SIAM Journal on Computing, Volume 53, Issue 4, Page 884-968, August 2024. Abstract. We consider the extension of [math] with quantifiers that state that the number of elements where a formula holds should belong to a given ultimately periodic set. We show that both satisfiability and finite satisfiability of the logic are decidable. We also show that the spectrum of any sentence, i.e., the set of the sizes of its finite models, is definable in Presburger arithmetic. In the process we present several refinements to the “biregular graph method.” In this method, decidability issues concerning two-variable logics are reduced to questions about Presburger definability of integer vectors associated with partitioned graphs, where nodes in a partition satisfy certain constraints on their in- and out-degrees.
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SIAM Journal on Computing, Volume 53, Issue 4, Page 825-883, August 2024. Abstract. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix [math] is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix [math] that is not testable. In the dense graph model, a similar picture has long been known [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451–476] despite the very different nature of the two models. In particular, we obtain our lower bound by an FO formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then use our class of FO definable bounded-degree expanders to answer a long-standing open problem for proximity-oblivious testers (POTs). POTs are a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SIAM J. Comput., 40 (2011), pp. 534–566] show that the graph properties that are constant-query proximity-oblivious testable in the bounded-degree model are precisely the properties that can be expressed as a generalized subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalized subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by showing that our FO definable property is GSF-local and propagating. Hence, in particular, our property does not admit a POT, despite being GSF-local. For this result we establish a new connection between FO properties and GSF-local properties via neighborhood profiles.
SIAM 计算期刊》，第 53 卷第 4 期，第 825-883 页，2024 年 8 月。 摘要。我们研究了有界度图模型和关系结构模型中一阶逻辑（FO）可定义属性的属性测试。我们证明，任何由带量词前缀[math]的公式定义的一阶逻辑属性都是可测试的（即以恒定的查询复杂度进行测试），而存在一个由带量词前缀[math]的公式表达的一阶逻辑属性是不可测试的。Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp.特别是，我们通过一个 FO 公式获得了下界，该公式定义了一类基于图的之字形乘积的有界度扩展器。我们希望这一点能引起我们的兴趣。然后，我们将利用我们这一类可定义的有界度扩展器来回答一个长期悬而未决的问题--近似盲测试器（POTs）。POT 是一类特别简单的测试算法，其中基本测试的执行次数可能取决于邻近度参数，但基本测试本身与邻近度参数无关。Goldreich 和 Ron [STOC 2009; SIAM J. Comput., 40 (2011), pp.至于非传播条件是否必要，目前还没有定论。事实上，将可表达为广义子图自由性属性的属性称为 GSF 局部属性，他们会问是否所有 GSF 局部属性都是非传播的。我们给出了否定的答案，证明了我们的 FO 可定义属性是 GSF 局部属性和传播属性。因此，特别是，尽管我们的属性是 GSF-local 属性，但它并不包含 POT。为此，我们通过邻域轮廓在 FO 性质和 GSF-local 性质之间建立了新的联系。
{"title":"On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing","authors":"Isolde Adler, Noleen Köhler, Pan Peng","doi":"10.1137/23m1556253","DOIUrl":"https://doi.org/10.1137/23m1556253","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 825-883, August 2024. <br/> Abstract. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix [math] is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix [math] that is not testable. In the dense graph model, a similar picture has long been known [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451–476] despite the very different nature of the two models. In particular, we obtain our lower bound by an FO formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then use our class of FO definable bounded-degree expanders to answer a long-standing open problem for proximity-oblivious testers (POTs). POTs are a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SIAM J. Comput., 40 (2011), pp. 534–566] show that the graph properties that are constant-query proximity-oblivious testable in the bounded-degree model are precisely the properties that can be expressed as a generalized subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalized subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by showing that our FO definable property is GSF-local and propagating. Hence, in particular, our property does not admit a POT, despite being GSF-local. For this result we establish a new connection between FO properties and GSF-local properties via neighborhood profiles.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"12 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Volume 53, Issue 4, Page 803-824, August 2024. Abstract. We consider the problem of finding an [math]-approximate stationary point of a smooth function on a compact domain of [math]. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where [math] is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension [math], improves upon the [math] oracle complexity of gradient descent. For example for [math], Vavasis’s approach obtains the complexity [math]. Moreover, for [math] he also proved a lower bound of [math] for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension [math], GFT closes the gap with Vavasis’s lower bound (up to a logarithmic factor), as we show that it has complexity [math]. In dimension [math], we show a complexity of [math], improving upon Vavasis’s [math]. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis’s algorithm. We augment this result with another algorithm, named cut and flow (CF), which improves upon Vavasis’s algorithm in any fixed dimension.
