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Hardness of Approximation in PSPACE and Separation Results for Pebble Games
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-4003-3168
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-1487-445X
2015 (English)In: Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, Institute of Electrical and Electronics Engineers (IEEE), 2015, 466-485 p.Conference paper, Published paper (Refereed)
Resource type
Text
Abstract [en]

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 deciding 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. © 2015 IEEE.

Place, publisher, year, edition, pages
Institute of Electrical and Electronics Engineers (IEEE), 2015. 466-485 p.
Keyword [en]
Dymond-Tompa game, pebbling, PSPACE-complete, PSPACE-hardness of approximation, Raz-Mc Kenzie game, resolution depth, reversible pebbling, separation
National Category
Electrical Engineering, Electronic Engineering, Information Engineering
Identifiers
URN: urn:nbn:se:kth:diva-186800DOI: 10.1109/FOCS.2015.36ISI: 000379204700027Scopus ID: 2-s2.0-84960371365ISBN: 9781467381918 (print)OAI: oai:DiVA.org:kth-186800DiVA: diva2:928552
Conference
56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, 17 October 2015 through 20 October 2015
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-13 Last updated: 2017-05-09Bibliographically approved
In thesis
1. Space in Proof Complexity
Open this publication in new window or tab >>Space in Proof Complexity
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

ropositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter.

Different approaches to reasoning are captured by corresponding proof systems. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution.

In resolution we can prove a space lower bound just by showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that lets us derive space lower bounds, and we use it to separate degree and space.

For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time.

We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is exploratory work, but we can also prove new results for the usual space measure.

We define a new proof system that aims to capture the power of current SAT solvers, and we show a landscape of length-space trade-offs comparable to those in resolution.

To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, very simple computational models that are useful for modelling space. In addition to results with applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate.

Another area is communication complexity, the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2017. 318 p.
Series
TRITA-CSC-A, ISSN 1653-5723 ; 2017:15
Keyword
proof complexity, resolution, polynomial calculus, cutting planes, space complexity, computational complexity, pebble games, communication complexity, CDCL
National Category
Computer Science
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-206571 (URN)978-91-7729-422-1 (ISBN)
Public defence
2017-06-09, E2, Lindstedtsvägen, 3, Stockholm, 14:00 (English)
Opponent
Supervisors
Funder
EU, FP7, Seventh Framework Programme, 279611
Note

QC 20170509

Available from: 2017-05-10 Created: 2017-05-09 Last updated: 2017-05-10Bibliographically approved

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