Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms最新文献_第9页

Secretary Problems: The Power of a Single Sample 秘书问题:单一样本的力量

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-08-19 DOI: 10.48550/arXiv.2208.09159

Pranav Nuti, Jan Vondr'ak

{"title":"Secretary Problems: The Power of a Single Sample","authors":"Pranav Nuti, Jan Vondr'ak","doi":"10.48550/arXiv.2208.09159","DOIUrl":"https://doi.org/10.48550/arXiv.2208.09159","url":null,"abstract":"In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers $X_i$ that come from distributions $mathcal{D}_i$, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample $Y_i$ from each distribution $mathcal{D}_i$. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem was first studied by Correa et al. [SODA 2020] who managed to construct an algorithm that achieves a probability of $0.4529$. In this paper, we improve this probability to $0.5009$, almost matching an upper bound of $simeq 0.5024$ which we show follows from earlier work. We also show that there is an algorithm which achieves the probability of $simeq 0.5024$ asymptotically if no particular distribution is especially likely to yield the largest number. For the adversarial order version of the problem, we show that we can select the maximum number with a probability of $1/4$, and that this is best possible. Our work demonstrates that unlike in the case of the expected value objective studied by Rubinstein et al. [ITCS 2020], knowledge of a single sample is not enough to recover the factor of success guaranteed by full knowledge of the distribution.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"31 1","pages":"2015-2029"},"PeriodicalIF":0.0,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80631017","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

Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring Lovász局部引理和边着色的改进分布式算法

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-08-18 DOI: 10.48550/arXiv.2208.08701

Peter Davies

{"title":"Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring","authors":"Peter Davies","doi":"10.48550/arXiv.2208.08701","DOIUrl":"https://doi.org/10.48550/arXiv.2208.08701","url":null,"abstract":"The Lov'asz Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have $n$ `bad events', each of which occurs with probability at most $p$ and is independent of all but $d$ other events, then under certain criteria on $p$ and $d$, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lov'asz Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM '10] also implied an efficient distributed algorithm for the problem, running in $O(log^2 n)$ rounds. For instances with low $d$, this was improved to $O(d^2+log^{O(1)}log n)$ by Fischer and Ghaffari [DISC '17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP '19]). We give an improved algorithm for the Lov'asz Local Lemma, providing a trade-off between the strength of the criterion relating $p$ and $d$, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari's algorithm, we improve the round complexity to $O(frac{d}{log d}+log^{O(1)}log n)$. At the other end of the trade-off, we obtain a $log^{O(1)}log n$ round complexity for a substantially wider regime than previously known. As our main application, we also give the first $log^{O(1)}log n$-round distributed algorithm for the problem of $Delta+o(Delta)$-edge coloring a graph of maximum degree $Delta$. This is an almost exponential improvement over previous results: no prior $log^{o(1)} n$-round algorithm was known even for $2Delta-2$-edge coloring.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"65 1","pages":"4273-4295"},"PeriodicalIF":0.0,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83956433","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

Mean estimation when you have the source code; or, quantum Monte Carlo methods 有源代码时的均值估计;或者，量子蒙特卡罗方法

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-08-16 DOI: 10.48550/arXiv.2208.07544

Robin Kothari, R. O'Donnell

引用次数: 9

Private Query Release via the Johnson-Lindenstrauss Transform 通过Johnson-Lindenstrauss变换释放私有查询

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-08-15 DOI: 10.48550/arXiv.2208.07410

Aleksandar Nikolov

引用次数: 6

Stronger Privacy Amplification by Shuffling for Rényi and Approximate Differential Privacy 基于rsamnyi和近似差分隐私的洗牌增强隐私放大

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-08-09 DOI: 10.48550/arXiv.2208.04591

V. Feldman, Audra McMillan, Kunal Talwar

引用次数: 20

Short Synchronizing Words for Random Automata 随机自动机的短同步词

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-07-28 DOI: 10.48550/arXiv.2207.14108

G. Chapuy, G. Perarnau

{"title":"Short Synchronizing Words for Random Automata","authors":"G. Chapuy, G. Perarnau","doi":"10.48550/arXiv.2207.14108","DOIUrl":"https://doi.org/10.48550/arXiv.2207.14108","url":null,"abstract":"We prove that a uniformly random automaton with $n$ states on a 2-letter alphabet has a synchronizing word of length $O(n^{1/2}log n)$ with high probability (w.h.p.). That is to say, w.h.p. there exists a word $omega$ of such length, and a state $v_0$, such that $omega$ sends all states to $v_0$. Prior to this work, the best upper bound was the quasilinear bound $O(nlog^3n)$ due to Nicaud (2016). The correct scaling exponent had been subject to various estimates by other authors between $0.5$ and $0.56$ based on numerical simulations, and our result confirms that the smallest one indeed gives a valid upper bound (with a log factor). Our proof introduces the concept of $w$-trees, for a word $w$, that is, automata in which the $w$-transitions induce a (loop-rooted) tree. We prove a strong structure result that says that, w.h.p., a random automaton on $n$ states is a $w$-tree for some word $w$ of length at most $(1+epsilon)log_2(n)$, for any $epsilon>0$. The existence of the (random) word $w$ is proved by the probabilistic method. This structure result is key to proving that a short synchronizing word exists.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"80 1","pages":"581-604"},"PeriodicalIF":0.0,"publicationDate":"2022-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77794651","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: 5

Improved Bounds for Sampling Solutions of Random CNF Formulas 随机CNF公式抽样解的改进界

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-07-25 DOI: 10.48550/arXiv.2207.11892