{"title":"How to Trap a Gradient Flow","authors":"Sébastien Bubeck, Dan Mikulincer","doi":"10.1137/21m1397854","DOIUrl":"https://doi.org/10.1137/21m1397854","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 803-824, August 2024. <br/> Abstract. We consider the problem of finding an [math]-approximate stationary point of a smooth function on a compact domain of [math]. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where [math] is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension [math], improves upon the [math] oracle complexity of gradient descent. For example for [math], Vavasis’s approach obtains the complexity [math]. Moreover, for [math] he also proved a lower bound of [math] for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension [math], GFT closes the gap with Vavasis’s lower bound (up to a logarithmic factor), as we show that it has complexity [math]. In dimension [math], we show a complexity of [math], improving upon Vavasis’s [math]. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis’s algorithm. We augment this result with another algorithm, named cut and flow (CF), which improves upon Vavasis’s algorithm in any fixed dimension.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"26 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. Counting graph homomorphisms and its generalizations such as the counting constraint satisfaction problem (CSP), variations of the counting CSP, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms [M. Dyer and C. Greenhill, Random Structures Algorithms, 17 (2000), pp. 260–289] and the counting CSP [A. A. Bulatov, J. ACM, 60 (2013), pp. 34:1–34:41, and M. E. Dyer and D. Richerby, SIAM J. Comput., 42 (2013), pp. 1245–1274] is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper, [J. Faben and M. Jerrum, Theory Comput., 11 (2015), pp. 35–57] suggested a conjecture stating that counting homomorphisms to a fixed graph [math] modulo a prime number is hard whenever it is hard to count exactly unless [math] has automorphisms of certain kind. In this paper, we confirm this conjecture. As a part of this investigation, we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.
SIAM 计算期刊》，提前印刷。 摘要自 Valiant 的开创性工作以来，人们一直在深入研究图同态计数及其广义问题，如计数约束满足问题 (CSP)、计数 CSP 变体以及一般计数问题。虽然图同态精确计数的复杂性 [M. Dyer and C. Greenhill, 2007]，但在计算图同态精确计数的复杂性方面却存在着巨大的差距。Dyer and C. Greenhill, Random Structures Algorithms, 17 (2000), pp.A. Bulatov, J. ACM, 60 (2013), pp.J. Faben 和 M. Jerrum，Theory Comput.，11 (2015)，pp. 35-57]在他们 2015 年的论文中提出了一个猜想，指出对固定图 [math] modulo 一个素数的同态计数是很难的，除非 [math] 有某类自动态。在本文中，我们证实了这一猜想。作为研究的一部分，我们开发了一些技术，这些技术拓宽了模块计数的还原范围，并适用于一般的 CSP，而不局限于图同态。
{"title":"Complexity Classification of Counting Graph Homomorphisms Modulo a Prime Number","authors":"Andrei Bulatov, Amirhossein Kazeminia","doi":"10.1137/23m1548633","DOIUrl":"https://doi.org/10.1137/23m1548633","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. Counting graph homomorphisms and its generalizations such as the counting constraint satisfaction problem (CSP), variations of the counting CSP, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms [M. Dyer and C. Greenhill, Random Structures Algorithms, 17 (2000), pp. 260–289] and the counting CSP [A. A. Bulatov, J. ACM, 60 (2013), pp. 34:1–34:41, and M. E. Dyer and D. Richerby, SIAM J. Comput., 42 (2013), pp. 1245–1274] is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper, [J. Faben and M. Jerrum, Theory Comput., 11 (2015), pp. 35–57] suggested a conjecture stating that counting homomorphisms to a fixed graph [math] modulo a prime number is hard whenever it is hard to count exactly unless [math] has automorphisms of certain kind. In this paper, we confirm this conjecture. As a part of this investigation, we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"25 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin
