On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

Couceiro, Miguel; Haddad, Lucien; Lagerqvist, Victor; Roy, Biman Kumar Saha

doi:10.1109/ismvl.2017.27

Cited by 3 publications

(3 citation statements)

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“…However, the principal motivation was to obtain dichotomy theorems for CSP-like problems incompatible with existential quantification, and the explicit connection to fine-grained time complexity of CSP was not realised until later by Jonsson et al [36]. However, continued advancements in understanding this inclusion structure revealed that even severely restricted classes of constraint languages have a very complicated structure [18,40]. In particular, it is known that any finite language Γ can be characterized only using an infinite number of partial polymorphisms [41]; in fact, for any finite set of purely partial polymorphisms 𝐹 , the language Γ = Inv(𝐹 ) contains 2 2 Θ(𝑛) relations of arity 𝑛, as opposed to finite languages Γ which can qfpp-define only 2 𝑛 𝑂 (1) relations of each arity.…”

Section: Related Workmentioning

confidence: 99%

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

Lagerkvist

Wahlström

2021

ACM Trans. Comput. Theory

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We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH) , showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c 0 > 1 such that the problem cannot be solved in time O ( c n ) for any c < c 0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O ( c n ) for some c < 2. Such lower bounds have proven extremely elusive, and except for cases where c 0 =2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations . Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.

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Section: Related Workmentioning

confidence: 99%

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

Lagerkvist

Wahlström

2021

ACM Trans. Comput. Theory

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“…Unfortunately, this theorem is difficult to apply in practice since it requires a good understanding of the structure of the closed sets pPol(Γ) for all possible choices of Γ. Despite advances made by several different researchers [12,13,35,51], no such classification is known even for Boolean Γ, and even less is known for Γ such that SAT(Γ) is NP-hard. Hence, we propose a method inspired by the rich algebraic toolbox developed for studying the classical complexity of CSP: does the SETH-hardness of SAT(Γ) and CSP(Γ) only depend on the identities satisfied by the partial polymorphisms of Γ?…”

Section: Theorem 1 Let γ and ∆ Be Finite Constraint Languages Over A ...mentioning

confidence: 99%

“…This problem was referred to as the easiest NP-complete SAT problem and was later generalized to a broad class of finite-domain CSPs [32]. However, continued advancements in understanding this inclusion structure revealed that even severely restricted classes of constraint languages had a very complicated structure [12,35]. In a similar vein of negative results it was also proven that (1) pPol(Γ) cannot be generated by any finite set of partial operations whenever Γ is finite and SAT(Γ) is NP-hard, and (2) if P is a finite set of partial operations such that Inv(P)-SAT is NP-hard, then any pp-definable relation over Inv(P) can be transformed into a pp-definition using only a linear number of existentially quantified variables [37].…”

Section: Related Workmentioning

confidence: 99%

Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

Lagerkvist,

Wahlström

2018

Preprint

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We study the complexity of SAT(Γ) problems for potentially infinite languages Γ closed under variable negation, which we refer to as sign-symmetric languages Γ. Via an algebraic connection, this reduces to the study of restricted partial polymorphisms we refer to as pSDI-operations (for partial, self-dual and idempotent), under which the language Γ is invariant. First, we focus on the language classes themselves. We classify the structure of the least restrictive pSDI-operations, corresponding to the most powerful languages Γ, and find that these operations can be divided into levels, corresponding to a rough notion of difficulty, where every level k has an easiest language class, containing the language for (k − 1)-SAT, and a hardest language class, containing (among other things) constraints encoded as roots of multivariate polynomials of degree (k − 1). Particular classes in each level correspond to the natural partially defined versions of previously studied total algebraic invariants. In particular, the easiest class on level k ≥ 3 corresponds to the partial k-ary near-unanimity (k-NU) operation, and a larger class corresponds to the partial k-edge operation. The largest class at each level corresponds to a partial operation u k we call k-universal. Furthermore, every sign-symmetric language Γ not preserved by u k implements all k-clauses, hence SAT(Γ) is at least as hard as k-SAT; and if Γ is not preserved by u k for any k, then SAT(Γ) is trivially SETH-hard (i.e., takes time O * (2 n ) under SETH).Second, we consider implications of this for the complexity of SAT(Γ). We find that particular classes in the hierarchy correspond to previously known algorithmic strategies. In particular, languages preseved by the partial 2-edge operation can be solved via S S -style meet in the middle, and languages preserved by the partial 3-NU operation can be solved via fast matrix multiplication. These results also hold for the correspondning non-Boolean CSP problems. We also find that symmetric 3-edge languages reduce to finding a monochromatic triangle in an edge-coloured graph, which can be done using algorithms for sparse matrix multiplication; and if the sunflower conjecture holds for sunflowers with k petals, then the partial k-NU language has an improved algorithm via Schöning-style local search.Complementing this, we show a lower bound, showing that for every level k there is a constant c k such that for every partial operation p on level k, the problem SAT(Γ) with Γ = Inv(p) cannot be solved faster than O * (c n k ) unless SETH fails. In particular, when Γ = Inv(2-edge), this gives us the first NP-hard SAT problem which simultaneously has non-trivial upper and lower bounds on the running time, assuming SETH. Finally, we note a possible conjecture: It is consistent with our present knowledge that SAT(Γ) admits an improved algorithm if and only if Γ is preserved by u k for some constant k. However, to show this in the positive poses some significant difficulty.

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