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Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.ORCID iD: 0000-0001-8923-1240
KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
2019 (English)In: Proceedings of the 34th Annual Computational Complexity Conference (CCC ’19), 2019, p. 18:1-18:16Conference paper, Published paper (Other academic)
Abstract [en]

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if an only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

Place, publisher, year, edition, pages
2019. p. 18:1-18:16
National Category
Computer Sciences
Identifiers
URN: urn:nbn:se:kth:diva-249609OAI: oai:DiVA.org:kth-249609DiVA, id: diva2:1304722
Conference
Computational Complexity Conference
Note

QC 20190527

Available from: 2019-04-12 Created: 2019-04-12 Last updated: 2019-05-27Bibliographically approved
In thesis
1. Lower Bounds and Trade-offs in Proof Complexity
Open this publication in new window or tab >>Lower Bounds and Trade-offs in Proof Complexity
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Propositional proof complexity is a field in theoretical computer science that analyses the resources needed to prove statements. In this thesis, we are concerned about the length of proofs and trade-offs between different resources, such as length and space.

A classical NP-hard problem in computational complexity is that of determining whether a graph has a clique of size k. We show that for all k ≪ n^(1/4) regular resolution requires length n^Ω(k) to establish that an Erdős–Rényi graph with n vertices and appropriately chosen edge density does not contain a k-clique. In particular, this implies an unconditional lower bound on the running time of state-of-the-artalgorithms for finding a maximum clique.

In terms of trading resources, we prove a length-space trade-off for the cutting planes proof system by first establishing a communication-round trade-off for real communication via a round-aware simulation theorem. The technical contri-bution of this result allows us to obtain a separation between monotone-AC^(i-1) and monotone-NC^i.

We also obtain a trade-off separation between cutting planes (CP) with unbounded coefficients and cutting planes where coefficients are at most polynomial in thenumber of variables (CP*). We show that there are formulas that have CP proofs in constant space and quadratic length, but any CP* proof requires either polynomial space or exponential length. This is the first example in the literature showing any type of separation between CP and CP*.

For the Nullstellensatz proof system, we prove a size-degree trade-off via a tight reduction of Nullstellensatz refutations of pebbling formulas to the reversible pebbling game. We show that for any directed acyclic graph G it holds that G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatzrefutation of the pebbling formula over G in size t + 1 and degree s.

Finally, we introduce the study of cumulative space in proof complexity, a measure that captures the space used throughout the whole proof and not only the peak space usage. We prove cumulative space lower bounds for the resolution proof system, which can be viewed as time-space trade-offs where, when time is bounded, space must be large a significant fraction of the time.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2019. p. 247
Series
TRITA-EECS-AVL ; 2019:47
Keywords
Proof complexity, trade-offs, lower bounds, size, length, space
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-249610 (URN)978-91-7873-191-6 (ISBN)
Public defence
2019-06-14, Kollegiesalen, Brinellvägen 8, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20190527

Available from: 2019-05-27 Created: 2019-05-24 Last updated: 2019-05-27Bibliographically approved

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Nordström, Jakob

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