Parameterized Complexity of DPLL Search Procedures

We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every Ramsey graph must contain a large subgraph with some of the statistical properties of the random graph.

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“…Does every resolution proof that there is no k-clique in G require size n Ω(k) ? For tree-like resolution this problem has been solved in [5].…”

Section: Open Problemsmentioning

confidence: 99%

The Complexity of Proving That a Graph Is Ramsey

Lauria

Pudlák

Rödl

et al. 2013

Automata, Languages, and Programming

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“…Tree Ordered Geometric Resolution (as mentioned in Section 1) is the subclass of Ordered Geometric Resolution that only re-uses the input gap boxes: in other words, if an intermediate box has to be used more than once, then all the set of resolutions leading up to the intermediate box has to be repeated. 10 Figure 2 summarizes the power and limitations of the three classes of resolution above. All of our lower bounds (which we present next) follow by constructing explicit hard examples with an empty output for various classes of resolution that we consider in this paper.…”

Section: Limitations Of Resolution Strategiesmentioning

confidence: 99%

“…This essentially corresponds to a unary representation of the domain and is a representation in proof complexity that has been studied before (see e.g. [10]). Below we state the reasons why the binary encoding that we use above is more reasonable than this unary encoding for our purposes.…”

Section: J2 Encoding Issuesmentioning

confidence: 99%

Joins via Geometric Resolutions

Khamis

Ngo

Ré

et al. 2015

Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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We present a simple geometric framework for the relational join. Using this framework, we design an algorithm that achieves the fractional hypertree-width bound, which generalizes classical and recent worst-case algorithmic results on computing joins. In addition, we use our framework and the same algorithm to show a series of what are colloquially known as beyond worst-case results. The framework allows us to prove results for data stored in Btrees, multidimensional data structures, and even multiple indices per table. A key idea in our framework is formalizing the inference one does with an index as a type of geometric resolution; transforming the algorithmic problem of computing joins to a geometric problem. Our notion of geometric resolution can be viewed as a geometric analog of logical resolution. In addition to the geometry and logic connections, our algorithm can also be thought of as backtracking search with memoization.

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“…It seems hard to prove p-Res(1) lower bounds for parameterized k-clique on a random graph [4], but we now introduce a contradiction that looks similar but for which lower bounds should be easier. It is a variant of the Pigeonhole principle which could give us another very natural separation of p-Res(1) from p-Res (2).…”

mentioning

confidence: 95%

Relativization makes contradictions harder for Resolution

Dantchev¹,

Martin²

2014

Annals of Pure and Applied Logic

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Barnaby MartinSchool of Science and Technology, Middlesex University, The Burroughs, Hendon, London NW4 4BT, U.K. AbstractWe provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d), as well as their tree-like versions, Res * (d). The contradictions we use are natural combinatorial principles: the Least number principle, LNP n and an ordered variant thereof, the Induction principle, IP n . LNP n is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNP n iterated d times provides a separation between Res(d) and Res(d + 1). We prove the same result for the iterated relativization of IP n , where the tree-like variant Res * (d) is considered instead of Res(d).We go on to provide separations between the parameterized versions of Res(1) and Res(2). Here we are able again to use the relativization of the LNP n , but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res * (1) and Res * (2). Here, the relativization of IP n will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming.

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Parameterized Complexity of DPLL Search Procedures

Cited by 15 publications

References 32 publications

The Complexity of Proving That a Graph Is Ramsey

The Complexity of Proving That a Graph Is Ramsey

Joins via Geometric Resolutions

Relativization makes contradictions harder for Resolution

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