We show that the combinatorial complexity of the. union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.
{"title":"Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions","authors":"Esther Ezra, M. Sharir","doi":"10.1109/FOCS.2007.9","DOIUrl":"https://doi.org/10.1109/FOCS.2007.9","url":null,"abstract":"We show that the combinatorial complexity of the. union of n \"fat\" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least .some fixed constant) of arbitrary sizes, is O(n2+epsiv),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3 having arbitrary side lengths, is O(n2+epsiv), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have, arbitrary sizes) in R3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.","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":"126742119","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}
Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, J. Harer, Dmitriy Morozov
We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.
{"title":"Inferring Local Homology from Sampled Stratified Spaces","authors":"Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, J. Harer, Dmitriy Morozov","doi":"10.1109/FOCS.2007.33","DOIUrl":"https://doi.org/10.1109/FOCS.2007.33","url":null,"abstract":"We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.","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":"124021720","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}
The complexity of the Black-White Pebbling Game has remained open for 30 years. It was devised to capture the power of non-deterministic space bounded computation. Since then it has been applied to problems in diverse areas of computer science including VLSI design and more recently propositional proof complexity. In this paper we show that the Black-While Pebbling Game is PSPACE-complete. We then use similar ideas in a more complicated reduction to prove the PSPACE-completeness of Resolution space. The reduction also yields a surprising exponential time/space speedup for Resolution in which an increase of 3 units of space results in an exponential decrease in proof-size.
{"title":"Exponential Time/Space Speedups for Resolution and the PSPACE-completeness of Black-White Pebbling","authors":"Philipp Hertel, T. Pitassi","doi":"10.1109/FOCS.2007.22","DOIUrl":"https://doi.org/10.1109/FOCS.2007.22","url":null,"abstract":"The complexity of the Black-White Pebbling Game has remained open for 30 years. It was devised to capture the power of non-deterministic space bounded computation. Since then it has been applied to problems in diverse areas of computer science including VLSI design and more recently propositional proof complexity. In this paper we show that the Black-While Pebbling Game is PSPACE-complete. We then use similar ideas in a more complicated reduction to prove the PSPACE-completeness of Resolution space. The reduction also yields a surprising exponential time/space speedup for Resolution in which an increase of 3 units of space results in an exponential decrease in proof-size.","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":"121272491","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}
Nishanth Chandran, Vipul Goyal, R. Ostrovsky, A. Sahai
In STOC'05, Aim, Hopper and Longford introduced the notion of covert computation. A covert computation protocol is one in which parties am run a protocol without knowing if other parties ore also participating in the protocol or not. At the end of the protocol, if all parties participated in the protocol and if the function output is favorable to all parties, then the output is revealed. Ahn et al. constructed a protocol for covert two-partv computation in the random oracle model In this paper, we offer a construction for covert multiparty computation. Our construction is in the standard model and does not require random oracles. In order to achieve this goal, we introduce a number of new techniques. Central to our work is the development of "zero-knowledge proofs to garbled circuits," which we believe could be of independent interest. Along the way, we also develop a definition of covert computation as per the Ideal/Real model simulation paradigm.
{"title":"Covert Multi-Party Computation","authors":"Nishanth Chandran, Vipul Goyal, R. Ostrovsky, A. Sahai","doi":"10.1109/FOCS.2007.21","DOIUrl":"https://doi.org/10.1109/FOCS.2007.21","url":null,"abstract":"In STOC'05, Aim, Hopper and Longford introduced the notion of covert computation. A covert computation protocol is one in which parties am run a protocol without knowing if other parties ore also participating in the protocol or not. At the end of the protocol, if all parties participated in the protocol and if the function output is favorable to all parties, then the output is revealed. Ahn et al. constructed a protocol for covert two-partv computation in the random oracle model In this paper, we offer a construction for covert multiparty computation. Our construction is in the standard model and does not require random oracles. In order to achieve this goal, we introduce a number of new techniques. Central to our work is the development of \"zero-knowledge proofs to garbled circuits,\" which we believe could be of independent interest. Along the way, we also develop a definition of covert computation as per the Ideal/Real model simulation paradigm.","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":"129009425","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 develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MODm o ANDisin log n cannot compute the MODq function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.
