CLAP: A New Algorithm for Promise CSPs

Ciardo, Lorenzo; Živný, Stanislav

doi:10.48550/arxiv.2107.05018

Cited by 2 publications

(3 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. We first observe that the condition M → Pol(A, B) is equivalent to the condition F M (A) → B by [12,Lemma 4.4] (see also [41,Lemma 22] for the proof for abstract minions).…”

Section: Minion Testsmentioning

confidence: 99%

Hierarchies of Minion Tests for PCSPs through Tensors

Ciardo¹,

Živný²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the "base level" of the hierarchy, and a geometric part -which we call tensorisation -inspired by multilinear algebra.We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, in order to analyse the Sum-of-Squares SDP hierarchy we also characterise the solvability of the standard SDP relaxation through a new minion. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.

show abstract

“…Proof. We first observe that the condition M → Pol(A, B) is equivalent to the condition F M (A) → B by [12,Lemma 4.4] (see also [41,Lemma 22] for the proof for abstract minions).…”

Section: Minion Testsmentioning

confidence: 99%

Hierarchies of Minion Tests for PCSPs through Tensors

Ciardo¹,

Živný²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…PCSPs were introduced by Austrin, Guruswami, and Håstad [5] and Brakensiek and Guruswami [16] as a general framework for studying approximability of perfectly satisfiable CSPs and have emerged as a new exciting direction in constraint satisfaction that requires different techniques than CSPs. 2 Recent works on PCSPs include those using analytical methods [12,13,17,22] and those building on algebraic methods [3,7,10,15,18,19,26,31,45,63] developed in [8]. However, most basic questions are still left open, including complexity classifications and applicability of different types of algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…This has led some researchers to believe that the algorithmic hierarchy based on BA could be a universal constraint-satisfaction solver -i.e., a constant level of the hierarchy could solve all tractable CSPs, cf. [18,31,34,58].…”

Section: Introductionmentioning

confidence: 99%