Graph colouring is hard for algorithms based on hilbert's nullstellensatz and gröbner bases
Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
We consider the graph k-colouring problem encoded as a set of polynomial equations in the standard way. We prove that there are bounded-degree graphs that do not have legal k-colourings but for which the polynomial calculus proof system defined in [Clegg et al. 1996, Alekhnovich et al. 2002] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gröbner bases solving graph k-colouring using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al. 2008, 2009, 2011, 2015] based on Hilbert's Nullstellensatz proofs for a slightly different encoding, thus resolving an open problem mentioned, e.g., in [De Loera et al. 2009] and [Li et al. 2016]. We obtain our results by combining the polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Miksa and Nordström 2015] with a reduction from FPHP to k-colouring derivable by polynomial calculus in constant degree.
Original language | English |
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Title of host publication | 32nd Computational Complexity Conference, CCC 2017 |
Editors | Ryan O'Donnell |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Publication date | 1 Jul 2017 |
Article number | 2 |
ISBN (Electronic) | 9783959770408 |
DOIs | |
Publication status | Published - 1 Jul 2017 |
Externally published | Yes |
Event | 32nd Computational Complexity Conference, CCC 2017 - Riga, Latvia Duration: 6 Jul 2017 → 9 Jul 2017 |
Conference
Conference | 32nd Computational Complexity Conference, CCC 2017 |
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Land | Latvia |
By | Riga |
Periode | 06/07/2017 → 09/07/2017 |
Sponsor | Microsoft Research, University of Latvia |
Series | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 79 |
ISSN | 1868-8969 |
- 3-colouring, Cutting planes, Gröbner basis, Nullstellensatz, Polynomial calculus, Proof complexity
Research areas
ID: 251867943