2010 IEEE 25th Annual Conference on Computational Complexity最新文献

{"title":"Relationless Completeness and Separations","authors":"P. Hrubes, A. Wigderson, A. Yehudayoff","doi":"10.1109/CCC.2010.34","DOIUrl":"https://doi.org/10.1109/CCC.2010.34","url":null,"abstract":"This paper extends Valiant’s work on VP and VNP to the settings in which variables are not multiplicatively commutative and/or associative. Our main result is a theory of completeness for these algebraic worlds. We define analogs of Valiant’s classes VP and VNP, as well as of the polynomials permanent and determinant, in these worlds. We then prove that even in a completely relationless world which assumes no commutativity nor associativity, permanent remains VNP-complete, and determinant can polynomially simulate any arithmetic formula, just as in the standard commutative, associative world of Valiant. In the absence of associativity, the completeness proof gives rise to the following combinatorial problem: what is the smallest binary tree which contains as minors all binary trees with n leaves. We give an explicit construction of such a universal tree of polynomial size, a result of possibly independent interest. Given that such non-trivial reductions are possible even without commutativity and associativity, we turn to lower bounds. In the non-associative, commutative world we prove exponential circuit lower bounds on explicit polynomials, separating the non-associative commutative analogs of VP and VNP. Obtaining such lower bounds and a separation in the complementary associative, non-commutative world has been open for about 30 years.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126576536","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}

{"title":"On the Power of Randomized Reductions and the Checkability of SAT","authors":"Mohammad Mahmoody, David Xiao","doi":"10.1109/CCC.2010.16","DOIUrl":"https://doi.org/10.1109/CCC.2010.16","url":null,"abstract":"We prove new results regarding the complexity of various complexity classes under randomized oracle reductions. We first prove that BPP^prSZK subseteq AM cap coAM, where prSZK is the class of promise problems having statistical zero knowledge proofs. This strengthens the previously known facts that prSZK is closed under NC^1 truth-table reductions (Sahai and Vadhan, J. ACM '03) and that P^prSZK subseteq AM cap coAM (Vadhan, personal communication). Our proof relies on showing that a certain class of real-valued functions that we call RTFAM can be approximated using an AM protocol. Then we investigate the power of randomized oracle reductions with relation to the notion of instance checking (Blum and Kannan, J. ACM '95). We observe that a theorem of Beigel implies that if any problem in TFNP such as Nash equilibrium is NP-hard under randomized oracle reductions, then SAT is checkable. We also observe that Beigel's theorem can be extended to an average-case setting by relating checking to the notion of program testing (Blum et al., JCSS '93). From this, we derive that if one-way functions can be based on NP-hardness via a randomized oracle reduction, then SAT is checkable. By showing that NP has a non-uniform tester, we also show that worst-case to average-case randomized oracle reduction for any relation (or language) R in NP implies that R has a non-uniform instance checker. These results hold even for adaptive randomized oracle reductions.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124958628","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}

{"title":"Parallel Repetition of Two Prover Games (Invited Survey)","authors":"R. Raz","doi":"10.1109/CCC.2010.9","DOIUrl":"https://doi.org/10.1109/CCC.2010.9","url":null,"abstract":"The parallel repetition theorem states that for any two-prover game with value smaller than 1, parallel repetition reduces the value of the game in an exponential rate. We give a short introduction to the problem of parallel repetition of two-prover games and some of its applications in theoretical computer science, mathematics and physics. We will concentrate mainly on recent results.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125715448","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}

