International Symposium on Mathematical Foundations of Computer Science最新文献_第10页

Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language 二维网格和Dyck语言的量子下界和上界

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-07-06 DOI: 10.4230/LIPIcs.MFCS.2020.8

A. Ambainis, K. Balodis, Janis Iraids, K. Khadiev, Vladislavs Klevickis, Krisjanis Prusis, Yixin Shen, Juris Smotrovs, J. Vihrovs

{"title":"Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language","authors":"A. Ambainis, K. Balodis, Janis Iraids, K. Khadiev, Vladislavs Klevickis, Krisjanis Prusis, Yixin Shen, Juris Smotrovs, J. Vihrovs","doi":"10.4230/LIPIcs.MFCS.2020.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2020.8","url":null,"abstract":"We study the quantum query complexity of two problems. \u0000First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. We call this the $Dyck_{k,n}$ problem. We prove a lower bound of $Omega(c^k sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. Here $n$ is the length of the word. When $k$ is a constant, this is interesting as a representative example of star-free languages for which a surprising $tilde{O}(sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When $k$ is not a constant, $Dyck_{k,n}$ is not context-free. We give an algorithm with $Oleft(sqrt{n}(log{n})^{0.5k}right)$ quantum queries for $Dyck_{k,n}$ for all $k$. This is better than the trival upper bound $n$ for $k=oleft(frac{log(n)}{loglog n}right)$. \u0000Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the \"balanced parentheses\" problem into the grid, we show a lower bound of $Omega(n^{1.5-epsilon})$ for the directed 2D grid and $Omega(n^{2-epsilon})$ for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"332 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130915637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 17

Simplified Game of Life: Algorithms and Complexity 简化的生命游戏:算法和复杂性

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-07-06 DOI: 10.4230/LIPIcs.MFCS.2020.22

K. Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, J. Svoboda

{"title":"Simplified Game of Life: Algorithms and Complexity","authors":"K. Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, J. Svoboda","doi":"10.4230/LIPIcs.MFCS.2020.22","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2020.22","url":null,"abstract":"Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114481885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Elimination distance to bounded degree on planar graphs 平面图上有界度消去距离

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-07-05 DOI: 10.4230/LIPIcs.MFCS.2020.65

Alexander Lindermayr, S. Siebertz, Alexandre Vigny

引用次数: 11

Fractional Covers of Hypergraphs with Bounded Multi-Intersection 有界多交超图的分数覆盖

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-07-01 DOI: 10.4230/LIPIcs.MFCS.2020.41

G. Gottlob, Matthias Lanzinger, R. Pichler, Igor Razgon

引用次数: 3

Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach 用线性伽罗群分解有限域上的多项式:一个加性组合方法

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-07-01 DOI: 10.4230/LIPIcs.MFCS.2020.42

Zeyu Guo

{"title":"Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach","authors":"Zeyu Guo","doi":"10.4230/LIPIcs.MFCS.2020.42","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2020.42","url":null,"abstract":"Let $tilde{f}(X)inmathbb{Z}[X]$ be a degree-$n$ polynomial such that $f(X):=tilde{f}(X)bmod p$ factorizes into $n$ distinct linear factors over $mathbb{F}_p$. We study the problem of deterministically factoring $f(X)$ over $mathbb{F}_p$ given $tilde{f}(X)$. Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of $f(X)$ in the case that the Galois group of $tilde{f}(X)$ is (permutation isomorphic to) a linear group $Gleq mathrm{GL}(V)$ on the set $S$ of roots of $tilde{f}(X)$, where $V$ is a finite-dimensional vector space over a finite field $mathbb{F}$ and $S$ is identified with a subset of $V$. In particular, when $|S|=|V|^{Omega(1)}$, the algorithm runs in time polynomial in $n^{log n/(loglogloglog n)^{1/3}}$ and the size of the input, improving Evdokimov's algorithm. Our result also applies to a general Galois group $G$ when combined with a recent algorithm of the author. \u0000To prove our main result, we introduce a family of objects called linear $m$-schemes and reduce the problem of factoring $f(X)$ to a combinatorial problem about these objects. We then apply techniques from additive combinatorics to obtain an improved bound. Our techniques may be of independent interest.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132238716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Ideal Membership Problem for Boolean Minority 布尔少数的理想隶属度问题

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-06-29 DOI: 10.4230/LIPIcs.MFCS.2021.16

