Computational complexity has already had plenty to say about the computation of economic equilibria. However, understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. In this talk we survey our main results presented at EC'15, which show that the existence of equilibria in markets is inextricably connected to the computational complexity of related optimization problems, such as revenue or welfare maximization. We demonstrate how this relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare maximization.
{"title":"Why prices need algorithms (invited talk)","authors":"T. Roughgarden, Inbal Talgam-Cohen","doi":"10.1145/3055399.3079077","DOIUrl":"https://doi.org/10.1145/3055399.3079077","url":null,"abstract":"Computational complexity has already had plenty to say about the computation of economic equilibria. However, understanding when equilibria are guaranteed to exist is a central theme in economic theory, seemingly unrelated to computation. In this talk we survey our main results presented at EC'15, which show that the existence of equilibria in markets is inextricably connected to the computational complexity of related optimization problems, such as revenue or welfare maximization. We demonstrate how this relationship implies, under suitable complexity assumptions, a host of impossibility results. We also suggest a complexity-theoretic explanation for the lack of useful extensions of the Walrasian equilibrium concept: such extensions seem to require the invention of novel polynomial-time algorithms for welfare maximization.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79782326","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}
A polynomial threshold function (PTF) of degree d is a boolean function of the form f=sgn(p), where p is a degree-d polynomial, and sgn is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is not close to a constant function, where a boolean function g is called δ-close to constant if, for some vε{1,-1}, we have g(x)=v for all but at most δ fraction of inputs. We show for every PTF f of degree d≥ 1, and parameters 0<δ, r≤ 1/16, that Pr∾ Rr [fρ is not δ-close to constant] ≤ √ #183;(logr-1· logδ-1)O(d2), where ρ ∾ Rr is a random restriction leaving each variable, independently, free with probability r, and otherwise assigning it 1 or -1 uniformly at random. In fact, we show a more general result for random block restrictions: given an arbitrary partitioning of input variables into m blocks, a random block restriction picks a uniformly random block ℓΕ [m] and assigns 1 or -1, uniformly at random, to all variable outside the chosen block ℓ. We prove the Block Restriction Lemma saying that a PTF f of degree d becomes δ-close to constant when hit with a random block restriction, except with probability at most m-1/2 #183; (logm#183; logδ-1)O(d2). As an application of our Restriction Lemma, we prove lower bounds against constant-depth circuits with PTF gates of any degree 1≤ d≪ √logn/loglogn, generalizing the recent bounds against constant-depth circuits with linear threshold gates (LTF gates) proved by Kane and Williams (STOC, 2016) and Chen, Santhanam, and Srinivasan (CCC, 2016). In particular, we show that there is an n-variate boolean function Fn Ε P such that every depth-2 circuit with PTF gates of degree d≥ 1 that computes Fn must have at least (n3/2+1/d)#183; (logn)-O(d2) wires. For constant depths greater than 2, we also show average-case lower bounds for such circuits with super-linear number of wires. These are the first super-linear bounds on the number of wires for circuits with PTF gates. We also give short proofs of the optimal-exponent average sensitivity bound for degree-d PTFs due to Kane (Computational Complexity, 2014), and the Littlewood-Offord type anticoncentration bound for degree-d multilinear polynomials due to Meka, Nguyen, and Vu (Theory of Computing, 2016). Finally, we give derandomized versions of our Block Restriction Lemma and Littlewood-Offord type anticoncentration bounds, using a pseudorandom generator for PTFs due to Meka and Zuckerman (SICOMP, 2013).