{"title":"A New Sampling Protocol and Applications to Basing Cryptographic Primitives on the Hardness of NP","authors":"Iftach Haitner, Mohammad Mahmoody, David Xiao","doi":"10.1109/CCC.2010.17","DOIUrl":"https://doi.org/10.1109/CCC.2010.17","url":null,"abstract":"We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, $Sam_d$ where $d$ is the recursion depth, is based on the identically-named oracle defined in the work of Haitner et al. (FOCS '07). Our main result is a constant-round public-coin protocol ``$AMSam$'' that allows an efficient verifier to emulate a $Sam_d$ oracle for any constant depth $d = O(1)$ with the help of a $BPP^NP$ prover. $AMSam$ allows us to conclude that if $L$ is decidable by a $k$-adaptive randomized oracle algorithm with access to a $Sam_{O(1)}$ oracle, then $L in AM[k] cap coAM[k]$. The above yields the following corollary: assume there exists an $O(1)$-adaptive reduction that bases constant-round statistically hiding commitment on $NP$-hardness, then $NP subseteq coAM$ and the polynomial hierarchy collapses. The same result holds for any primitive that can be broken by $Sam_{O(1)}$ including collision-resistant hash functions and $O(1)$-round oblivious transfer where security holds statistically for one of the parties. We also obtain non-trivial (though weaker) consequences for $k$-adaptive reductions for any $k = poly(n)$. Prior to our work, most results in this research direction either applied only to non-adaptive reductions (citeauthor{BogdanovT06}, SIAM J. of Comp. '06 and citeauthor{AkaviaGGM06}, FOCS '06) or to one-way permutations (citeauthor{Brassard79} FOCS '79). The main technical tool we use to prove the above is a new constant-round public-coin protocol ($SWS$), which we believe to be of interest in its own right, that guarantees the following: given an efficient function $f$ on $n$ bits, let $D$ be the output distribution $D = f(U_n)$, then $SWS$ allows an efficient verifier Arthur to use an all-powerful prover Merlin's help to sample a random $y getsr D$ along with a good multiplicative approximation of the probability $p_y = Pr_{y' getsr D}[y' = y]$. The crucial feature of $SWS$ is that it extends even to distributions of the form $D = f(U_cs)$, where $U_cs$ is the uniform distribution on an efficiently decidable subset $cs subseteq zo^n$ (such $D$ are called efficiently samplable with emph{post-selection}), as long as the verifier is also given a good approximation of the value $|cs|$.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122141541","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}

{"title":"On the Relative Strength of Pebbling and Resolution","authors":"Jakob Nordström","doi":"10.1145/2159531.2159538","DOIUrl":"https://doi.org/10.1145/2159531.2159538","url":null,"abstract":"The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic black-white pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This paper contributes to both these approaches. First, we design a restricted form of black-white pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat black-only pebbling, and in particular that the space lower bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in [Ben-Sasson and Nordstrom 2009].","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115140609","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}

{"title":"Derandomizing from Random Strings","authors":"H. Buhrman, L. Fortnow, M. Koucký, B. Loff","doi":"10.1109/CCC.2010.15","DOIUrl":"https://doi.org/10.1109/CCC.2010.15","url":null,"abstract":"In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K has high Kolmogorov complexity. As a partial converse to our claim we show that strings of very high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130944197","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}

{"title":"No Strong Parallel Repetition with Entangled and Non-signaling Provers","authors":"J. Kempe, O. Regev","doi":"10.1109/CCC.2010.10","DOIUrl":"https://doi.org/10.1109/CCC.2010.10","url":null,"abstract":"We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the emph{parallel repetition question}: If the game is played $ell$ times in parallel, does the maximum winning probability decay exponentially in $ell$? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form $(1-Theta(eps))^ell$ where $1-eps$ is the value of the game and $ell$ is the number of repetitions. This question is known as the emph{strong parallel repetition question} and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does emph{not} hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact emph{perfect}) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form $(1-Theta(eps^2))^ell$. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123594465","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}

{"title":"The Partition Bound for Classical Communication Complexity and Query Complexity","authors":"Rahul Jain, H. Klauck","doi":"10.1109/CCC.2010.31","DOIUrl":"https://doi.org/10.1109/CCC.2010.31","url":null,"abstract":"We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the γ2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than the approximate polynomial degree and adversary bounds.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132473897","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}

{"title":"A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions","authors":"Ilias Diakonikolas, R. Servedio, Li-Yang Tan, Andrew Wan","doi":"10.4086/toc.2014.v010a002","DOIUrl":"https://doi.org/10.4086/toc.2014.v010a002","url":null,"abstract":"We give a \"regularity lemma\" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {−1,1}^n. Roughly speaking, this result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a \"regular\" PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d >= 1, eps > 0, every degree-d PTF over n variables can be approximated to accuracy eps by a constant degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to logarithmic factors.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129754859","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}

{"title":"On the Unique Games Conjecture (Invited Survey)","authors":"Subhash Khot","doi":"10.1109/SFCS.2005.61","DOIUrl":"https://doi.org/10.1109/SFCS.2005.61","url":null,"abstract":"This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry.","PeriodicalId":328781,"journal":{"name":"2010 IEEE 25th Annual Conference on Computational Complexity","volume":"268 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122437397","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}