SIAM Journal on Computing, Volume 53, Issue 3, Page 764-802, June 2024. Abstract. We consider proof complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this proof complexity with the normal unary encoding in several refutation systems, based on Resolution and Sherali–Adams. We first consider [math], which is an extension of Resolution working on [math]-DNFs (Disjunctive Normal Form formulas). We prove an exponential lower bound of [math] for the size of refutations of the binary version of the [math]-Clique Principle in [math], where [math] and [math] is a doubly exponential function. Our result improves that of Lauria et al., who proved a similar lower bound for [math], i.e., Resolution. For the [math]-Clique and other principles we study, we show how lower bounds in Resolution for the unary version follow from lower bounds in [math] for the binary version, so we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the (weak) Pigeonhole Principle [math]. We prove that for any [math], [math] requires refutations of size [math] in [math] for [math]. Our lower bound cannot be improved substantially with the same method since for [math] we can prove there are [math] size refutations of [math] in [math]. This is a consequence of the same upper bound for the unary weak Pigeonhole Principle of Buss and Pitassi. We contrast unary versus binary encoding in the Sherali–Adams (SA) refutation system where we prove lower bounds for both rank and size. For the unary encoding of the Pigeonhole Principle and the Ordering Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the (weak) Pigeonhole Principle [math] requires exponentially sized (in [math]) SA refutations, whereas the binary encoding of the Ordering Principle admits logarithmic rank, polynomially sized SA refutations. We continue by considering a natural refutation system we call “SA+Squares,” which is intermediate between SA and Lasserre (Sum-of-Squares). This has been studied under the name static-[math] by Grigoriev et al. In this system, the unary encoding of the Linear Ordering Principle [math] requires [math] rank while the unary encoding of the Pigeonhole Principle becomes constant rank. Since Potechin has shown that the rank of [math] in Lasserre is [math], we uncover an almost quadratic separation between SA+Squares and Lasserre in terms of rank. Grigoriev et al. noted that the unary Pigeonhole Principle has rank 2 in SA+Squares and therefore polynomial size. Since we show the same applies to the binary [math], we deduce an exponential separation for size between SA and SA+Squares.
SIAM 计算期刊》，第 53 卷第 3 期，第 764-802 页，2024 年 6 月。 摘要。我们根据某些组合原理的不寻常二进制编码来考虑证明复杂性。我们将这种证明复杂性与基于解析（Resolution）和谢拉利-亚当斯（Sherali-Adams）的几个驳斥系统中的正常一元编码进行对比。我们首先考虑[math]，它是在[math]-DNFs（Disjunctive Normal Form form 公式）上运行的 Resolution 的扩展。我们证明了[math]中二元版[math]-Clique 原则的反驳大小的指数下限，其中[math]和[math]是双指数函数。我们的结果改进了劳里亚等人的结果，他们证明了[math]的类似下界，即分辨率。对于我们研究的[math]-Clique 和其他原理，我们展示了一元版本的解析下界是如何与二元版本的[math]下界相一致的，因此我们开始系统地研究基于解析的系统对二元编码给出的矛盾族进行证明的复杂性。我们接着考虑二进制版本的（弱）鸽洞原理[math]。我们证明，对于任何 [math]，[math] 都需要在 [math] 中对 [math] 进行大小为 [math] 的反驳。我们的下界无法用同样的方法大幅提高，因为对于[math]，我们可以证明在[math]中存在[math]大小的[math]反驳。这是布斯和皮塔西的一元弱鸽子洞原理的同一上界的结果。我们对比了谢拉利-亚当斯（Sherali-Adams，SA）反驳系统中的一元编码和二元编码，证明了秩和大小的下界。对于鸽子洞原理和排序原理的一元编码，众所周知，SA 中的驳斥需要线性秩，尽管两者都允许多项式大小的驳斥。我们证明，（弱）鸽洞原理[math]的二进制编码需要指数大小（在[math]中）的 SA 反驳，而排序原理的二进制编码允许对数等级、多项式大小的 SA 反驳。我们将继续考虑一种我们称之为 "SA+Squares "的自然驳斥系统，它介于 SA 和 Lasserre（Sum-of-Squares）之间。在这个系统中，线性排序原理[math]的一元编码需要[math]秩，而鸽子洞原理的一元编码则需要恒定秩。由于波钦（Potechin）已经证明拉塞尔方程中[math]的秩为[math]，因此我们发现 SA+Squares 与拉塞尔方程在秩方面几乎存在二次分隔。格里戈里耶夫等人指出，一元鸽洞原理在 SA+Squares 中的秩为 2，因此大小为多项式。由于我们证明二元[math]也是如此，因此我们推导出 SA 和 SA+Squares 的大小呈指数级分离。
