Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1209.0263

A strong direct product theorem in terms of the smooth rectangle bound

Rahul Jain, Penghui Yao

Published Sep 3, 2012• 19 citations
arXivHTMLPDF

Abstract

A strong direct product theorem states that, in order to solve k instances of a problem, if we provide less than k times the resource required to compute one instance, then the probability of overall success is exponentially small in k. In this paper, we consider the model of two-way public-coin communication complexity and show a strong direct product theorem for all relations in terms of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower bound method in this model. Our result therefore uniformly implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound, for example Inner Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result also implies near optimal direct product results for several important functions and relations used to show exponential separations between classical and quantum communication complexity, for which near optimal lower bounds are provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008] and Klartag and Regev [2011]. In fact we are not aware of any relation for which it is known that the smooth rectangle bound does not provide an optimal lower bound. This lower bound subsumes many of the other lower bound methods, for example the rectangle bound (a.k.a the corruption bound), the smooth discrepancy bound (a.k.a the γ_2 bound) which in turn subsumes the discrepancy bound, the subdistribution bound and the conditional min-entropy bound. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [2012], in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012].

DOI
10.48550/arxiv.1209.0263

Related papers

New strong direct product results in communication complexity.

Rahul Jain

We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and e > 0 be a constant. Let R1,pube(f) represent the communication complexity of f, with worst-case error e in this model. We show that if for computing fk (k independent copies of f) in this model, o(k c R1, pub1/3(f)) communication is used, then the success is exponentially small in k. We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest.Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.

Electronic colloquium on computational complexity•Jan 1, 2011•16 citations

A Direct Product Theorem for the Two-Party Bounded-Round Public-Coin Communication Complexity

Rahul Jain, Attila Pereszlényi, Penghui Yao

A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if we provide resource which is less than k times the resource required for computing one instance of the problem with constant success probability, then the probability of correctly computing all the k instances together, is exponentially small in k. In this paper, we consider the model of two-party bounded-round public-coin randomized communication complexity. We show a direct product theorem for the communication complexity of any relation in this model. In particular, our result implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations. As an immediate application of our result, we get a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. Our result generalizes the result of Jain [2011] which can be regarded as the special case when t=1. Our result can be considered as an important progress towards settling the strong direct product conjecture for the two-party public-coin communication complexity, a major open question in this area. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in Jain~\cite{Jain:2011}. %, where a strong direct product theorem for the %two-party one-way public-coin communication complexity of all %relations is shown (that is the special case of our result when $t=1$). One key tool used in our work and also in Jain~\cite{Jain:2011} is a message compression technique due to Braver man and Rao~\cite{Braverman2011}, who used it to show a {\em direct sum} theorem in the same model of communication complexity as considered by us. Another important tool that we use is a correlated sampling protocol, which for example, has been used in Holenstein~\cite{Holenstein2007} for proving a parallel repetition theorem for two-prover games.

Annual Symposium on Foundations of Computer Science•Oct 1, 2012•31 citations

New Strong Direct Product Results in Communication Complexity

Rahul Jain

We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and ϵ > 0 be a constant. Let R 1,pub ϵ ( f ) represent the communication complexity of f , with worst-case error ϵ in this model. We show that if for computing f k ( k independent copies of f ) in this model, o ( k ċ R 1, pub 1/3 ( f )) communication is used, then the success is exponentially small in k . We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest. Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω( n ) for the set-disjointness problem on n -bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.

Journal of the ACM•Jun 30, 2015•22 citations

Direct product theorems for classical communication complexity via subdistribution bounds

Rahul Jain, Hartmut Klauck, Ashwin Nayak

A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound , which is a relaxation of the well-studied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the one-way version of this bound tightly captures the one-way public-coin randomized communication complexity of any relation, and the two-way version bounds the two-way public-coin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one- and two-way protocols, and the weak direct product property under arbitrary distributions for two-way protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The simplicity and broad applicability of our technique is perhaps an indication of its potential to solve yet more challenging questions regarding the direct product problem.

ACM Symposium on Theory of Computing (STOC)•May 17, 2008•33 citations

A strong direct product theorem for two-way public coin communication complexity

Rahul Jain

We show a direct product result for two-way public coin communication complexity of all relations in terms of a new complexity measure that we define. Our new measure is a generalization to non-product distributions of the two-way product subdistribution bound of [J, Klauck and Nayak 08], thereby our result implying their direct product result in terms of the two-way product subdistribution bound. We show that our new complexity measure gives tight lower bound for the set-disjointness problem, as a result we reproduce strong direct product result for this problem, which was previously shown by [Klauck 00].

arXiv:1010.0846•Oct 5, 2010•4 citations

A direct product theorem for bounded-round public-coin randomized communication complexity

