1. 
 Chan, S. M., et al.
(författare)

Hardness of Approximation in PSPACE and Separation Results for Pebble Games
 2015

Ingår i: Proceedings  Annual IEEE Symposium on Foundations of Computer Science, FOCS.  : Institute of Electrical and Electronics Engineers (IEEE).  9781467381918 ; , s. 466485

Konferensbidrag (refereegranskat)abstract
 We consider the pebble game on DAGs with bounded fanin 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 deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACEcomplete, 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 PSPACEhard 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 PSPACEhard 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.


2. 
 Filmus, Yuval, et al.
(författare)

Space complexity in polynomial calculus
 2015

Ingår i: SIAM journal on computing (Print).  : Society for Industrial & Applied Mathematics (SIAM).  00975397 . 10957111. ; 44:4, s. 11191153

Tidskriftsartikel (refereegranskat)abstract
 During the last 10 to 15 years, an active line of research in proof complexity has been to study space complexity and timespace tradeoffs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably wellunderstood from this point of view. For other proof systems of interest, however, such as polynomial calculus or cutting planes, progress has been more limited. Essentially nothing has been known about space complexity in cutting planes, and for polynomial calculus the only lower bound has been for conjunctive normal form (CNF) formulas of unbounded width in [Alekhnovich et al., SIAM J. Comput., 31 (2002), pp. 11841211], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any kCNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.: (1) We prove an omega(n) space lower bound in PC for the canonical 3CNF version of the pigeonhole principle formulas PHPmn with m pigeons and n holes, and show that this is tight. (2) For PCR, we prove an omega(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over Alekhnovich et al. measured in the width of the formulas. (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an omega(n) space lower bound in PCR for these formulas as well. (4) Finally, we prove that any kCNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other wellstudied formula families in proof complexity.