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SIAM Journal on Computing, Volume 53, Issue 3, Page 701-763, June 2024. Abstract. Three decades ago, Karp, Vazirani, and Vazirani [Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 352–358] defined the online matching problem and gave an optimal [math]-competitive algorithm. Fifteen years later, Mehta et al. [J. ACM, 54 (2007), pp. 22:1–22:19] introduced the first generalization called AdWords driven by online advertising and obtained the optimal [math] competitive ratio in the special case of small bids. It has been open ever since whether there is an algorithm for general bids better than the 0.5-competitive greedy algorithm. This paper presents a 0.5016-competitive algorithm for AdWords, answering this open question on the positive end. The algorithm builds on several ingredients, including a combination of the online primal dual framework and the configuration linear program of matching problems recently explored by Huang and Zhang [Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020], a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach et al. [Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020], which we call the panoramic online correlated selection.
SIAM 计算期刊》，第 53 卷第 3 期，第 701-763 页，2024 年 6 月。 摘要三十年前，Karp、Vazirani 和 Vazirani [Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 1990, pp.15 年后，Mehta 等人 [J. ACM, 54 (2007), pp.从那时起，是否存在比 0.5 竞争贪婪算法更好的一般出价算法一直是个未知数。本文提出了一种适用于 AdWords 的 0.5016 竞争性算法，从正面回答了这一开放性问题。该算法基于多个要素，包括黄和张（Huang and Zhang）[第 52 届 ACM 计算理论研讨会论文集，2020 年]最近探索的在线原始对偶框架与匹配问题的配置线性程序的结合、我们称之为全景视图的 AdWords 的新表述，以及 Fahrbach 等人[第 61 届 IEEE 计算机科学基础研讨会论文集，2020 年]对在线相关选择的概括，我们称之为全景在线相关选择。
{"title":"AdWords in a Panorama","authors":"Zhiyi Huang, Qiankun Zhang, Yuhao Zhang","doi":"10.1137/22m1478896","DOIUrl":"https://doi.org/10.1137/22m1478896","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 3, Page 701-763, June 2024. <br/> Abstract. Three decades ago, Karp, Vazirani, and Vazirani [Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, 1990, pp. 352–358] defined the online matching problem and gave an optimal [math]-competitive algorithm. Fifteen years later, Mehta et al. [J. ACM, 54 (2007), pp. 22:1–22:19] introduced the first generalization called AdWords driven by online advertising and obtained the optimal [math] competitive ratio in the special case of small bids. It has been open ever since whether there is an algorithm for general bids better than the 0.5-competitive greedy algorithm. This paper presents a 0.5016-competitive algorithm for AdWords, answering this open question on the positive end. The algorithm builds on several ingredients, including a combination of the online primal dual framework and the configuration linear program of matching problems recently explored by Huang and Zhang [Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020], a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach et al. [Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020], which we call the panoramic online correlated selection.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"55 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. We prove an optimal [math] lower bound on a spectral gap of the Glauber dynamics for antiferromagnetic two-spin systems with [math] vertices in the tree uniqueness regime. This spectral gap holds for any, including unbounded, maximum degree [math]. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack [math]: [math] mixing time for the hardcore model with fugacity [math] and [math] mixing time for the Ising model with edge activity [math], where the maximum degree [math] may depend on the number of vertices [math] and [math] depends only on [math]. Our proof is built on the recently developed connections between the Glauber dynamics for spin systems and high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called complete spectral independence, and use a novel Markov chain, called field dynamics, to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.
