We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any-boolean 2-CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clause-by-clause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are "hard" to round in a certain sense, we obtain a Unique Games-based inapproximability matching this "hardness" of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the "hardness" of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2-CSPs. As an application, we show that Max 2-AND is hard to approximate within 0.87435. This improves upon the best previous hardness of alphaGW + epsi ap 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of Max 2-AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor alphaGW.
我们继续研究基于半定规划的近似算法和唯一博弈猜想之间的联系。给定任意-布尔2-CSP(或者更一般地说，两个布尔变量上的任意非负目标函数)，我们展示了如何将搜索良好的不可逼近性结果减少到某个数值最小化问题。我们分析的主要对象是在对基于半定规划的算法进行逐句分析时产生的向量三元组。给定一组具有一定限制类型的加权三元组，即在某种意义上“难以”四舍五入，我们将获得一种独特的基于游戏的不逼近性，与这种四舍五入向量三元组的“硬度”相匹配。相反，任何具有SDP解的实例都可以看作是一组向量三元组，并且我们表明，我们总能找到一个分配给实例的值，它至少与相应向量三元组的四舍五入的“硬度”一样好。我们推测，硬度结果所需的限制类型实际上是没有限制，这意味着这些上限和下界完全匹配。这一猜想得到了特定2- csp的所有现有结果的支持。作为一个应用，我们表明Max 2-AND很难在0.87435范围内近似。这在alphaGW + epsi ap 0.87856的最佳硬度基础上得到了改进，并且非常接近于匹配已知最佳算法的近似比率0.87401。它还确定了max2 - and的平衡实例，即每个变量正负相等地经常出现的实例，并不是最难近似的，因为这些可以在因子alphaGW内近似。
{"title":"Towards Sharp Inapproximability For Any 2-CSP","authors":"Per Austrin","doi":"10.1137/070711670","DOIUrl":"https://doi.org/10.1137/070711670","url":null,"abstract":"We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any-boolean 2-CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clause-by-clause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are \"hard\" to round in a certain sense, we obtain a Unique Games-based inapproximability matching this \"hardness\" of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the \"hardness\" of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2-CSPs. As an application, we show that Max 2-AND is hard to approximate within 0.87435. This improves upon the best previous hardness of alphaGW + epsi ap 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of Max 2-AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor alphaGW.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117060741","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}
An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol perp, which means "I don't know," A central question in average-case complexity is whether every distributional decision problem in N P has an errorless heuristic scheme: This is an algorithm that, for every delta > 0, runs in time polynomial in the instance size and | / delta and answers perp only on a delta fraction of instances. We study the question from the standpoint of hardness amplification and show that If every problem in (NP,U) has errorless heuristic circuits that output the correct answer on n -2/9+omicron(1)-fraction of inputs, then (NP,U) has non-uniform errorless heuristic schemes. If every problem in (NP,U) has randomized errorless heuristic algorithms that output the correct answer on (log n)-1/10+omicron(1)-fraction of inputs, then (NP.W) has randomized errorless heuristic schemes. In both cases, the low-end amplification is achieved by analyzing a new sensitivity property of monotone boolean Junctions in NP. In the non-uniform setting we use a " holographic Junction" introduced by Benjamini, Schramm, and Wilson (STOC 2005). For the uniform setting we introduce a new Junction that can be viewed as an efficient version of Talagrand's "random DNF".
{"title":"Hardness Amplification for Errorless Heuristics","authors":"Andrej Bogdanov, S. Safra","doi":"10.1109/FOCS.2007.25","DOIUrl":"https://doi.org/10.1109/FOCS.2007.25","url":null,"abstract":"An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol perp, which means \"I don't know,\" A central question in average-case complexity is whether every distributional decision problem in N P has an errorless heuristic scheme: This is an algorithm that, for every delta > 0, runs in time polynomial in the instance size and | / delta and answers perp only on a delta fraction of instances. We study the question from the standpoint of hardness amplification and show that If every problem in (NP,U) has errorless heuristic circuits that output the correct answer on n -2/9+omicron(1)-fraction of inputs, then (NP,U) has non-uniform errorless heuristic schemes. If every problem in (NP,U) has randomized errorless heuristic algorithms that output the correct answer on (log n)-1/10+omicron(1)-fraction of inputs, then (NP.W) has randomized errorless heuristic schemes. In both cases, the low-end amplification is achieved by analyzing a new sensitivity property of monotone boolean Junctions in NP. In the non-uniform setting we use a \" holographic Junction\" introduced by Benjamini, Schramm, and Wilson (STOC 2005). For the uniform setting we introduce a new Junction that can be viewed as an efficient version of Talagrand's \"random DNF\".","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128746957","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 a variant of the classic multi-armed bandit problem (MAB), which we call feedback MAB, where the reward obtained by playing each of n independent arms varies according to an underlying on/off Markov process with known parameters. The evolution of the Markov chain happens irrespective of whether the arm is played, and furthermore, the exact state of the Markov chain is only revealed to the player when the arm is played and the reward observed. At most one arm (or in general, M arms) can be played any time step. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is an instance of a partially observable Markov decision process (POMDP), and a special case of the notoriously intractable "restless bandit" problem. Unlike the stochastic MAB problem, the feedback MAB problem does not admit to greedy index-based optimal policies. Vie state of the system at any time step encodes the beliefs about the states of different arms, and the policy decisions change these beliefs - this aspect complicates the design and analysis of simple algorithms. We design a constant factor approximation to the feedback MAB problem by solving and rounding a natural LP relaxation to this problem. As far as we are aware, this is the first approximation algorithm for a POMDP problem.
