For any real number p > 0, we nearly completely characterize the space complexity of estimating ||A||pp = ∑i=1n σip for n × n matrices A in which each row and each column has O(1) non-zero entries and whose entries are presented one at a time in a data stream model. Here the σi are the singular values of A, and when p ≥ 1, ||A||pp is the p-th power of the Schatten p-norm. We show that when p is not an even integer, to obtain a (1+є)-approximation to ||A||pp with constant probability, any 1-pass algorithm requires n1−g(є) bits of space, where g(є) → 0 as є → 0 and є > 0 is a constant independent of n. However, when p is an even integer, we give an upper bound of n1−2/p (є−1logn) bits of space, which holds even in the turnstile data stream model. The latter is optimal up to (є−1 logn) factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices A: the previous best lower bound was Ω(logn) for p∈ (0,1), Ω(n1/p−1/2/logn) for p∈ [1,2) and Ω(n1−2/p) for p∈ (2,∞). We note for p ∈ (2, ∞), while our lower bound for even integers is the same, for other p in this range our lower bound is n1−g(є), which is considerably stronger than the previous n1−2/p for small enough constant є > 0. We obtain similar near-linear lower bounds for Ky-Fan norms, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work.
{"title":"On approximating functions of the singular values in a stream","authors":"Yi Li, David P. Woodruff","doi":"10.1145/2897518.2897581","DOIUrl":"https://doi.org/10.1145/2897518.2897581","url":null,"abstract":"For any real number p > 0, we nearly completely characterize the space complexity of estimating ||A||pp = ∑i=1n σip for n × n matrices A in which each row and each column has O(1) non-zero entries and whose entries are presented one at a time in a data stream model. Here the σi are the singular values of A, and when p ≥ 1, ||A||pp is the p-th power of the Schatten p-norm. We show that when p is not an even integer, to obtain a (1+є)-approximation to ||A||pp with constant probability, any 1-pass algorithm requires n1−g(є) bits of space, where g(є) → 0 as є → 0 and є > 0 is a constant independent of n. However, when p is an even integer, we give an upper bound of n1−2/p (є−1logn) bits of space, which holds even in the turnstile data stream model. The latter is optimal up to (є−1 logn) factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices A: the previous best lower bound was Ω(logn) for p∈ (0,1), Ω(n1/p−1/2/logn) for p∈ [1,2) and Ω(n1−2/p) for p∈ (2,∞). We note for p ∈ (2, ∞), while our lower bound for even integers is the same, for other p in this range our lower bound is n1−g(є), which is considerably stronger than the previous n1−2/p for small enough constant є > 0. We obtain similar near-linear lower bounds for Ky-Fan norms, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128780331","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}
Sayan Bhattacharya, M. Henzinger, Danupon Nanongkai
We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2+є)-approximate maximum matching in general graphs with O(poly(logn, 1/є)) update time. (2) An algorithm that maintains an αK approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1≤ αK < 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.
{"title":"New deterministic approximation algorithms for fully dynamic matching","authors":"Sayan Bhattacharya, M. Henzinger, Danupon Nanongkai","doi":"10.1145/2897518.2897568","DOIUrl":"https://doi.org/10.1145/2897518.2897568","url":null,"abstract":"We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2+є)-approximate maximum matching in general graphs with O(poly(logn, 1/є)) update time. (2) An algorithm that maintains an αK approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1≤ αK < 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134497648","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}
We consider the problem of computing a k-sparse approximation to the Fourier transform of a length N signal. Our main result is a randomized algorithm for computing such an approximation (i.e. achieving ℓ2/ℓ2 sparse recovery guarantees using Fourier measurements) using Od(klogNloglogN) samples of the signal in time domain and Od(klogd+3 N) runtime, where d≥ 1 is the dimensionality of the Fourier transform. The sample complexity matches the Ω(klog(N/k)) lower bound for non-adaptive algorithms due to [DIPW] for any k≤ N1−δ for a constant δ>0 up to an O(loglogN) factor. Prior to our work a result with comparable sample complexity klogN logO(1)logN and sublinear runtime was known for the Fourier transform on the line [IKP], but for any dimension d≥ 2 previously known techniques either suffered from a (logN) factor loss in sample complexity or required Ω(N) runtime.
