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Solving CSPs Using Weak Local Consistency
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2021-07-27 , DOI: 10.1137/18m117577x Marcin Kozik
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2021-07-27 , DOI: 10.1137/18m117577x Marcin Kozik
SIAM Journal on Computing, Volume 50, Issue 4, Page 1263-1286, January 2021.
The characterization of all the constraint satisfaction problems solvable by local consistency checking (also known as CSPs of bounded width) was proposed by Feder and Vardi [SIAM J. Comput., 28 (1998), pp. 57--104]. It was confirmed by two independent proofs by Bulatov [Bounded Relational Width, manuscript, 2009] and Barto and Kozik [L. Barto and M. Kozik, 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, pp. 595--603], [L. Barto and M. Kozik, J. ACM, 61 (2014), 3]. Later Barto [J. Logic Comput., 26 (2014), pp. 923--943] proved a collapse of the hierarchy of local consistency notions by showing that (2,3) minimality solves all the CSPs of bounded width. In this paper we present a new consistency notion, jpq consistency, which also solves all the CSPs of bounded width. Our notion is strictly weaker than (2,3) consistency, (2,3) minimality, path consistency, and singleton arc consistency (SAC). This last fact allows us to answer the question of Chen, Dalmau, and Grußien [J. Logic Comput., 23 (2013), pp. 87--108] by confirming that SAC solves all the CSPs of bounded width. Moreover, as known algorithms work faster for SAC, the result implies that CSPs of bounded width can be, in practice, solved more efficiently. The definition of jpq consistency is closely related to a consistency condition obtained from the rounding of an SDP relaxation of a CSP instance. In fact, the main result of this paper is used by Dalmau et al. [Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2017, pp. 340--357] to show that CSPs with near unanimity polymorphisms admit robust approximation algorithms with polynomial loss. Finally, an algebraic characterization of some term conditions satisfied in algebras associated with templates of bounded width, first proved by Brady, is a direct consequence of our result.
中文翻译：
使用弱局部一致性解决 CSP
SIAM Journal on Computing，第 50 卷，第 4 期，第 1263-1286 页，2021 年 1 月。
Feder 和 Vardi [SIAM J. Comput., 28 (1998), pp. 57--104] 提出了可通过局部一致性检查（也称为有界宽度的 CSP）解决的所有约束满足问题的表征。Bulatov [Bounded Relational Width，手稿，2009] 和 Barto 和 Kozik [L. Barto 和 M. Kozik，第 50 届 IEEE 计算机科学基础研讨会，2009 年，第 595--603 页]，[L. Barto 和 M. Kozik, J. ACM, 61 (2014), 3]。后来巴托 [J. Logic Comput., 26 (2014), pp. 923--943] 通过显示 (2,3) 极小性解决所有有界宽度的 CSP，证明了局部一致性概念的层次结构的崩溃。在本文中，我们提出了一个新的一致性概念 jpq 一致性，它也解决了所有有界宽度的 CSP。我们的概念严格弱于（2,3）一致性，（2, 3）极小性、路径一致性和单例弧一致性（SAC）。最后一个事实使我们能够回答 Chen、Dalmau 和 Grußien 的问题 [J. Logic Comput., 23 (2013), pp. 87--108] 通过确认 SAC 解决了所有有界宽度的 CSP。此外，由于已知算法对 SAC 的工作速度更快，结果意味着在实践中可以更有效地解决有界宽度的 CSP。jpq 一致性的定义与 CSP 实例的 SDP 松弛四舍五入得到的一致性条件密切相关。事实上，本文的主要结果被 Dalmau 等人使用。[第 28 届年度 ACM-SIAM 离散算法研讨会论文集，SIAM，费城，ACM，纽约，2017 年，第 340--357 页] 表明具有接近一致多态性的 CSP 承认具有多项式损失的稳健逼近算法。最后，
更新日期：2021-10-03
The characterization of all the constraint satisfaction problems solvable by local consistency checking (also known as CSPs of bounded width) was proposed by Feder and Vardi [SIAM J. Comput., 28 (1998), pp. 57--104]. It was confirmed by two independent proofs by Bulatov [Bounded Relational Width, manuscript, 2009] and Barto and Kozik [L. Barto and M. Kozik, 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, pp. 595--603], [L. Barto and M. Kozik, J. ACM, 61 (2014), 3]. Later Barto [J. Logic Comput., 26 (2014), pp. 923--943] proved a collapse of the hierarchy of local consistency notions by showing that (2,3) minimality solves all the CSPs of bounded width. In this paper we present a new consistency notion, jpq consistency, which also solves all the CSPs of bounded width. Our notion is strictly weaker than (2,3) consistency, (2,3) minimality, path consistency, and singleton arc consistency (SAC). This last fact allows us to answer the question of Chen, Dalmau, and Grußien [J. Logic Comput., 23 (2013), pp. 87--108] by confirming that SAC solves all the CSPs of bounded width. Moreover, as known algorithms work faster for SAC, the result implies that CSPs of bounded width can be, in practice, solved more efficiently. The definition of jpq consistency is closely related to a consistency condition obtained from the rounding of an SDP relaxation of a CSP instance. In fact, the main result of this paper is used by Dalmau et al. [Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2017, pp. 340--357] to show that CSPs with near unanimity polymorphisms admit robust approximation algorithms with polynomial loss. Finally, an algebraic characterization of some term conditions satisfied in algebras associated with templates of bounded width, first proved by Brady, is a direct consequence of our result.
中文翻译：
使用弱局部一致性解决 CSP
SIAM Journal on Computing，第 50 卷，第 4 期，第 1263-1286 页，2021 年 1 月。
Feder 和 Vardi [SIAM J. Comput., 28 (1998), pp. 57--104] 提出了可通过局部一致性检查（也称为有界宽度的 CSP）解决的所有约束满足问题的表征。Bulatov [Bounded Relational Width，手稿，2009] 和 Barto 和 Kozik [L. Barto 和 M. Kozik，第 50 届 IEEE 计算机科学基础研讨会，2009 年，第 595--603 页]，[L. Barto 和 M. Kozik, J. ACM, 61 (2014), 3]。后来巴托 [J. Logic Comput., 26 (2014), pp. 923--943] 通过显示 (2,3) 极小性解决所有有界宽度的 CSP，证明了局部一致性概念的层次结构的崩溃。在本文中，我们提出了一个新的一致性概念 jpq 一致性，它也解决了所有有界宽度的 CSP。我们的概念严格弱于（2,3）一致性，（2, 3）极小性、路径一致性和单例弧一致性（SAC）。最后一个事实使我们能够回答 Chen、Dalmau 和 Grußien 的问题 [J. Logic Comput., 23 (2013), pp. 87--108] 通过确认 SAC 解决了所有有界宽度的 CSP。此外，由于已知算法对 SAC 的工作速度更快，结果意味着在实践中可以更有效地解决有界宽度的 CSP。jpq 一致性的定义与 CSP 实例的 SDP 松弛四舍五入得到的一致性条件密切相关。事实上，本文的主要结果被 Dalmau 等人使用。[第 28 届年度 ACM-SIAM 离散算法研讨会论文集，SIAM，费城，ACM，纽约，2017 年，第 340--357 页] 表明具有接近一致多态性的 CSP 承认具有多项式损失的稳健逼近算法。最后，
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