{"title":"Approximation Algorithms for Partial-Information Based Stochastic Control with Markovian Rewards","authors":"S. Guha, Kamesh Munagala","doi":"10.1109/FOCS.2007.12","DOIUrl":"https://doi.org/10.1109/FOCS.2007.12","url":null,"abstract":"We consider a variant of the classic multi-armed bandit problem (MAB), which we call feedback MAB, where the reward obtained by playing each of n independent arms varies according to an underlying on/off Markov process with known parameters. The evolution of the Markov chain happens irrespective of whether the arm is played, and furthermore, the exact state of the Markov chain is only revealed to the player when the arm is played and the reward observed. At most one arm (or in general, M arms) can be played any time step. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is an instance of a partially observable Markov decision process (POMDP), and a special case of the notoriously intractable \"restless bandit\" problem. Unlike the stochastic MAB problem, the feedback MAB problem does not admit to greedy index-based optimal policies. Vie state of the system at any time step encodes the beliefs about the states of different arms, and the policy decisions change these beliefs - this aspect complicates the design and analysis of simple algorithms. We design a constant factor approximation to the feedback MAB problem by solving and rounding a natural LP relaxation to this problem. As far as we are aware, this is the first approximation algorithm for a POMDP problem.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128146193","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}
Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d. > l deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linear-space representation of graphs which enables us to determine how a batch of edge updates can impact the graph. Given a set of d edge updates, in time O(d polylg n) we can obtain the number of connected components, the size of each component, and a fast oracle for answering connectivity queries in the updated graph. The initial representation is polynomial-time constructible.
{"title":"Planning for Fast Connectivity Updates","authors":"M. Patrascu, Mikkel Thorup","doi":"10.1109/FOCS.2007.54","DOIUrl":"https://doi.org/10.1109/FOCS.2007.54","url":null,"abstract":"Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d. > l deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linear-space representation of graphs which enables us to determine how a batch of edge updates can impact the graph. Given a set of d edge updates, in time O(d polylg n) we can obtain the number of connected components, the size of each component, and a fast oracle for answering connectivity queries in the updated graph. The initial representation is polynomial-time constructible.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121453009","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}
Pub Date : 2007-10-21DOI: 10.4086/toc.2010.v006a005
D. Marx
It is well-known that constraint satisfaction problems (CSP) can be solved in time nO(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidth-based algorithm, even if we restrict the problem to some special class of primal graphs. Formally, let g be an arbitrary class of graphs and assume that there is an algorithm A solving binary CSP for instances whose primal graph is in g. We prove that if the running lime of A is f(G)nO(k/logk), where k is the treewidth of the primal graph G and f is an arbitrary function, then the Exponential Time Hypothesis fails. We prove the result also in the more general framework of the homomorphism problem for bounded-arity relational structures. For this problem, the treewidth of the core of the left-hand side structure plays the same role as the. treewidth of the primal graph above.
{"title":"Can you beat treewidth?","authors":"D. Marx","doi":"10.4086/toc.2010.v006a005","DOIUrl":"https://doi.org/10.4086/toc.2010.v006a005","url":null,"abstract":"It is well-known that constraint satisfaction problems (CSP) can be solved in time nO(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidth-based algorithm, even if we restrict the problem to some special class of primal graphs. Formally, let g be an arbitrary class of graphs and assume that there is an algorithm A solving binary CSP for instances whose primal graph is in g. We prove that if the running lime of A is f(G)nO(k/logk), where k is the treewidth of the primal graph G and f is an arbitrary function, then the Exponential Time Hypothesis fails. We prove the result also in the more general framework of the homomorphism problem for bounded-arity relational structures. For this problem, the treewidth of the core of the left-hand side structure plays the same role as the. treewidth of the primal graph above.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116357973","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}
Sofya Raskhodnikova, D. Ron, Amir Shpilka, Adam D. Smith
We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least 1/n. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length n. For both problems, we prove a nearly linear in n lower bound on the query complexity, applicable even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables. X1 and X2, with very different expectations and the following condition on the first k moments: E[X1]/E[X2] = E[X12]/E[X22] = ... = E[X1k]/E[X2k]. Our lower bound method is also applicable to other problems. In particular, it gives new lower bounds for the sample complexity of (1) approximating the entropy of a distribution and (2) approximating how well a given string is compressed by the Lempel-Ziv scheme.