{"title":"Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits","authors":"A. Chattopadhyay","doi":"10.1109/FOCS.2007.30","DOIUrl":"https://doi.org/10.1109/FOCS.2007.30","url":null,"abstract":"We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MODm o ANDisin log n cannot compute the MODq function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.","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":"124381000","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}
The zero testing and sign determination problems of real algebraic numbers of high extension degree are important in computational complexity and numerical analysis. In this paper we concentrate an sparse cyclotomic integers. Given an integer n and a sparse polynomial f(x) = Ckxe(k) + ck-1xe(k-1) + ... + c1xe(1)over Z, we present a deterministic polynomial time algorithm to decide whether f(wn) is zero or not, where f(wn) denotes the n-th primitive root of unity e2piradic(-1/n). All previously known algorithms are either randomized, or do not run in polynomial time. As a side result, we prove that if n is free of prime factors less than k + 1, there exist k field automorphisms sigma1, sigma2, ... , sigmak in the Galois group Gal (Q(wn)/Q) such that for any nonzero integers c1, c2 ... , ck and for any integers 0 les e1 < e2 < ... < ek < n, there exists i so that |sigmai(ckwnek + ck-1wne(k-1) + ... + c1wne(1)) | ges 1/2(k(2)logn+klogk).
{"title":"Derandomization of Sparse Cyclotomic Integer Zero Testing","authors":"Qi Cheng","doi":"10.1109/FOCS.2007.23","DOIUrl":"https://doi.org/10.1109/FOCS.2007.23","url":null,"abstract":"The zero testing and sign determination problems of real algebraic numbers of high extension degree are important in computational complexity and numerical analysis. In this paper we concentrate an sparse cyclotomic integers. Given an integer n and a sparse polynomial f(x) = C<sub>k</sub>x<sup>e(k)</sup> + c<sub>k-1</sub>x<sup>e(k-1)</sup> + ... + c<sub>1</sub>x<sup>e(1)</sup>over Z, we present a deterministic polynomial time algorithm to decide whether f(w<sub>n</sub>) is zero or not, where f(w<sub>n</sub>) denotes the n-th primitive root of unity e<sup>2piradic(-1/n)</sup>. All previously known algorithms are either randomized, or do not run in polynomial time. As a side result, we prove that if n is free of prime factors less than k + 1, there exist k field automorphisms sigma<sub>1</sub>, sigma<sub>2</sub>, ... , sigma<sub>k</sub> in the Galois group Gal (Q(w<sub>n</sub>)/Q) such that for any nonzero integers c<sub>1</sub>, c<sub>2</sub> ... , c<sub>k</sub> and for any integers 0 les e<sub>1</sub> < e<sub>2</sub> < ... < e<sub>k</sub> < n, there exists i so that |sigma<sub>i</sub>(c<sub>k</sub>w<sub>n</sub> <sup>ek</sup> + c<sub>k-1</sub>w<sub>n</sub> <sup>e(k-1)</sup> + ... + c<sub>1</sub>w<sub>n</sub> <sup>e(1)</sup>) | ges 1/2<sup>(k(2)log</sup> <sup>n+klogk)</sup>.","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":"123275880","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}
Suppose that every k points in a metric space X are D-distortion embeddable into lscr 1. We give upper and lower bounds on the distortion required to embed the entire space X into lscr 1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into lscr 1 with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.
{"title":"Local Global Tradeoffs in Metric Embeddings","authors":"M. Charikar, K. Makarychev, Yury Makarychev","doi":"10.1137/070712080","DOIUrl":"https://doi.org/10.1137/070712080","url":null,"abstract":"Suppose that every k points in a metric space X are D-distortion embeddable into lscr 1. We give upper and lower bounds on the distortion required to embed the entire space X into lscr 1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into lscr 1 with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.","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":"133919720","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 paper we present a new approximation algorithm for the Max Acyclic Subgraph problem. Given an instance where the maximum acyclic subgraph contains 1/2 + delta fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Omega(delta/ log n) fraction of all edges.