{"title":"A polynomial restriction lemma with applications","authors":"Valentine Kabanets, D. Kane, Zhenjian Lu","doi":"10.1145/3055399.3055470","DOIUrl":"https://doi.org/10.1145/3055399.3055470","url":null,"abstract":"A polynomial threshold function (PTF) of degree d is a boolean function of the form f=sgn(p), where p is a degree-d polynomial, and sgn is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is not close to a constant function, where a boolean function g is called δ-close to constant if, for some vε{1,-1}, we have g(x)=v for all but at most δ fraction of inputs. We show for every PTF f of degree d≥ 1, and parameters 0<δ, r≤ 1/16, that Pr∾ Rr [fρ is not δ-close to constant] ≤ √ #183;(logr-1· logδ-1)O(d2), where ρ ∾ Rr is a random restriction leaving each variable, independently, free with probability r, and otherwise assigning it 1 or -1 uniformly at random. In fact, we show a more general result for random block restrictions: given an arbitrary partitioning of input variables into m blocks, a random block restriction picks a uniformly random block ℓΕ [m] and assigns 1 or -1, uniformly at random, to all variable outside the chosen block ℓ. We prove the Block Restriction Lemma saying that a PTF f of degree d becomes δ-close to constant when hit with a random block restriction, except with probability at most m-1/2 #183; (logm#183; logδ-1)O(d2). As an application of our Restriction Lemma, we prove lower bounds against constant-depth circuits with PTF gates of any degree 1≤ d≪ √logn/loglogn, generalizing the recent bounds against constant-depth circuits with linear threshold gates (LTF gates) proved by Kane and Williams (STOC, 2016) and Chen, Santhanam, and Srinivasan (CCC, 2016). In particular, we show that there is an n-variate boolean function Fn Ε P such that every depth-2 circuit with PTF gates of degree d≥ 1 that computes Fn must have at least (n3/2+1/d)#183; (logn)-O(d2) wires. For constant depths greater than 2, we also show average-case lower bounds for such circuits with super-linear number of wires. These are the first super-linear bounds on the number of wires for circuits with PTF gates. We also give short proofs of the optimal-exponent average sensitivity bound for degree-d PTFs due to Kane (Computational Complexity, 2014), and the Littlewood-Offord type anticoncentration bound for degree-d multilinear polynomials due to Meka, Nguyen, and Vu (Theory of Computing, 2016). Finally, we give derandomized versions of our Block Restriction Lemma and Littlewood-Offord type anticoncentration bounds, using a pseudorandom generator for PTFs due to Meka and Zuckerman (SICOMP, 2013).","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81295156","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 randomized algorithms that solve Subset Sum and Knapsack instances with n items in O*(20.86n) time, where the O*(·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(nk-0.5(n)) time and O(logn) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(logn) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.
{"title":"Faster space-efficient algorithms for subset sum and k-sum","authors":"N. Bansal, S. Garg, Jesper Nederlof, Nikhil Vyas","doi":"10.1145/3055399.3055467","DOIUrl":"https://doi.org/10.1145/3055399.3055467","url":null,"abstract":"We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O*(20.86n) time, where the O*(·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(nk-0.5(n)) time and O(logn) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(logn) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83873595","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}
Real-word networks are often prone to failures. A reliable network needs to cope with this situation and must provide a backup communication channel. This motivates the study of survivable network design, which has been a focus of research for a few decades. To date, survivable network design problems on undirected graphs are well-understood. For example, there is a 2 approximation in the case of edge failures [Jain, FOCS'98/Combinatorica'01]. The problems on directed graphs, in contrast, have seen very little progress. Most techniques for the undirected case like primal-dual and iterative rounding methods do not seem to extend to the directed case. Almost no non-trivial approximation algorithm is known even for a simple case where we wish to design a network that tolerates a single failure. In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an n-vertex weighted directed graph, a root r, and a set of h terminals S, find a min-cost subgraph H that has two edge/vertex disjoint paths from r to any tε S. 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by Feldman et al., [SODA'09; JCSS] and has then been studied by Cheriyan et al. [SODA'12; TALG] and Laekhanukit [SODA'14]. However, no positive result was known except for the special case of a D-shallow instance [Laekhanukit, ICALP'16]. We present an O(D3logD#183; h2/D#183; logn) approximation algorithm for 2-DST that runs in time O(nO(D)), for any Dε[log2h]. This implies a polynomial-time O(hεlogn) approximation for any constant ε>0, and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is O(log2-εn)-hard to approximate [Halperin and Krauthgamer, STOC'03]. As a by product, we obtain an algorithm with the same approximation guarantee for the 2-Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are 2-edge/vertex connected. Our approximation algorithm is based on a careful combination of several techniques. In more detail, we decompose an optimal solution into two (possibly not edge disjoint) divergent trees that induces two edge disjoint paths from the root to any given terminal. These divergent trees are then embedded into a shallow tree by means of Zelikovsky's height reduction theorem. On the latter tree we solve a 2-Connected Group Steiner Tree problem and then map back this solution to the original graph. Crucially, our tree embedding is achieved via a probabilistic mapping guided by an LP: This is the main technical novelty of our approach, and might be useful for future work.
