m-NCⁱ⁺¹, different from Raz-McKenzie (Combinatorica 1999). The enumerative-combinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.

The two-player pebble game of Dymond-Tompa is identified as a barrier for existing techniques to save space or to speed up parallel algorithms for evaluation problems. Many combinatorial lower bounds to study I versus NI and NC versus P under different restricted settings scale in the same way as the pebbling algorithm of Dymond-Tompa. These lower bounds include, (1) the monotone separation of m-I from m-NI by studying the size of monotone switching networks in Potechin '10; (2) a new semantic separation of NC from P and of NCi from NCi+1 by studying circuit depth, based on the techniques developed for the semantic separation of NC1 from NC2 by the universal composition relation in Edmonds-Impagliazzo-Rudich-Sgall '01 and in Hastad- Wigderson '97; and (3) the monotone separation of m-NC from m-P and of m-NCi from m-NCi+1 by studying (a) the depth of monotone circuits in Raz-McKenzie '99; and (b) the size of monotone switching networks in Chan- Potechin '12. This supports the attempt to separate NC from P by focusing on depth complexity, and suggests the study of combinatorial invariants shaped by pebbling for proving lower bounds. An application to proof complexity gives tight bounds for the size and the depth of some refinements of resolution refutations.

We prove tight size bounds on monotone switching networks for the k-clique problem, and for an explicit monotone problem by analyzing the generation problem with a pyramid structure of height h. This gives alternative proofs of the separations of m-NC from m-P and of m-NCi from m-NCi+1, different from Raz-McKenzie (Combinatorica '99). The enumerative-combinatorial and Fourier analytic techniques in this work are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.