Rahul Jain, Attila Pereszlényi, Penghui Yao

In this paper, we show a direct product theorm in the model of two-party bounded-round public-coin randomized communication complexity. For a relation f subset of X times Y times Z (X,Y,Z are finite sets), let R^{(t), pub}_e (f) denote the two-party t-message public-coin communication complexity of f with worst case error e. We show that for any relation f and positive integer k: R^{(t), pub}_{1 - 2^{-Omega(k/t^2)}}(f^k) = Omega(k/t (R^{(t), pub}_{1/3}(f) - O(t^2))) . In particular, it implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations f. Our result for example implies a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in [Jain 2011], where a strong direct product theorem for the two-party one-way public-coin communication complexity of all relations is shown (that is the special case of our result when t=1). One key tool used in our work and also in [Jain 2011] is a message compression technique due to [Braverman and Rao 2011], who used it to show a direct sum theorem for the two-party bounded-round public-coin randomized communication complexity of all relations. Another important tool that we use is a correlated sampling protocol, which for example, has been used in [Holenstein 2007] for proving a parallel repetition theorem for two-prover games.

arXiv:1201.1666•Jan 8, 2012•11 citations
A strong direct product theorem states that, in order to solve k instances of a problem, if we provide less than k times the resource required to compute one instance, then the probability of overall success is exponentially small in k. In this paper, we consider the model of two-way public-coin communication complexity and show a strong direct product theorem for all relations in terms of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower bound method in this model. Our result therefore uniformly implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound, for example Inner Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result also implies near optimal direct product results for several important functions and relations used to show exponential separations between classical and quantum communication complexity, for which near optimal lower bounds are provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008] and Klartag and Regev [2011]. In fact we are not aware of any relation for which it is known that the smooth rectangle bound does not provide an optimal lower bound. This lower bound subsumes many of the other lower bound methods, for example the rectangle bound (a.k.a the corruption bound), the smooth discrepancy bound (a.k.a the γ_2 bound) which in turn subsumes the discrepancy bound, the subdistribution bound and the conditional min-entropy bound. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [2012], in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012].
We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and e > 0 be a constant. Let R1,pube(f) represent the communication complexity of f, with worst-case error e in this model. We show that if for computing fk (k independent copies of f) in this model, o(k c R1, pub1/3(f)) communication is used, then the success is exponentially small in k. We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest.Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω(n) for the set-disjointness problem on n-bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.
A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if we provide resource which is less than k times the resource required for computing one instance of the problem with constant success probability, then the probability of correctly computing all the k instances together, is exponentially small in k. In this paper, we consider the model of two-party bounded-round public-coin randomized communication complexity. We show a direct product theorem for the communication complexity of any relation in this model. In particular, our result implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations. As an immediate application of our result, we get a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. Our result generalizes the result of Jain [2011] which can be regarded as the special case when t=1. Our result can be considered as an important progress towards settling the strong direct product conjecture for the two-party public-coin communication complexity, a major open question in this area. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in Jain~\cite{Jain:2011}. %, where a strong direct product theorem for the %two-party one-way public-coin communication complexity of all %relations is shown (that is the special case of our result when t=1t=1t=1). One key tool used in our work and also in Jain~\cite{Jain:2011} is a message compression technique due to Braver man and Rao~\cite{Braverman2011}, who used it to show a {\em direct sum} theorem in the same model of communication complexity as considered by us. Another important tool that we use is a correlated sampling protocol, which for example, has been used in Holenstein~\cite{Holenstein2007} for proving a parallel repetition theorem for two-prover games.
We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and ϵ > 0 be a constant. Let R 1,pub ϵ ( f ) represent the communication complexity of f , with worst-case error ϵ in this model. We show that if for computing f k ( k independent copies of f ) in this model, o ( k ċ R 1, pub 1/3 ( f )) communication is used, then the success is exponentially small in k . We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest. Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω( n ) for the set-disjointness problem on n -bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.
A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound , which is a relaxation of the well-studied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the one-way version of this bound tightly captures the one-way public-coin randomized communication complexity of any relation, and the two-way version bounds the two-way public-coin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one- and two-way protocols, and the weak direct product property under arbitrary distributions for two-way protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The simplicity and broad applicability of our technique is perhaps an indication of its potential to solve yet more challenging questions regarding the direct product problem.
We show a direct product result for two-way public coin communication complexity of all relations in terms of a new complexity measure that we define. Our new measure is a generalization to non-product distributions of the two-way product subdistribution bound of [J, Klauck and Nayak 08], thereby our result implying their direct product result in terms of the two-way product subdistribution bound. We show that our new complexity measure gives tight lower bound for the set-disjointness problem, as a result we reproduce strong direct product result for this problem, which was previously shown by [Klauck 00].
In this paper, we show a direct product theorm in the model of two-party bounded-round public-coin randomized communication complexity. For a relation f subset of X times Y times Z (X,Y,Z are finite sets), let R^{(t), pub}_e (f) denote the two-party t-message public-coin communication complexity of f with worst case error e. We show that for any relation f and positive integer k: R^{(t), pub}_{1 - 2^{-Omega(k/t^2)}}(f^k) = Omega(k/t (R^{(t), pub}_{1/3}(f) - O(t^2))) . In particular, it implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations f. Our result for example implies a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in [Jain 2011], where a strong direct product theorem for the two-party one-way public-coin communication complexity of all relations is shown (that is the special case of our result when t=1). One key tool used in our work and also in [Jain 2011] is a message compression technique due to [Braverman and Rao 2011], who used it to show a direct sum theorem for the two-party bounded-round public-coin randomized communication complexity of all relations. Another important tool that we use is a correlated sampling protocol, which for example, has been used in [Holenstein 2007] for proving a parallel repetition theorem for two-prover games.