Robust Algorithms with Polynomial Loss for Near-Unanimity CSPs

Dalmau, Víctor; Kozik, Marcin; Krokhin, Andrei; Makarychev, Konstantin; Makarychev, Yury; Opršal, Jakub

doi:10.1137/18m1163932

Cited by 9 publications

(7 citation statements)

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“…show that, modulo P NP, if Min CSP(Γ) admits a constant-factor-approximation algorithm then Γ must have a near-unanimity (NU) polymorphism (recall the definition of an NU operation from Section 2.2). NU polymorphisms have been well studied in universal algebra [47] and have been applied in CSP [43,40,7,38]. For example, every relation invariant under an n-ary NU operation is uniquely determined by its (n − 1)-ary projections [47], and NU polymorphisms characterize CSPs of "bounded strict width" [7].…”

Section: Np-hardness Resultsmentioning

confidence: 99%

Towards a characterization of constant-factor approximable finite-valued CSPs

Dalmau

Krokhin

Manokaran

2018

Journal of Computer and System Sciences

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In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. An instance of VCSP is given by a finite set of variables and a sum of functions belonging to Γ and depending on a subset of the variables. Each function takes values in [0, 1] specifying costs of assignments of labels to its variables, and the goal is to find an assignment of labels to the variables that minimizes the sum. A recent result of Ene et al. says that, under the mild technical condition that Γ contains the function corresponding to the equality relation, the basic LP relaxation is optimal for constant-factor approximation for VCSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP(Γ). We also show how these algebraic conditions can in principle be used to round solutions of the basic LP relaxation, and how this leads to efficient constant-factor approximation algorithms for several examples that cover all previously known cases that are NP-hard to solve to optimality but admit constant-factor approximation. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation. Thus, our results strongly indicate where the boundary of constant-factor approximability for VCSPs lies. ✩ This article is an extended version of a paper published in the proceedings of SODA'15. (Rajsekar Manokaran) 21, 32]). Much work around the Unique Games Conjecture (UGC) directly concerns CSPs [21]. This conjecture states that, for any ǫ > 0, there is a large enough number k = k(ǫ) such that it NP-hard to tell ǫ-satisfiable from (1 − ǫ)-satisfiable instances of CSP(Γ k ), where Γ k consists of all graphs of bijections on a k-element set. Many approximation algorithms for classical optimization problems have been shown optimal assuming the UGC [21,32]. Raghavendra proved [17] that one SDP-based algorithm provides optimal approximation for all problems GCSP(Γ) assuming the UGC. In this paper, we investigate problems VCSP(Γ) and Min CSP(Γ) on an arbitrary finite domain that belong to APX, i.e. admit a (polynomial-time) constant-factor approximation algorithm, proving some results that strongly indicate where the boundary of this property lies.Related Work. Note that each problem Max CSP(Γ) trivially admits a constantfactor approximation algorithm because a random assignment of values to the variables is guaranteed to satisfy a constant fraction of constraints; this can be derandomized by the standard method of conditional probabilities. The same also holds for GCSP. Clearly, for Min CSP(Γ) to admit a constant-factor approximation algorithm, CSP(Γ) must be polynomial-time solvable.The approximability of problems VCSP(Γ) has been studied, mostly for Min CSPs in the Boolean case (i.e., with domain {0, 1}, such CSPs are sometimes ...

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Section: Np-hardness Resultsmentioning

confidence: 99%

Towards a characterization of constant-factor approximable finite-valued CSPs

Dalmau

Krokhin

Manokaran

2018

Journal of Computer and System Sciences

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“…a linear program. In the context of CSPs, convex relaxations have been studied for robust solvability [40,26,4,25]. Convex relaxations have been also successfully applied to the study of the three extensions of CSPs.…”

Section: :3mentioning

confidence: 99%

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Viola

Živný

2021

ACM Trans. Algorithms

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Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

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“…What is the minimum number of simultaneously unsatisfied constraints [48,67]? Given an almost satisfiable instance, can one find a somewhat satisfying solution [12,46,47]? In this paper, we will focus on the following question:…”

Section: Introductionmentioning

confidence: 99%

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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Robust Algorithms with Polynomial Loss for Near-Unanimity CSPs

Cited by 9 publications

References 35 publications

Towards a characterization of constant-factor approximable finite-valued CSPs

Towards a characterization of constant-factor approximable finite-valued CSPs

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

The Sherali-Adams Hierarchy for Promise CSPs through Tensors

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