Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Butti, Silvia; Dalmau, Víctor

doi:10.4230/lipics.mfcs.2021.27

Cited by 7 publications

(24 citation statements)

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“…Essentially, the k-th level of the hierarchy is obtained by enforcing consistency of probability distributions over partial assignments on up to k variables in the input structure. We shall follow the definition as presented in [36]. 2 Given two σ-structures X, A, we introduce a variable λ V (f ) for every subset V ⊆ X with 1 ≤ |V | ≤ k and every function f : V → A, and a variable λ R,x (f ) for every R ∈ σ, every x ∈ R X , and every f : {x} → A.…”

Section: Sherali-adams Lp Hierarchymentioning

confidence: 99%

“…In the context of (exact solvability of) CSPs, [86] characterised the power of Sherali-Adams for valued CSPs, which implies a characterisation for CSPs: The k-th level, for k ≥ 3, solves a CSP if and only if the third level solves it, and this coincides with the condition that characterises the power of the local consistency algorithm [11,31], where the collapse to the third level was shown [8]. Butti and Dalmau [36] recently characterised for CSPs when the k-th level of the Sherali-Adams linear programming hierarchy accepts in terms of a construction different from the one introduced in this work. Unlike the tensorisation, the construction considered in [36] yields a relational structure whose domain includes the set of constraints of the original structure.…”

Section: Introductionmentioning

confidence: 97%

“…The work of Barto, Bulín, Krokhin, and Opršal [10] lifted and greatly extended the algebraic framework developed for CSPs [15,32,64] to the realm of PCSPs. Subsequently, there has been a series of recent works on the computational complexity of PCSPs building on [10], including applicability of convex relaxations [3,22,23,27,36,41] and complexity of fragments of PCSPs [2,9,13,25,28,60]. Other strong results on PCSPs have also been established via other techniques than those in [10], e.g., hardness of various (hyper)graph colourings [6,51,62,68] and other PCSPs [17,18,26,30].…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to describe the functioning of the algorithms in the hierarchy by describing the geometry of a space of tensors -which can be accomplished by using multilinear algebra. As far as we know, this approach has not appeared in the literature on Sherali-Adams (and related algorithmic techniques such as local consistency [11] and the high-dimensional Weisfeiler-Leman algorithm [4,36]).…”

Section: Introductionmentioning

confidence: 99%

“…Butti and Dalmau [36] recently characterised for CSPs when the k-th level of the Sherali-Adams linear programming hierarchy accepts in terms of a construction different from the one introduced in this work. Unlike the tensorisation, the construction considered in [36] yields a relational structure whose domain includes the set of constraints of the original structure.…”

Section: Introductionmentioning

confidence: 99%

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The Sherali-Adams Hierarchy for Promise CSPs through Tensors

Ciardo¹,

Živný²

2022

Preprint

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We study the Sherali-Adams linear programming hierarchy in the context of promise constraint satisfaction problems (PCSPs). We characterise when a level of the hierarchy accepts an instance in terms of a homomorphism problem for an appropriate multilinear structure obtained through a tensor power of the constraint language. The geometry of this structure, which consists in a space of tensors satisfying certain symmetries, allows then to establish non-solvability of the approximate graph colouring problem via constantly many levels of Sherali-Adams.Besides this primary application, our tensorisation construction introduces a new tool to the study of hierarchies of algorithmic relaxations for computational problems within (and, possibly, beyond) the context of constraint satisfaction. In particular, we see it as a key step towards the algebraic characterisation of the power of Sherali-Adams for PCSPs. * 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.constant level of the Sherali-Adams linear programming relaxation otherwise [7]. This classification was later extended to problems over arbitrary finite domains by Brandts et al. [28].Another example of a PCSP, identified in [24], is finding a "not-all-equal" assignment to a monotone 3-CNF formula given that a "1-in-3" assignment is promised to exist; i.e., given a 3-CNF formula with positive literals only and the promise that an assignment exists that satisfies exactly one literal in each clause, the task is to find an assignment that satisfies one or two literals in each clause. This problem is solvable in polynomial time via a constant level of the Sherali-Adams linear programming relaxation [24] but not via a reduction to finite-domain CSPs [10].A third example of a PCSP is the well-known approximate graph colouring problem: Given a c-colourable graph, find a d-colouring of it, for c ≤ d. In the decision version, the problem asks to distinguish graphs that are c-colourable from graphs that are not even d-colourable. This corresponds to PCSP(K c , K d ), where K p is the clique on p vertices. Contrary to the two examples above -and despite a long history dating back to 1976 [56] -the complexity of this problem is still unknown in general; it is widely believed that it is NP-hard for any constant values of c and d with 3 ≤ c ≤ d. For c = d, it becomes the classic c-colouring problem, which appeared on Karp's original list of 21 NP-complete problems [65]. The case c = 3, d = 4 was only proved to be NP-hard in 2000 by Khanna, Linial, and Safra [66] (cf. also [59]); more generally, they showed hardness of the case d = c + 2⌊c/3⌋ − 1. This was improved to d = 2c − 2 in 2016 [21], and recently to d = 2c − 1 in [10]. In pa...

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Section: Sherali-adams Lp Hierarchymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Sherali-Adams Hierarchy for Promise CSPs through Tensors

Ciardo¹,

Živný²

2022

Preprint

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Approximate Graph Colouring and the Hollow Shadow

Ciardo

Živný

2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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The Weisfeiler-Leman Dimension of Conjunctive Queries

Göbel,

Goldberg,

Roth

2024

Proc. ACM Manag. Data

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A graph parameter is a function f on graphs with the property that, for any pair of isomorphic graphs G 1 and G 2 , f(G 1 )=f(G 2 ). The Weisfeiler--Leman (WL) dimension of f is the minimum k such that, if G 1 and G 2 are indistinguishable by the k-dimensional WL-algorithm then f(G 1 )=f(G 2 ). The WL-dimension of f is ∞ if no such k exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ, we quantify the WL-dimension of the function that maps every graph G to the number of answers of φ in G. The works of Dvorak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ, the WL-dimension is equal to the treewidth of the Gaifman graph of φ. In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query φ, we prove that its WL-dimension is equal to the semantic extension width sew(φ), a novel width measure that can be thought of as a combination of the treewidth of φ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ cannot be computed by GNNs of order smaller than sew(φ). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query φ with semantic extension width k, we consider a graph F of treewidth k obtained from the Gaifman graph of φ by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Furer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs χ(F) and ^χ(F) that are indistinguishable by the (k-1)-dimensional WL-algorithm since F has treewidth k. Finally, in the technical heart of the paper, we show that φ has a different number of answers in χ(F) and ^χ(F). Thus, φ can distinguish two graphs that cannot be distinguished by the (k-1)-dimensional WL-algorithm, so the WL-dimension of φ is at least k.

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Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Cited by 7 publications

References 0 publications

The Sherali-Adams Hierarchy for Promise CSPs through Tensors

The Sherali-Adams Hierarchy for Promise CSPs through Tensors

Approximate Graph Colouring and the Hollow Shadow

The Weisfeiler-Leman Dimension of Conjunctive Queries

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