{"title":"Rapid Mixing of Glauber Dynamics via Spectral Independence for All Degrees","authors":"Xiaoyu Chen, Weiming Feng, Yitong Yin, Xinyuan Zhang","doi":"10.1137/22m1474734","DOIUrl":"https://doi.org/10.1137/22m1474734","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We prove an optimal [math] lower bound on a spectral gap of the Glauber dynamics for antiferromagnetic two-spin systems with [math] vertices in the tree uniqueness regime. This spectral gap holds for any, including unbounded, maximum degree [math]. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack [math]: [math] mixing time for the hardcore model with fugacity [math] and [math] mixing time for the Ising model with edge activity [math], where the maximum degree [math] may depend on the number of vertices [math] and [math] depends only on [math]. Our proof is built on the recently developed connections between the Glauber dynamics for spin systems and high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called complete spectral independence, and use a novel Markov chain, called field dynamics, to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"51 1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yaroslav Alekseev, Dima Grigoriev, Edward A. Hirsch, Iddo Tzameret
SIAM Journal on Computing, Volume 53, Issue 3, Page 648-700, June 2024. Abstract. We introduce the binary value principle, which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, based on a well-known hypothesis by Shub and Smale about the hardness of computing factorials, where IPS is the strong algebraic proof system introduced by Grochow and Pitassi [J. ACM, 65 (2018), 37]. Conversely, we show that short IPS refutations of this instance bridge the gap between sufficiently strong algebraic and semialgebraic proof systems. Our results extend to unrestricted IPS the paradigm introduced by Forbes, Shpilka, Tzameret, and Wigderson [Theory Comput., 17 (2021), pp. 1–88], whereby lower bounds against subsystems of IPS were obtained using restricted algebraic circuit lower bounds, and demonstrate that the binary value principle captures the advantage of semialgebraic over algebraic reasoning, for sufficiently strong systems. Specifically, we show the following. (1) Conditional IPS lower bounds: The Shub–Smale hypothesis [Duke Math. J., 81 (1995), pp. 47–54] implies a superpolynomial lower bound on the size of IPS refutations of the binary value principle over the rationals defined as the unsatisfiable linear equation [math] for Boolean [math]’s. Further, the related and more widely known [math]-conjecture [Duke Math. J., 81 (1995), pp. 47–54] implies a superpolynomial lower bound on the size of IPS refutations of a variant of the binary value principle over the ring of rational functions. No prior conditional lower bounds were known for IPS or apparently weaker propositional proof systems such as Frege systems (though our lower bounds do not translate to Frege lower bounds since the hard instances are not Boolean formulas). (2) Algebraic versus semialgebraic proofs: Admitting short refutations of the binary value principle is necessary for any algebraic proof system to fully simulate any known semialgebraic proof system, and for strong enough algebraic proof systems it is also sufficient. In particular, we introduce a very strong proof system that simulates all known semialgebraic proof systems (and most other known concrete propositional proof systems), under the name Cone Proof System (CPS), as a semialgebraic analogue of the IPS: CPS establishes the unsatisfiability of collections of polynomial equalities and inequalities over the reals, by representing sum-of-squares proofs (and extensions) as algebraic circuits. We prove that IPS polynomially simulates CPS iff IPS admits polynomial-size refutations of the binary value principle (for the language of systems of equations that have no 0/1-solutions), over both [math] and [math].