{"title":"Sparse fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time","authors":"M. Kapralov","doi":"10.1145/2897518.2897650","DOIUrl":"https://doi.org/10.1145/2897518.2897650","url":null,"abstract":"We consider the problem of computing a k-sparse approximation to the Fourier transform of a length N signal. Our main result is a randomized algorithm for computing such an approximation (i.e. achieving ℓ2/ℓ2 sparse recovery guarantees using Fourier measurements) using Od(klogNloglogN) samples of the signal in time domain and Od(klogd+3 N) runtime, where d≥ 1 is the dimensionality of the Fourier transform. The sample complexity matches the Ω(klog(N/k)) lower bound for non-adaptive algorithms due to [DIPW] for any k≤ N1−δ for a constant δ>0 up to an O(loglogN) factor. Prior to our work a result with comparable sample complexity klogN logO(1)logN and sublinear runtime was known for the Fourier transform on the line [IKP], but for any dimension d≥ 2 previously known techniques either suffered from a (logN) factor loss in sample complexity or required Ω(N) runtime.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117106221","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}
We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.
{"title":"Beyond matroids: secretary problem and prophet inequality with general constraints","authors":"A. Rubinstein","doi":"10.1145/2897518.2897540","DOIUrl":"https://doi.org/10.1145/2897518.2897540","url":null,"abstract":"We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"172 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120863801","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}
We introduce the notion of balance for directed graphs: a weighted directed graph is α-balanced if for every cut S ⊆ V, the total weight of edges going from S to V∖ S is within factor α of the total weight of edges going from V∖ S to S. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with α = 1) and residual graphs of (1+є)-approximate undirected maximum flows (with α=O(1/є)). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an O(α · log3 n / loglogn) competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is Ω(n) in general; this result improves upon the long-standing best known lower bound of Ω(√n) by Hajiaghayi et al. We also show that our restriction to single-source instances is necessary by showing an Ω(√n) lower bound for multiple-source oblivious routing in Eulerian graphs. We also study the maximum flow problem in balanced directed graphs with arbitrary capacities. We develop an efficient algorithm that finds an (1+є)-approximate maximum flows in α-balanced graphs in time O(m α2 / є2). We show that, using our approximate maximum flow algorithm, we can efficiently determine whether a given directed graph is α-balanced. Additionally, we give an application to the directed sparsest cut problem.
{"title":"Routing under balance","authors":"Alina Ene, G. Miller, J. Pachocki, Aaron Sidford","doi":"10.1145/2897518.2897654","DOIUrl":"https://doi.org/10.1145/2897518.2897654","url":null,"abstract":"We introduce the notion of balance for directed graphs: a weighted directed graph is α-balanced if for every cut S ⊆ V, the total weight of edges going from S to V∖ S is within factor α of the total weight of edges going from V∖ S to S. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with α = 1) and residual graphs of (1+є)-approximate undirected maximum flows (with α=O(1/є)). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an O(α · log3 n / loglogn) competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is Ω(n) in general; this result improves upon the long-standing best known lower bound of Ω(√n) by Hajiaghayi et al. We also show that our restriction to single-source instances is necessary by showing an Ω(√n) lower bound for multiple-source oblivious routing in Eulerian graphs. We also study the maximum flow problem in balanced directed graphs with arbitrary capacities. We develop an efficient algorithm that finds an (1+є)-approximate maximum flows in α-balanced graphs in time O(m α2 / є2). We show that, using our approximate maximum flow algorithm, we can efficiently determine whether a given directed graph is α-balanced. Additionally, we give an application to the directed sparsest cut problem.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134246261","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}
We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an α-approximate set cover (for α= o(√n)) via a single-pass streaming algorithm, we show that Θ(mn/α) space is both sufficient and necessary (up to an O(logn) factor); here m denotes number of the sets and n denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of α saving in the space is achievable in this case and that this is the best possible. In other words, we show that Θ(mn/α2) space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of α. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.
{"title":"Tight bounds for single-pass streaming complexity of the set cover problem","authors":"Sepehr Assadi, S. Khanna, Yang Li","doi":"10.1145/2897518.2897576","DOIUrl":"https://doi.org/10.1145/2897518.2897576","url":null,"abstract":"We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an α-approximate set cover (for α= o(√n)) via a single-pass streaming algorithm, we show that Θ(mn/α) space is both sufficient and necessary (up to an O(logn) factor); here m denotes number of the sets and n denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of α saving in the space is achievable in this case and that this is the best possible. In other words, we show that Θ(mn/α2) space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of α. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122792028","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}
We study the classical Node-Disjoint Paths (NDP) problem: given an n-vertex graph G and a collection =(s1,t1),…,(sk,tk) of pairs of vertices of G called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of O(√n) on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than Ω(log1/2−δn)-approximation for any constant δ, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is Ω(√n) for NDP, even in planar graphs. In this paper, we break the barrier of O(√n) on the approximability of NDP in planar graphs and obtain an Õ(n9/19)-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.