{"title":"Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem","authors":"Sofya Raskhodnikova, D. Ron, Amir Shpilka, Adam D. Smith","doi":"10.1109/FOCS.2007.67","DOIUrl":"https://doi.org/10.1109/FOCS.2007.67","url":null,"abstract":"We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least 1/n. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length n. For both problems, we prove a nearly linear in n lower bound on the query complexity, applicable even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables. X<sub>1</sub> and X<sub>2</sub>, with very different expectations and the following condition on the first k moments: E[X<sub>1</sub>]/E[X<sub>2</sub>] = E[X<sub>1</sub> <sup>2</sup>]/E[X<sub>2</sub> <sup>2</sup>] = ... = E[X<sub>1</sub> <sup>k</sup>]/E[X<sub>2</sub> <sup>k</sup>]. Our lower bound method is also applicable to other problems. In particular, it gives new lower bounds for the sample complexity of (1) approximating the entropy of a distribution and (2) approximating how well a given string is compressed by the Lempel-Ziv scheme.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125097533","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. Ambainis, Andrew M. Childs, B. Reichardt, R. Spalek, Shengyu Zhang
For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or "approximately balanced," formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
{"title":"Any AND-OR Formula of Size N can be Evaluated in time N^{1/2 + o(1)} on a Quantum Computer","authors":"A. Ambainis, Andrew M. Childs, B. Reichardt, R. Spalek, Shengyu Zhang","doi":"10.1137/080712167","DOIUrl":"https://doi.org/10.1137/080712167","url":null,"abstract":"For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or \"approximately balanced,\" formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131181781","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}
N. Bansal, H. Chan, R. Khandekar, K. Pruhs, C. Stein, B. Schieber
We give the first O(l)-speed O(l) approximation polynomial-time algorithms for several nonpreemptive min-sum scheduling problems where jobs arrive over time and must be processed on one machine. More precisely, we give the first O(l)-speed O(l)-approximations for the non-preemptive scheduling problems; l|rj| SigmawjFj (weighted flow time), l |rj| SigmaTj (total tardiness), the broadcast version of 1 |rj| SigmawjFj , an O(I)-speed, 1-approximation for l |rj| Sigma U macrj (throughput maximization), and an O(l)-machine, O(l)-speed O(1)-approximation for l |rj| SigmawjTj (weighted tardiness). Our main contribution is an integer programming formulation whose relaxation is sufficiently close to the integer optimum, and which can be transformed to a schedule on a faster machine.
针对一些作业随时间到达且必须在一台机器上处理的非抢占式最小和调度问题，给出了第一种O(l)速度O(l)逼近多项式时间算法。更准确地说，我们给出了非抢占调度问题的第一个O(l)-速度O(l)-近似;l|rj| SigmawjFj(加权流时间)，l|rj| SigmawjFj(总延迟)，1 |rj| SigmawjFj的广播版本，l|rj| Sigma U macrj(吞吐量最大化)的O(I)-速度，1-逼近，l|rj| SigmawjTj(加权延迟)的O(l)-机器，O(l)-速度，O(1)-逼近。我们的主要贡献是一个整数规划公式，它的松弛足够接近整数最优，并且可以在更快的机器上转换为调度。
{"title":"Non-Preemptive Min-Sum Scheduling with Resource Augmentation","authors":"N. Bansal, H. Chan, R. Khandekar, K. Pruhs, C. Stein, B. Schieber","doi":"10.1109/FOCS.2007.46","DOIUrl":"https://doi.org/10.1109/FOCS.2007.46","url":null,"abstract":"We give the first O(l)-speed O(l) approximation polynomial-time algorithms for several nonpreemptive min-sum scheduling problems where jobs arrive over time and must be processed on one machine. More precisely, we give the first O(l)-speed O(l)-approximations for the non-preemptive scheduling problems; l|r<sub>j</sub>| Sigmaw<sub>j</sub>F<sub>j</sub> (weighted flow time), l |r<sub>j</sub>| SigmaT<sub>j</sub> (total tardiness), the broadcast version of 1 |r<sub>j</sub>| Sigmaw<sub>j</sub>F<sub>j</sub> , an O(I)-speed, 1-approximation for l |r<sub>j</sub>| Sigma U macr<sub>j</sub> (throughput maximization), and an O(l)-machine, O(l)-speed O(1)-approximation for l |r<sub>j</sub>| Sigmaw<sub>j</sub>T<sub>j</sub> (weighted tardiness). Our main contribution is an integer programming formulation whose relaxation is sufficiently close to the integer optimum, and which can be transformed to a schedule on a faster machine.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114056499","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 a new primitive called intrusion-resilient secret sharing (IRSS), whose security proof exploits the fact that there exist functions which can be efficiently computed interactively using low communication complexity in k, but not in k-1 rounds. IRSS is a means of sharing a secret message amongst a set of players which comes with a very strong security guarantee. The shares in an IRSS are made artificially large so that it is hard to retrieve them completely, and the reconstruction procedure is interactive requiring the players to exchange k short messages. The adversaries considered can attack the scheme in rounds, where in each round the adversary chooses some player to corrupt and some function, and retrieves the output of that function applied to the share of the corrupted player. This model captures for example computers connected to a network which can