{"title":"Surviving in directed graphs: a quasi-polynomial-time polylogarithmic approximation for two-connected directed Steiner tree","authors":"F. Grandoni, Bundit Laekhanukit","doi":"10.1145/3055399.3055445","DOIUrl":"https://doi.org/10.1145/3055399.3055445","url":null,"abstract":"Real-word networks are often prone to failures. A reliable network needs to cope with this situation and must provide a backup communication channel. This motivates the study of survivable network design, which has been a focus of research for a few decades. To date, survivable network design problems on undirected graphs are well-understood. For example, there is a 2 approximation in the case of edge failures [Jain, FOCS'98/Combinatorica'01]. The problems on directed graphs, in contrast, have seen very little progress. Most techniques for the undirected case like primal-dual and iterative rounding methods do not seem to extend to the directed case. Almost no non-trivial approximation algorithm is known even for a simple case where we wish to design a network that tolerates a single failure. In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an n-vertex weighted directed graph, a root r, and a set of h terminals S, find a min-cost subgraph H that has two edge/vertex disjoint paths from r to any tε S. 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by Feldman et al., [SODA'09; JCSS] and has then been studied by Cheriyan et al. [SODA'12; TALG] and Laekhanukit [SODA'14]. However, no positive result was known except for the special case of a D-shallow instance [Laekhanukit, ICALP'16]. We present an O(D3logD#183; h2/D#183; logn) approximation algorithm for 2-DST that runs in time O(nO(D)), for any Dε[log2h]. This implies a polynomial-time O(hεlogn) approximation for any constant ε>0, and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is O(log2-εn)-hard to approximate [Halperin and Krauthgamer, STOC'03]. As a by product, we obtain an algorithm with the same approximation guarantee for the 2-Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are 2-edge/vertex connected. Our approximation algorithm is based on a careful combination of several techniques. In more detail, we decompose an optimal solution into two (possibly not edge disjoint) divergent trees that induces two edge disjoint paths from the root to any given terminal. These divergent trees are then embedded into a shallow tree by means of Zelikovsky's height reduction theorem. On the latter tree we solve a 2-Connected Group Steiner Tree problem and then map back this solution to the original graph. Crucially, our tree embedding is achieved via a probabilistic mapping guided by an LP: This is the main technical novelty of our approach, and might be useful for future work.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76115583","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 two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2/logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn/logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by Kapron et al. To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms.
{"title":"Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time","authors":"Danupon Nanongkai, Thatchaphol Saranurak","doi":"10.1145/3055399.3055447","DOIUrl":"https://doi.org/10.1145/3055399.3055447","url":null,"abstract":"We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2/logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn/logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by Kapron et al. To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81093573","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}
In this talk we will discuss a general framework to solve certain sums of products of functions over semi-rings. This captures many well-known problems in disparate areas such as CSPs, Probabilistic Graphical Models, Databases, Logic and Matrix Operations. This talk is based on joint work titled FAQ: Questions Asked Frequently with Mahmoud Abo Khamis and Hung Q. Ngo, which appeared in PODS 2016.
{"title":"Answering FAQs in CSPs, probabilistic graphical models, databases, logic and matrix operations (invited talk)","authors":"A. Rudra","doi":"10.1145/3055399.3079073","DOIUrl":"https://doi.org/10.1145/3055399.3079073","url":null,"abstract":"In this talk we will discuss a general framework to solve certain sums of products of functions over semi-rings. This captures many well-known problems in disparate areas such as CSPs, Probabilistic Graphical Models, Databases, Logic and Matrix Operations. This talk is based on joint work titled FAQ: Questions Asked Frequently with Mahmoud Abo Khamis and Hung Q. Ngo, which appeared in PODS 2016.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81094054","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 define a concept class ℱ to be time-space hard (or memory-samples hard) if any learning algorithm for ℱ requires either a memory of size super-linear in n or a number of samples super-polynomial in n, where n is the length of one sample. A recent work shows that the class of all parity functions is time-space hard [Raz, FOCS'16]. Building on [Raz, FOCS'16], we show that the class of all sparse parities of Hamming weight ℓ is time-space hard, as long as ℓ ≥ ω(logn / loglogn). Consequently, linear-size DNF Formulas, linear-size Decision Trees and logarithmic-size Juntas are all time-space hard. Our result is more general and provides time-space lower bounds for learning any concept class of parity functions. We give applications of our results in the field of bounded-storage cryptography. For example, for every ωlogn) ≤ k ≤ n, we obtain an encryption scheme that requires a private key of length k, and time complexity of n per encryption/decryption of each bit, and is provably and unconditionally secure as long as the attacker uses at most o(nk) memory bits and the scheme is used at most 2o(k) times. Previously, this was known only for k=n [Raz, FOCS'16].
