Foundations of Software Technology and Theoretical Computer Science最新文献_第3页

Tight Chang's-lemma-type bounds for Boolean functions 布尔函数的紧张引理型边界

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-12-03 DOI: 10.4230/LIPIcs.FSTTCS.2021.10

S. Chakraborty, Nikhil S. Mande, R. Mittal, Tulasimohan Molli, Manaswi Paraashar, Swagato Sanyal

{"title":"Tight Chang's-lemma-type bounds for Boolean functions","authors":"S. Chakraborty, Nikhil S. Mande, R. Mittal, Tulasimohan Molli, Manaswi Paraashar, Swagato Sanyal","doi":"10.4230/LIPIcs.FSTTCS.2021.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2021.10","url":null,"abstract":"Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For each positive threshold $t$, Chang's lemma provides a lower bound on $wt(f):=Pr[f(x)=-1]$ in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least $1/t$. We examine the tightness of Chang's lemma w.r.t. the following three natural settings of the threshold: - the Fourier sparsity of $f$, denoted $k(f)$, - the Fourier max-supp-entropy of $f$, denoted $k'(f)$, defined to be $max {1/|hat{f}(S)| : hat{f}(S) neq 0}$, - the Fourier max-rank-entropy of $f$, denoted $k''(f)$, defined to be the minimum $t$ such that characters whose Fourier coefficients are at least $1/t$ in absolute value span a space of dimension $r(f)$. We prove new lower bounds on $wt(f)$ in terms of these measures. One of our lower bounds subsumes and refines the previously best known upper bound on $r(f)$ in terms of $k(f)$ by Sanyal (ToC, 2019). Another lower bound is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of the absolute values of the level-$1$ Fourier coefficients. We also show that Chang's lemma for the these choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions (which are modifications of the Addressing function) witnessing their tightness. Finally we construct Boolean functions $f$ for which - our lower bounds asymptotically match $wt(f)$, and - for any choice of the threshold $t$, the lower bound obtained from Chang's lemma is asymptotically smaller than $wt(f)$.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129180818","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: 3

Harmonic algorithms for packing d-dimensional cuboids into bins 将d维长方体装入箱子的调和算法

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-11-22 DOI: 10.4230/LIPIcs.FSTTCS.2021.32

Eklavya Sharma

{"title":"Harmonic algorithms for packing d-dimensional cuboids into bins","authors":"Eklavya Sharma","doi":"10.4230/LIPIcs.FSTTCS.2021.32","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2021.32","url":null,"abstract":"We study harmonic-based algorithms for the $d$-dimensional ($d$D) generalizations of three classical geometric packing problems: geometric bin packing (BP), strip packing (SP), and geometric knapsack (KS). Caprara (MOR 2008) studied a harmonic-based algorithm $mathtt{HDH}_k$, that has an asymptotic approximation ratio of $T_{infty}^{d-1}$ (where $T_{infty} approx 1.691$) for $d$D BP and $d$D SP when items are not allowed to be rotated. We give fast and simple harmonic-based algorithms with asymptotic approximation ratios of $T_{infty}^{d-1}$, $T_{infty}^{d}$ and $(1-epsilon)3^{-d}$ for $d$D SP, $d$D BP and $d$D KS, respectively, when orthogonal rotations are allowed about all or a subset of axes. This gives the first approximation algorithm for $d$D KS for $d > 3$. Furthermore, we provide a more sophisticated harmonic-based algorithm, which we call $mathtt{HGaP}_k$, that is $T_{infty}^{d-1}(1+epsilon)$-asymptotic-approximate for $d$D BP for the rotational case. This gives an approximation ratio of $2.860 + epsilon$ for 3D BP with rotations, which improves upon the current best-known algorithm. \u0000In addition, we study multiple-choice packing problems that generalize the rotational case. Here we are given $n$ sets of $d$D cuboidal items and we have to choose exactly one (resp. at most one for the knapsack variant) item from each set and then pack the chosen items. All our algorithms also work for multiple-choice packing problems.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117153990","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

Ambiguity through the lens of measure theory 测度理论视角下的歧义

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-11-20 DOI: 10.4230/LIPIcs.FSTTCS.2022.34

Olivier Carton

引用次数: 0

Generalizations of Length Limited Huffman Coding for Hierarchical Memory Settings 分级存储器设置中有限长度霍夫曼编码的推广

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-10-10 DOI: 10.4230/LIPIcs.FSTTCS.2021.8

Shashwat Banchhor, Rishikesh R. Gajjala, Yogish Sabharwal, Sandeep Sen

{"title":"Generalizations of Length Limited Huffman Coding for Hierarchical Memory Settings","authors":"Shashwat Banchhor, Rishikesh R. Gajjala, Yogish Sabharwal, Sandeep Sen","doi":"10.4230/LIPIcs.FSTTCS.2021.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2021.8","url":null,"abstract":"In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the {em soft-length limited Huffman coding} problem. In this version, there is a penalty associated with each of the $n$ characters of the alphabet whose encodings exceed a specified bound $D$($leq n$), where the penalty increases linearly with the length of the encoding beyond $D$. The goal of the problem is to find a prefix-free encoding having minimum average code length and total penalty within a pre-specified bound ${cal P}$. This generalizes the LLHC problem. We present an algorithm to solve this problem that runs in time $O( nD )$. We study a further generalization in which the penalty function and the objective function can both be arbitrary monotonically non-decreasing functions of the codeword length. We provide dynamic programming based exact and PTAS algorithms for this setting.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127831956","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