{"title":"Semialgebraic Proofs, IPS Lower Bounds, and the [math]-Conjecture: Can a Natural Number be Negative?","authors":"Yaroslav Alekseev, Dima Grigoriev, Edward A. Hirsch, Iddo Tzameret","doi":"10.1137/20m1374523","DOIUrl":"https://doi.org/10.1137/20m1374523","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 3, Page 648-700, June 2024. <br/> Abstract. We introduce the binary value principle, which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, based on a well-known hypothesis by Shub and Smale about the hardness of computing factorials, where IPS is the strong algebraic proof system introduced by Grochow and Pitassi [J. ACM, 65 (2018), 37]. Conversely, we show that short IPS refutations of this instance bridge the gap between sufficiently strong algebraic and semialgebraic proof systems. Our results extend to unrestricted IPS the paradigm introduced by Forbes, Shpilka, Tzameret, and Wigderson [Theory Comput., 17 (2021), pp. 1–88], whereby lower bounds against subsystems of IPS were obtained using restricted algebraic circuit lower bounds, and demonstrate that the binary value principle captures the advantage of semialgebraic over algebraic reasoning, for sufficiently strong systems. Specifically, we show the following. (1) Conditional IPS lower bounds: The Shub–Smale hypothesis [Duke Math. J., 81 (1995), pp. 47–54] implies a superpolynomial lower bound on the size of IPS refutations of the binary value principle over the rationals defined as the unsatisfiable linear equation [math] for Boolean [math]’s. Further, the related and more widely known [math]-conjecture [Duke Math. J., 81 (1995), pp. 47–54] implies a superpolynomial lower bound on the size of IPS refutations of a variant of the binary value principle over the ring of rational functions. No prior conditional lower bounds were known for IPS or apparently weaker propositional proof systems such as Frege systems (though our lower bounds do not translate to Frege lower bounds since the hard instances are not Boolean formulas). (2) Algebraic versus semialgebraic proofs: Admitting short refutations of the binary value principle is necessary for any algebraic proof system to fully simulate any known semialgebraic proof system, and for strong enough algebraic proof systems it is also sufficient. In particular, we introduce a very strong proof system that simulates all known semialgebraic proof systems (and most other known concrete propositional proof systems), under the name Cone Proof System (CPS), as a semialgebraic analogue of the IPS: CPS establishes the unsatisfiability of collections of polynomial equalities and inequalities over the reals, by representing sum-of-squares proofs (and extensions) as algebraic circuits. We prove that IPS polynomially simulates CPS iff IPS admits polynomial-size refutations of the binary value principle (for the language of systems of equations that have no 0/1-solutions), over both [math] and [math].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"74 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski, Paul Seymour
SIAM Journal on Computing, Volume 53, Issue 3, Page 624-647, June 2024. Abstract. A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By [math], we denote a path on [math] vertices. In this paper, we give polynomial-time algorithms for the following problems: the maximum weight independent set problem in long-hole–free graphs and the feedback vertex set problem in [math]-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended [math] is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let [math] be the class of graphs excluding an extended [math] and holes of length at least 6 as induced subgraphs; [math] contains all long-hole–free graphs and all [math]-free graphs. We show that, given an [math]-vertex graph [math] with vertex weights and an integer [math], one can, in time, [math] find a maximum-weight induced subgraph of [math] of treewidth less than [math]. This implies both aforementioned results. To achieve this goal, we extend the framework of potential maximal cliques (PMCs) to containers. Developed by Bouchitté and Todinca [SIAM J. Comput., 31 (2001), pp. 212–232] and extended by Fomin, Todinca, and Villanger [SIAM J. Comput., 44 (2015), pp. 54–87], this framework allows us to solve a wide variety of tasks, including finding a maximum-weight induced subgraph of treewidth less than [math] for fixed [math], in time polynomial in the size of the graph and the number of potential maximal cliques. Further developments, tailored to solve the maximum weight independent set problem within this framework (e.g., for [math]-free [Lokshtanov, Vatshelle, and Villanger, SODA 2014, pp. 570–581] or [math]-free graphs [Grzesik, Klimošová, Pilipczuk, and Pilipczuk, ACM Trans. Algorithms, 18 (2022), pp. 4:1–4:57]), enumerate only a specifically chosen subset of all PMCs of a graph. In all aforementioned works, the final step is an involved dynamic programming algorithm whose state space is based on the considered list of PMCs. Here, we modify the dynamic programming algorithm and show that it is sufficient to consider only a container for each PMC: a superset of the maximal clique that intersects the sought solution only in the vertices of the PMC. This strengthening of the framework not only allows us to obtain our main result but also leads to significant simplifications of the reasoning in previous papers.