Arpitha P. Bharathi, M. Mastrolilli

{"title":"Ideal Membership Problem for Boolean Minority","authors":"Arpitha P. Bharathi, M. Mastrolilli","doi":"10.4230/LIPIcs.MFCS.2021.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2021.16","url":null,"abstract":"The Ideal Membership Problem (IMP) tests if an input polynomial $fin mathbb{F}[x_1,dots,x_n]$ with coefficients from a field $mathbb{F}$ belongs to a given ideal $I subseteq mathbb{F}[x_1,dots,x_n]$. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). A dichotomy result between ``hard'' (NP-hard) and ``easy'' (polynomial time) IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Bulatov FOCS'17, Zhuk FOCS'17] (this is equivalent to IMP$_0$) and IMP$_d$ for the Boolean domain [Mastrolilli SODA'19], both based on the classification of the IMP through functions called polymorphisms. The complexity of the IMP$_d$ for five polymorphisms has been solved in [Mastrolilli SODA'19] whereas for the ternary minority polymorphism it was incorrectly declared to have been resolved by a previous result. As a matter of fact the complexity of the IMP$_d$ for the ternary minority polymorphism is open. In this paper we provide the missing link by proving that the IMP$_d$ for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This result, along with the results in [Mastrolilli SODA'19], completes the identification of the precise borderline of tractability for the IMP$_d$ for constrained problems over the Boolean domain. This paper is motivated by the pursuit of understanding the issue of bit complexity of Sum-of-Squares proofs raised by O'Donnell [ITCS'17]. Raghavendra and Weitz [ICALP'17] show how the IMP$_d$ tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133835339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Existential Theory of the Reals Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games 完全信息随机对策平稳纳什均衡实完备性的存在论

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-06-15 DOI: 10.4230/LIPIcs.MFCS.2020.45

Kristoffer Arnsfelt Hansen, Steffan Christ Sølvsten

引用次数: 2

Reducing graph transversals via edge contractions 通过边收缩减少图的截线

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-05-04 DOI: 10.4230/LIPIcs.MFCS.2020.64

Paloma T. Lima, V. F. D. Santos, Ignasi Sau, U. Souza

引用次数: 5

Continuous and monotone machines 连续和单调机器

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-05-04 DOI: 10.4230/LIPIcs.MFCS.2020.56

M. Konečný, Florian Steinberg, Holger Thies

{"title":"Continuous and monotone machines","authors":"M. Konečný, Florian Steinberg, Holger Thies","doi":"10.4230/LIPIcs.MFCS.2020.56","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2020.56","url":null,"abstract":"We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. \u0000Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq. Continuous machines, their equivalence to the standard encoding and correctness of basic operations are now part of Incone, a Coq library for computable analysis. While the correctness proofs use a classical meta-theory with countable choice, the translations and algorithms that are proven correct are all fully executable. Along the way we formally prove some known results such as existence of a self-modulating moduli of continuity for partial continuous operators on Baire space. \u0000To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics. We also connect continuous machines to the construction of precompletions and completions of represented spaces, topics that have recently caught the attention of the computable analysis community.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116287893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle 计算第二个哈密顿循环的精确和近似算法

International Symposium on Mathematical Foundations of Computer Science Pub Date : 2020-04-13 DOI: 10.4230/LIPIcs.MFCS.2020.27

Argyrios Deligkas, G. Mertzios, P. Spirakis, V. Zamaraev

{"title":"Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle","authors":"Argyrios Deligkas, G. Mertzios, P. Spirakis, V. Zamaraev","doi":"10.4230/LIPIcs.MFCS.2020.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.MFCS.2020.27","url":null,"abstract":"In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph $G$ and a Hamiltonian cycle $C_0$ of $G$, how can we compute a second Hamiltonian cycle $C_1 neq C_0$ of $G$? Cedric Smith proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is an algorithm which computes the second Hamiltonian cycle in time $O(n cdot 2^{(0.3-varepsilon)n})$ time, for some positive constant $varepsilon>0$, and in polynomial space, thus improving the state of the art running time for solving this problem. Our algorithm is based on a fundamental structural property of Thomason's lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a Hamiltonian graph $G$ with a given Hamiltonian cycle $C_0$ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least $n - 4alpha (sqrt{n}+2alpha)+8$, where $alpha = frac{Delta-2}{delta-2}$ and $delta,Delta$ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.","PeriodicalId":369104,"journal":{"name":"International Symposium on Mathematical Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130440260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2