Weighted Tiling Systems for Graphs: Evaluation Complexity 图的加权平铺系统:评估复杂性

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-09-30 DOI: 10.4230/LIPIcs.FSTTCS.2020.34

C. Aiswarya, P. Gastin

引用次数: 1

Uncertainty Reasoning for Probabilistic Petri Nets via Bayesian Networks 基于贝叶斯网络的概率Petri网不确定性推理

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-09-30 DOI: 10.4230/LIPIcs.FSTTCS.2020.38

Rebecca Bernemann, Benjamin Cabrera, R. Heckel, B. König

引用次数: 2

Constructing Large Matchings via Query Access to a Maximal Matching Oracle 通过查询访问构建最大匹配Oracle的大匹配

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-09-28 DOI: 10.4230/LIPIcs.FSTTCS.2020.26

Lidiya Khalidah binti Khalil, C. Konrad

{"title":"Constructing Large Matchings via Query Access to a Maximal Matching Oracle","authors":"Lidiya Khalidah binti Khalil, C. Konrad","doi":"10.4230/LIPIcs.FSTTCS.2020.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2020.26","url":null,"abstract":"Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-epsilon)$-approximation streaming algorithms that run in $O(text{poly} frac{1}{epsilon})$ passes in bipartite graphs and in $O( (frac{1}{epsilon})^{frac{1}{epsilon}})$ or $O(text{poly} (frac{1}{epsilon}) cdot log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of $2$-pass small space streaming algorithms with approximation factor $0.999$ has not yet been ruled out. \u0000The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? \u0000In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that $3$ passes are necessary and sufficient to improve on the trivial approximation factor of $1/2$: We give a lower bound of $0.6$ on the approximation ratio of such algorithms, which is optimal. We further show that $Omega( frac{1}{epsilon})$ passes are required for computing a $(1-epsilon)$-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that $Omega(n)$ passes are required in order to improve on the trivial approximation factor of $1/2$.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115473595","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: 6

Dynamic network congestion games 动态网络拥塞游戏

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-09-28 DOI: 10.4230/LIPIcs.FSTTCS.2020.40

N. Bertrand, N. Markey, S. Sadhukhan, O. Sankur

{"title":"Dynamic network congestion games","authors":"N. Bertrand, N. Markey, S. Sadhukhan, O. Sankur","doi":"10.4230/LIPIcs.FSTTCS.2020.40","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2020.40","url":null,"abstract":"Congestion games are a classical type of games studied in game theory, in which n players choose a resource, and their individual cost increases with the number of other players choosing the same resource. In network congestion games (NCGs), the resources correspond to simple paths in a graph, e.g. representing routing options from a source to a target. In this paper, we introduce a variant of NCGs, referred to as dynamic NCGs: in this setting, players take transitions synchronously, they select their next transitions dynamically, and they are charged a cost that depends on the number of players simultaneously using the same transition. \u0000We study, from a complexity perspective, standard concepts of game theory in dynamic NCGs: social optima, Nash equilibria, and subgame perfect equilibria. Our contributions are the following: the existence of a strategy profile with social cost bounded by a constant is in PSPACE and NP-hard. (Pure) Nash equilibria always exist in dynamic NCGs; the existence of a Nash equilibrium with bounded cost can be decided in EXPSPACE, and computing a witnessing strategy profile can be done in doubly-exponential time. The existence of a subgame perfect equilibrium with bounded cost can be decided in 2EXPSPACE, and a witnessing strategy profile can be computed in triply-exponential time.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122798483","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: 3

Reachability for Updatable Timed Automata made faster and more effective 可更新定时自动机的可达性使其更快、更有效

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-09-28 DOI: 10.4230/LIPIcs.FSTTCS.2020.47

P. Gastin, S. Mukherjee, B. Srivathsan

引用次数: 4

Process Symmetry in Probabilistic Transducers 概率传感器的过程对称性

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2020-07-23 DOI: 10.4230/LIPIcs.FSTTCS.2020.35

Shaull Almagor

{"title":"Process Symmetry in Probabilistic Transducers","authors":"Shaull Almagor","doi":"10.4230/LIPIcs.FSTTCS.2020.35","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSTTCS.2020.35","url":null,"abstract":"Model checking is the process of deciding whether a system satisfies a given specification. Often, when the setting comprises multiple processes, the specifications are over sets of input and output signals that correspond to individual processes. Then, many of the properties one wishes to specify are symmetric with respect to the processes identities. In this work, we consider the problem of deciding whether the given system exhibits symmetry with respect to the processes' identities. When the system is symmetric, this gives insight into the behaviour of the system, as well as allows the designer to use only representative specifications, instead of iterating over all possible process identities. \u0000Specifically, we consider probabilistic systems, and we propose several variants of symmetry. We start with precise symmetry, in which, given a permutation $pi$, the system maintains the exact distribution of permuted outputs, given a permuted inputs. We proceed to study approximate versions of symmetry, including symmetry induced by small $L_infty$ norm, variants of Parikh-image based symmetry, and qualitative symmetry. For each type of symmetry, we consider the problem of deciding whether a given system exhibits this type of symmetry.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115462712","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