occasionally he infected by malicious software like viruses, which can compute any function on the infected machine, but cannot sent out a huge amount of data. Using methods from the bounded-retrieval model, we construct an IRSS scheme which is secure against any computationally unbounded adversary as long as the total amount of information retrieved by the adversary is somewhat less than the length of the shares, and the adversary makes at most k-1 corruption rounds (as described above, where k rounds are necessary for reconstruction). We extend our basic scheme in several ways in order to allow the shares sent by the dealer to be short (the players then blow them up locally) and to handle even stronger adversaries who can learn some of the shares completely. As mentioned, there is an obvious connection between IRSS schemes and the fact that there exist functions with an exponential gap in their communication complexity for k and k-1 rounds. Our scheme implies such a separation which is in several aspects stronger than the previously known ones.
{"title":"Intrusion-Resilient Secret Sharing","authors":"Stefan Dziembowski, Krzysztof Pietrzak","doi":"10.1109/FOCS.2007.63","DOIUrl":"https://doi.org/10.1109/FOCS.2007.63","url":null,"abstract":"We introduce a new primitive called intrusion-resilient secret sharing (IRSS), whose security proof exploits the fact that there exist functions which can be efficiently computed interactively using low communication complexity in k, but not in k-1 rounds. IRSS is a means of sharing a secret message amongst a set of players which comes with a very strong security guarantee. The shares in an IRSS are made artificially large so that it is hard to retrieve them completely, and the reconstruction procedure is interactive requiring the players to exchange k short messages. The adversaries considered can attack the scheme in rounds, where in each round the adversary chooses some player to corrupt and some function, and retrieves the output of that function applied to the share of the corrupted player. This model captures for example computers connected to a network which can occasionally he infected by malicious software like viruses, which can compute any function on the infected machine, but cannot sent out a huge amount of data. Using methods from the bounded-retrieval model, we construct an IRSS scheme which is secure against any computationally unbounded adversary as long as the total amount of information retrieved by the adversary is somewhat less than the length of the shares, and the adversary makes at most k-1 corruption rounds (as described above, where k rounds are necessary for reconstruction). We extend our basic scheme in several ways in order to allow the shares sent by the dealer to be short (the players then blow them up locally) and to handle even stronger adversaries who can learn some of the shares completely. As mentioned, there is an obvious connection between IRSS schemes and the fact that there exist functions with an exponential gap in their communication complexity for k and k-1 rounds. Our scheme implies such a separation which is in several aspects stronger than the previously known ones.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127454331","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}
Nicole Immorlica, Anna R. Karlin, Mohammad Mahdian, Kunal Talwar
We study the power of ascending auctions in a scenario in which a seller is selling a collection of identical items to anonymous unit'demand bidders. We show that even with full knowledge of the set of bidders' private valuations for the items, if the bidders are ex-ante identical, no ascending auction can extract more than a constant. times the revenue of the best fixed-price scheme. This problem is equivalent to the problem of coming up with an optimal strategy for blowing up indistinguishable balloons with known capacities in order to maximize the amount of contained, air. We show that the algorithm which simply inflates all balloons to a fixed volume is close to optimal in this setting.
{"title":"Balloon Popping With Applications to Ascending Auctions","authors":"Nicole Immorlica, Anna R. Karlin, Mohammad Mahdian, Kunal Talwar","doi":"10.1109/FOCS.2007.15","DOIUrl":"https://doi.org/10.1109/FOCS.2007.15","url":null,"abstract":"We study the power of ascending auctions in a scenario in which a seller is selling a collection of identical items to anonymous unit'demand bidders. We show that even with full knowledge of the set of bidders' private valuations for the items, if the bidders are ex-ante identical, no ascending auction can extract more than a constant. times the revenue of the best fixed-price scheme. This problem is equivalent to the problem of coming up with an optimal strategy for blowing up indistinguishable balloons with known capacities in order to maximize the amount of contained, air. We show that the algorithm which simply inflates all balloons to a fixed volume is close to optimal in this setting.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128024046","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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