{"title":"Improved approximation for node-disjoint paths in planar graphs","authors":"Julia Chuzhoy, David H. K. Kim, Shi Li","doi":"10.1145/2897518.2897538","DOIUrl":"https://doi.org/10.1145/2897518.2897538","url":null,"abstract":"We study the classical Node-Disjoint Paths (NDP) problem: given an n-vertex graph G and a collection =(s1,t1),…,(sk,tk) of pairs of vertices of G called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of O(√n) on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than Ω(log1/2−δn)-approximation for any constant δ, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is Ω(√n) for NDP, even in planar graphs. In this paper, we break the barrier of O(√n) on the approximability of NDP in planar graphs and obtain an Õ(n9/19)-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114531543","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}
We consider the problem of finding the kth highest element in a totally ordered set of n elements (Select), and partitioning a totally ordered set into the top k and bottom n − k elements (Partition) using pairwise comparisons. Motivated by settings like peer grading or crowdsourcing, where multiple rounds of interaction are costly and queried comparisons may be inconsistent with the ground truth, we evaluate algorithms based both on their total runtime and the number of interactive rounds in three comparison models: noiseless (where the comparisons are correct), erasure (where comparisons are erased with probability 1 − γ), and noisy (where comparisons are correct with probability 1/2 + γ/2 and incorrect otherwise). We provide numerous matching upper and lower bounds in all three models. Even our results in the noiseless model, which is quite well-studied in the TCS literature on parallel algorithms, are novel.
{"title":"Parallel algorithms for select and partition with noisy comparisons","authors":"M. Braverman, Jieming Mao, S. Weinberg","doi":"10.1145/2897518.2897642","DOIUrl":"https://doi.org/10.1145/2897518.2897642","url":null,"abstract":"We consider the problem of finding the kth highest element in a totally ordered set of n elements (Select), and partitioning a totally ordered set into the top k and bottom n − k elements (Partition) using pairwise comparisons. Motivated by settings like peer grading or crowdsourcing, where multiple rounds of interaction are costly and queried comparisons may be inconsistent with the ground truth, we evaluate algorithms based both on their total runtime and the number of interactive rounds in three comparison models: noiseless (where the comparisons are correct), erasure (where comparisons are erased with probability 1 − γ), and noisy (where comparisons are correct with probability 1/2 + γ/2 and incorrect otherwise). We provide numerous matching upper and lower bounds in all three models. Even our results in the noiseless model, which is quite well-studied in the TCS literature on parallel algorithms, are novel.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131099235","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}
We present the hosted coloring framework for studying al- gorithmic and hardness results for the k-coloring problem. There is a class H of host graphs. One selects a graph H ∈ H and plants in it a balanced k-coloring (by partitioning the vertex set into k roughly equal parts, and removing all edges within each part). The resulting graph G is given as input to a polynomial time algorithm that needs to k-color G (any legal k-coloring would do – the algorithm is not required to recover the planted k-coloring). Earlier planted models correspond to the case that H is the class of all n-vertex d-regular graphs, a member H ∈ H is chosen at random, and then a balanced k-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when d = n δ (for 0 < δ ≤ 1), and Alon and Kahale [1997] managed to do so even when d is a sufficiently large constant. The new aspect in our framework is that it need not in- volve randomness. In one model within the framework (with k = 3) H is a d regular spectral expander (meaning that ex- cept for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than d) chosen by an adversary, and the planted 3-coloring is ran- dom. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In an- other model H is a random d-regular graph but the planted balanced 3-coloring is chosen by an adversary, after seeing H. We show that for a certain range of average degrees somewhat below √ n, finding a 3-coloring is NP-hard. To- gether these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.