Kun He, Kewen Wu, Kuan Yang

{"title":"Improved Bounds for Sampling Solutions of Random CNF Formulas","authors":"Kun He, Kewen Wu, Kuan Yang","doi":"10.48550/arXiv.2207.11892","DOIUrl":"https://doi.org/10.48550/arXiv.2207.11892","url":null,"abstract":"Let $Phi$ be a random $k$-CNF formula on $n$ variables and $m$ clauses, where each clause is a disjunction of $k$ literals chosen independently and uniformly. Our goal is to sample an approximately uniform solution of $Phi$ (or equivalently, approximate the partition function of $Phi$). Let $alpha=m/n$ be the density. The previous best algorithm runs in time $n^{mathsf{poly}(k,alpha)}$ for any $alphalesssim2^{k/300}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves both bounds by providing an almost-linear time sampler for any $alphalesssim2^{k/3}$. The density $alpha$ captures the emph{average degree} in the random formula. In the worst-case model with bounded emph{maximum degree}, current best efficient sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound. Our result is the first progress towards establishing the intuition that the solvability of the average-case model (random $k$-CNF formula with bounded average degree) is better than the worst-case model (standard $k$-CNF formula with bounded maximal degree) in terms of sampling solutions.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"13 1","pages":"3330-3361"},"PeriodicalIF":0.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80447444","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: 9

Spatial mixing and the random-cluster dynamics on lattices 格上的空间混合和随机簇动力学

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-07-22 DOI: 10.1137/1.9781611977554.ch174

Reza Gheissari, A. Sinclair

{"title":"Spatial mixing and the random-cluster dynamics on lattices","authors":"Reza Gheissari, A. Sinclair","doi":"10.1137/1.9781611977554.ch174","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch174","url":null,"abstract":"An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"186 1","pages":"4606-4621"},"PeriodicalIF":0.0,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73939736","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

A simple and sharper proof of the hypergraph Moore bound 超图摩尔界的一个简单而清晰的证明

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-07-22 DOI: 10.48550/arXiv.2207.10850

Jun-Ting Hsieh, Pravesh Kothari, Sidhanth Mohanty

{"title":"A simple and sharper proof of the hypergraph Moore bound","authors":"Jun-Ting Hsieh, Pravesh Kothari, Sidhanth Mohanty","doi":"10.48550/arXiv.2207.10850","DOIUrl":"https://doi.org/10.48550/arXiv.2207.10850","url":null,"abstract":"The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of hyperedges in a hypergraph. For graphs (i.e., $2$-uniform hypergraphs), a bound tight up to the leading constant was proven in a classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity $k>2$, an appropriate generalization was conjectured by Feige [Fei08]. The conjecture was settled up to an additional $log^{4k+1} n$ factor in the size in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument relies on a connection between the existence of short even covers and the spectrum of a certain randomly signed Kikuchi matrix. Their analysis, especially for the case of odd $k$, is significantly complicated. In this work, we present a substantially simpler and shorter proof of the hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi matrix and an edge deletion step that allows us to drop several involved steps in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi matrix and the use of the Schudy-Sviridenko polynomial concentration. Our simpler proof also obtains tighter parameters: in particular, the argument gives a new proof of the classical Moore bound of [AHL02] with no loss (the proof in [GKM21] loses a $log^3 n$ factor), and loses only a single logarithmic factor for all $k>2$-uniform hypergraphs. As in [GKM21], our ideas naturally extend to yield a simpler proof of the full trade-off for strongly refuting smoothed instances of constraint satisfaction problems with similarly improved parameters.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"2002 1","pages":"2324-2344"},"PeriodicalIF":0.0,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88321311","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

Tight Bounds for Monotone Minimal Perfect Hashing 单调最小完美哈希的紧界

Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms Pub Date : 2022-07-21 DOI: 10.48550/arXiv.2207.10556

Sepehr Assadi, Martín Farach-Colton, William Kuszmaul

{"title":"Tight Bounds for Monotone Minimal Perfect Hashing","authors":"Sepehr Assadi, Martín Farach-Colton, William Kuszmaul","doi":"10.48550/arXiv.2207.10556","DOIUrl":"https://doi.org/10.48550/arXiv.2207.10556","url":null,"abstract":"The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set $S= {s_1,ldots,s_n}$ of $n$ distinct keys from a universe $U$ of size $u$, create a data structure $DS$ that answers the following query: [ RankOp(q) = text{rank of } q text{ in } S text{ for all } qin S ~text{ and arbitrary answer otherwise.} ] Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes $DS$ in $O(nlogloglog u)$ bits and performs queries in $O(log u)$ time. It has been an open problem to either improve the space upper bound or to show that this somewhat odd looking bound is tight. In this paper, we show the latter: specifically that any data structure (deterministic or randomized) for monotone minimal perfect hashing of any collection of $n$ elements from a universe of size $u$ requires $Omega(n cdot logloglog{u})$ expected bits to answer every query correctly. We achieve our lower bound by defining a graph $mathbf{G}$ where the nodes are the possible ${u choose n}$ inputs and where two nodes are adjacent if they cannot share the same $DS$. The size of $DS$ is then lower bounded by the log of the chromatic number of $mathbf{G}$. Finally, we show that the fractional chromatic number (and hence the chromatic number) of $mathbf{G}$ is lower bounded by $2^{Omega(n logloglog u)}$.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"6 1","pages":"456-476"},"PeriodicalIF":0.0,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75030417","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