{"title":"On the Advantage over Random for Maximum Acyclic Subgraph","authors":"M. Charikar, K. Makarychev, Yury Makarychev","doi":"10.1109/FOCS.2007.47","DOIUrl":"https://doi.org/10.1109/FOCS.2007.47","url":null,"abstract":"In this paper we present a new approximation algorithm for the Max Acyclic Subgraph problem. Given an instance where the maximum acyclic subgraph contains 1/2 + delta fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Omega(delta/ log n) fraction of all edges.","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":"129434647","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 polynomial reconstruction problem, for low-degree multivariate polynomials over F[2]. In this problem, we are given a set of points x epsi {0, 1}n and target values f(x) epsi {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 - epsiv fraction of the points. Our goal is to find agree d polynomial that has good-agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 - 2-d + delta fraction of the points for any epsiv, delta > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, wherejis the algorithm is allowed to find a polynomial of degree d. Previously the only known, hardness of approximation (or even NP-completeness) was for the case when d = I, which follows from a celebrated result of Has tad. In the setting of computational learning, our result shows the hardness of (non-proper) agnostic learning of parities, where the learner is allowed, a low-degree polynomial over F[2] as a hypothesis. This is the first non-proper hardness result for this central problem in computational learning. Our results extend-to multivariate polynomial reconstruction over any finite field.
{"title":"Hardness of Reconstructing Multivariate Polynomials over Finite Fields","authors":"Parikshit Gopalan, Subhash Khot, Rishi Saket","doi":"10.1137/070705258","DOIUrl":"https://doi.org/10.1137/070705258","url":null,"abstract":"We study the polynomial reconstruction problem, for low-degree multivariate polynomials over F[2]. In this problem, we are given a set of points x epsi {0, 1}n and target values f(x) epsi {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 - epsiv fraction of the points. Our goal is to find agree d polynomial that has good-agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 - 2-d + delta fraction of the points for any epsiv, delta > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, wherejis the algorithm is allowed to find a polynomial of degree d. Previously the only known, hardness of approximation (or even NP-completeness) was for the case when d = I, which follows from a celebrated result of Has tad. In the setting of computational learning, our result shows the hardness of (non-proper) agnostic learning of parities, where the learner is allowed, a low-degree polynomial over F[2] as a hypothesis. This is the first non-proper hardness result for this central problem in computational learning. Our results extend-to multivariate polynomial reconstruction over any finite field.","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":"128699180","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 show that any partition-problem of hypergraphs has an O(n) time approximate partitioning algorithm and an efficient property tester. This extends the results of Goldreich, Goldwasser and Ron who obtained similar algorithms for the special case of graph partition problems in their seminal paper (1998). The partitioning algorithm is used to obtain the following results: ldr We derive a surprisingly simple O(n) time algorithmic version of Szemeredi's regularity lemma. Unlike all the previous approaches for this problem which only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists; ldr For any r ges 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n2r-1) time (deterministic) algorithms. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/isin) queries and constant running time.
{"title":"Approximate Hypergraph Partitioning and Applications","authors":"E. Fischer, A. Matsliah, A. Shapira","doi":"10.1109/FOCS.2007.11","DOIUrl":"https://doi.org/10.1109/FOCS.2007.11","url":null,"abstract":"We show that any partition-problem of hypergraphs has an O(n) time approximate partitioning algorithm and an efficient property tester. This extends the results of Goldreich, Goldwasser and Ron who obtained similar algorithms for the special case of graph partition problems in their seminal paper (1998). The partitioning algorithm is used to obtain the following results: ldr We derive a surprisingly simple O(n) time algorithmic version of Szemeredi's regularity lemma. Unlike all the previous approaches for this problem which only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists; ldr For any r ges 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n2r-1) time (deterministic) algorithms. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/isin) queries and constant running time.","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":"128009872","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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