我们提出了用于研究k-着色问题的算法和硬度结果的宿主着色框架。有一类主图H。选择一个图H∈H，并在其中种植一个平衡的k着色(通过将顶点集划分为k个大致相等的部分，并去除每个部分内的所有边)。结果图G作为一个多项式时间算法的输入，该算法需要对G进行k-着色(任何合法的k-着色都可以——该算法不需要恢复已植入的k-着色)。早期的种植模型对应于H是所有n顶点d正则图的类，随机选择一个成员H∈H，然后随机种植一个平衡的k着色。Blum和Spencer[1995]在d = n δ (0 < δ≤1)时为该模型设计了算法，Alon和Kahale[1997]即使d是一个足够大的常数也能做到这一点。在我们的框架的新方面是，它不需要在进化的随机性。在框架内的一个模型(k = 3)中，H是对手选择的d正则谱扩展器(意味着除了其邻接矩阵的最大特征值外，其他特征值的绝对值远小于d)，并且种植的3-着色是随机的。我们证明Alon和Kahale[1997]的3-着色算法可以被修改以适用于这种情况。在另一个模型中，H是一个随机的d规则图，但在看到H之后，对手选择了种植的平衡3-着色。我们表明，在一定的平均度范围内，略低于√n，找到3-着色是np困难的。综上所述，这些结果(以及我们已有的其他结果)有助于澄清种植着色模型中随机性的哪些方面是成功的3着色算法的关键。
{"title":"On the effect of randomness on planted 3-coloring models","authors":"R. David, U. Feige","doi":"10.1145/2897518.2897561","DOIUrl":"https://doi.org/10.1145/2897518.2897561","url":null,"abstract":"We present the hosted coloring framework for studying al- gorithmic and hardness results for the k-coloring problem. There is a class H of host graphs. One selects a graph H ∈ H and plants in it a balanced k-coloring (by partitioning the vertex set into k roughly equal parts, and removing all edges within each part). The resulting graph G is given as input to a polynomial time algorithm that needs to k-color G (any legal k-coloring would do – the algorithm is not required to recover the planted k-coloring). Earlier planted models correspond to the case that H is the class of all n-vertex d-regular graphs, a member H ∈ H is chosen at random, and then a balanced k-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when d = n δ (for 0 < δ ≤ 1), and Alon and Kahale [1997] managed to do so even when d is a sufficiently large constant. The new aspect in our framework is that it need not in- volve randomness. In one model within the framework (with k = 3) H is a d regular spectral expander (meaning that ex- cept for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than d) chosen by an adversary, and the planted 3-coloring is ran- dom. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In an- other model H is a random d-regular graph but the planted balanced 3-coloring is chosen by an adversary, after seeing H. We show that for a certain range of average degrees somewhat below √ n, finding a 3-coloring is NP-hard. To- gether these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127298001","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}
This paper studies a new online problem, referred to as min-cost perfect matching with delays (MPMD), defined over a finite metric space (i.e., a complete graph with positive edge weights obeying the triangle inequality) M that is known to the algorithm in advance. Requests arrive in a continuous time online fashion at the points of M and should be served by matching them to each other. The algorithm is allowed to delay its request matching commitments, but this does not come for free: the total cost of the algorithm is the sum of metric distances between matched requests plus the sum of times each request waited since it arrived until it was matched. A randomized online MPMD algorithm is presented whose competitive ratio is O (log2 n + logΔ), where n is the number of points in M and Δ is its aspect ratio. The analysis is based on a machinery developed in the context of a new stochastic process that can be viewed as two interleaved Poisson processes; surprisingly, this new process captures precisely the behavior of our algorithm. A related problem in which the algorithm is allowed to clear any unmatched request at a fixed penalty is also addressed. It is suggested that the MPMD problem is merely the tip of the iceberg for a general framework of online problems with delayed service that captures many more natural problems.
本文研究了一个新的在线问题，称为最小代价带延迟完美匹配(MPMD)，该问题定义在一个有限度量空间(即一个边权为正且服从三角不等式的完全图)M上，且算法事先已知。请求以连续时间在线方式到达M点，并且应该通过相互匹配来提供服务。该算法允许延迟其请求匹配承诺，但这并不是免费的:算法的总成本是匹配请求之间的度量距离之和加上每个请求从到达到匹配为止等待的时间之和。提出了一种随机在线MPMD算法，其竞争比为O (log2 n + logΔ)，其中n为M中的点数，Δ为其纵横比。分析是基于在一个新的随机过程的背景下开发的机器，可以看作是两个交错的泊松过程;令人惊讶的是，这个新过程精确地捕捉到了我们算法的行为。一个相关的问题，其中算法被允许清除任何不匹配的请求在一个固定的惩罚也解决了。有人建议，MPMD问题仅仅是包含延迟服务的在线问题的总体框架的冰山一角，它包含了许多更自然的问题。
{"title":"Online matching: haste makes waste!","authors":"Y. Emek, S. Kutten, Roger Wattenhofer","doi":"10.1145/2897518.2897557","DOIUrl":"https://doi.org/10.1145/2897518.2897557","url":null,"abstract":"This paper studies a new online problem, referred to as min-cost perfect matching with delays (MPMD), defined over a finite metric space (i.e., a complete graph with positive edge weights obeying the triangle inequality) M that is known to the algorithm in advance. Requests arrive in a continuous time online fashion at the points of M and should be served by matching them to each other. The algorithm is allowed to delay its request matching commitments, but this does not come for free: the total cost of the algorithm is the sum of metric distances between matched requests plus the sum of times each request waited since it arrived until it was matched. A randomized online MPMD algorithm is presented whose competitive ratio is O (log2 n + logΔ), where n is the number of points in M and Δ is its aspect ratio. The analysis is based on a machinery developed in the context of a new stochastic process that can be viewed as two interleaved Poisson processes; surprisingly, this new process captures precisely the behavior of our algorithm. A related problem in which the algorithm is allowed to clear any unmatched request at a fixed penalty is also addressed. It is suggested that the MPMD problem is merely the tip of the iceberg for a general framework of online problems with delayed service that captures many more natural problems.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130594784","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}
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