{"title":"Induced Subgraphs of Bounded Treewidth and the Container Method","authors":"Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski, Paul Seymour","doi":"10.1137/20m1383732","DOIUrl":"https://doi.org/10.1137/20m1383732","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 3, Page 624-647, June 2024. <br/> Abstract. A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By [math], we denote a path on [math] vertices. In this paper, we give polynomial-time algorithms for the following problems: the maximum weight independent set problem in long-hole–free graphs and the feedback vertex set problem in [math]-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended [math] is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let [math] be the class of graphs excluding an extended [math] and holes of length at least 6 as induced subgraphs; [math] contains all long-hole–free graphs and all [math]-free graphs. We show that, given an [math]-vertex graph [math] with vertex weights and an integer [math], one can, in time, [math] find a maximum-weight induced subgraph of [math] of treewidth less than [math]. This implies both aforementioned results. To achieve this goal, we extend the framework of potential maximal cliques (PMCs) to containers. Developed by Bouchitté and Todinca [SIAM J. Comput., 31 (2001), pp. 212–232] and extended by Fomin, Todinca, and Villanger [SIAM J. Comput., 44 (2015), pp. 54–87], this framework allows us to solve a wide variety of tasks, including finding a maximum-weight induced subgraph of treewidth less than [math] for fixed [math], in time polynomial in the size of the graph and the number of potential maximal cliques. Further developments, tailored to solve the maximum weight independent set problem within this framework (e.g., for [math]-free [Lokshtanov, Vatshelle, and Villanger, SODA 2014, pp. 570–581] or [math]-free graphs [Grzesik, Klimošová, Pilipczuk, and Pilipczuk, ACM Trans. Algorithms, 18 (2022), pp. 4:1–4:57]), enumerate only a specifically chosen subset of all PMCs of a graph. In all aforementioned works, the final step is an involved dynamic programming algorithm whose state space is based on the considered list of PMCs. Here, we modify the dynamic programming algorithm and show that it is sufficient to consider only a container for each PMC: a superset of the maximal clique that intersects the sought solution only in the vertices of the PMC. This strengthening of the framework not only allows us to obtain our main result but also leads to significant simplifications of the reasoning in previous papers.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"93 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Tao
SIAM Journal on Computing, Volume 53, Issue 3, Page 573-587, June 2024. Abstract. We show [math]. Here the class [math] consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse [math] by Fearnley et al. (STOC 2021). We also prove a companion result [math], where [math] is the class associated with the Sink-of-Potential-Line problem.
{"title":"Further Collapses in [math]","authors":"Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Tao","doi":"10.1137/22m1498346","DOIUrl":"https://doi.org/10.1137/22m1498346","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 3, Page 573-587, June 2024. <br/> Abstract. We show [math]. Here the class [math] consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse [math] by Fearnley et al. (STOC 2021). We also prove a companion result [math], where [math] is the class associated with the Sink-of-Potential-Line problem.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"56 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140841491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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