{"title":"Time-space hardness of learning sparse parities","authors":"Gillat Kol, R. Raz, Avishay Tal","doi":"10.1145/3055399.3055430","DOIUrl":"https://doi.org/10.1145/3055399.3055430","url":null,"abstract":"We define a concept class ℱ to be time-space hard (or memory-samples hard) if any learning algorithm for ℱ requires either a memory of size super-linear in n or a number of samples super-polynomial in n, where n is the length of one sample. A recent work shows that the class of all parity functions is time-space hard [Raz, FOCS'16]. Building on [Raz, FOCS'16], we show that the class of all sparse parities of Hamming weight ℓ is time-space hard, as long as ℓ ≥ ω(logn / loglogn). Consequently, linear-size DNF Formulas, linear-size Decision Trees and logarithmic-size Juntas are all time-space hard. Our result is more general and provides time-space lower bounds for learning any concept class of parity functions. We give applications of our results in the field of bounded-storage cryptography. For example, for every ωlogn) ≤ k ≤ n, we obtain an encryption scheme that requires a private key of length k, and time complexity of n per encryption/decryption of each bit, and is provably and unconditionally secure as long as the attacker uses at most o(nk) memory bits and the scheme is used at most 2o(k) times. Previously, this was known only for k=n [Raz, FOCS'16].","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87922048","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}
G. Italiano, Adam Karczmarz, Jakub Lacki, P. Sankowski
In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in O(nlog2nloglogn) total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only O(log2 n loglogn), which improves upon a previously known O(√n) solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs.
本文给出了一种求解有向平面图中递减单源可达性问题的新算法。它在O(nlog2nloglogn)总时间内处理任何边删除序列，并显式维护从固定源顶点可到达的顶点集。因此，如果所有边最终都被删除，处理每条边删除的平摊时间仅为O(log2 n loglog)，这比之前已知的O(√n)解决方案有所改进。我们还展示了一种具有相同总更新时间的有向平面图强连接分量的递减维护算法。这些结果构成了这两个问题的第一个几乎最优的算法(直到多对数因子)。据我们所知，这些是第一个在一般有向平面图上具有多对数更新时间的动态算法，用于非平凡可达型问题，在一般图中只有多项式界是已知的。
{"title":"Decremental single-source reachability in planar digraphs","authors":"G. Italiano, Adam Karczmarz, Jakub Lacki, P. Sankowski","doi":"10.1145/3055399.3055480","DOIUrl":"https://doi.org/10.1145/3055399.3055480","url":null,"abstract":"In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in O(nlog2nloglogn) total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only O(log2 n loglogn), which improves upon a previously known O(√n) solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80316552","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 graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G. We use the framework of graph motif parameters to obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G. More precisely, for graphs H on k edges, we show how to count subgraph copies of H in time kO(k)· n0.174k + o(k) by a surprisingly simple algorithm. This improves upon previously known running times, such as O(n0.91k + c) time for k-edge matchings or O(n0.46k + c) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ε C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds. Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy: For vertex-colored graphs H and G, where H is from a fixed class of graphs, we want to count color-preserving H-copies in G. We show that this problem is either polynomial-time solvable or FPT or #W[1]-hard, and that the FPT cases indeed need FPT time under reasonable assumptions.
{"title":"Homomorphisms are a good basis for counting small subgraphs","authors":"Radu Curticapean, Holger Dell, D. Marx","doi":"10.1145/3055399.3055502","DOIUrl":"https://doi.org/10.1145/3055399.3055502","url":null,"abstract":"We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G. We use the framework of graph motif parameters to obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G. More precisely, for graphs H on k edges, we show how to count subgraph copies of H in time kO(k)· n0.174k + o(k) by a surprisingly simple algorithm. This improves upon previously known running times, such as O(n0.91k + c) time for k-edge matchings or O(n0.46k + c) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ε C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds. Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy: For vertex-colored graphs H and G, where H is from a fixed class of graphs, we want to count color-preserving H-copies in G. We show that this problem is either polynomial-time solvable or FPT or #W[1]-hard, and that the FPT cases indeed need FPT time under reasonable assumptions.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82107389","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 quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex v, outputs the children of v. We construct a quantum algorithm which, given such access to a search tree of depth at most n, estimates the size of the tree T within a factor of 1± δ in Õ(√nT) steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which examines T nodes of a search tree into an Õ(√Tn3/2) time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro (arXiv:1509.02374). b)We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games) which evaluates formulas of size T and depth To(1) in time O(T1/2+o(1)). Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.
{"title":"Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games","authors":"A. Ambainis, M. Kokainis","doi":"10.1145/3055399.3055444","DOIUrl":"https://doi.org/10.1145/3055399.3055444","url":null,"abstract":"We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex v, outputs the children of v. We construct a quantum algorithm which, given such access to a search tree of depth at most n, estimates the size of the tree T within a factor of 1± δ in Õ(√nT) steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which examines T nodes of a search tree into an Õ(√Tn3/2) time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro (arXiv:1509.02374). b)We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games) which evaluates formulas of size T and depth To(1) in time O(T1/2+o(1)). Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74152392","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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