5 research outputs found
Streaming Verification of Graph Properties
Streaming interactive proofs (SIPs) are a framework for outsourced
computation. A computationally limited streaming client (the verifier) hands
over a large data set to an untrusted server (the prover) in the cloud and the
two parties run a protocol to confirm the correctness of result with high
probability. SIPs are particularly interesting for problems that are hard to
solve (or even approximate) well in a streaming setting. The most notable of
these problems is finding maximum matchings, which has received intense
interest in recent years but has strong lower bounds even for constant factor
approximations.
In this paper, we present efficient streaming interactive proofs that can
verify maximum matchings exactly. Our results cover all flavors of matchings
(bipartite/non-bipartite and weighted). In addition, we also present streaming
verifiers for approximate metric TSP. In particular, these are the first
efficient results for weighted matchings and for metric TSP in any streaming
verification model.Comment: 26 pages, 2 figure, 1 tabl
Rational Proofs with Multiple Provers
Interactive proofs (IP) model a world where a verifier delegates computation
to an untrustworthy prover, verifying the prover's claims before accepting
them. IP protocols have applications in areas such as verifiable computation
outsourcing, computation delegation, cloud computing. In these applications,
the verifier may pay the prover based on the quality of his work. Rational
interactive proofs (RIP), introduced by Azar and Micali (2012), are an
interactive-proof system with payments, in which the prover is rational rather
than untrustworthy---he may lie, but only to increase his payment. Rational
proofs leverage the provers' rationality to obtain simple and efficient
protocols. Azar and Micali show that RIP=IP(=PSAPCE). They leave the question
of whether multiple provers are more powerful than a single prover for rational
and classical proofs as an open problem.
In this paper, we introduce multi-prover rational interactive proofs (MRIP).
Here, a verifier cross-checks the provers' answers with each other and pays
them according to the messages exchanged. The provers are cooperative and
maximize their total expected payment if and only if the verifier learns the
correct answer to the problem. We further refine the model of MRIP to
incorporate utility gap, which is the loss in payment suffered by provers who
mislead the verifier to the wrong answer.
We define the class of MRIP protocols with constant, noticeable and
negligible utility gaps. We give tight characterization for all three MRIP
classes. We show that under standard complexity-theoretic assumptions, MRIP is
more powerful than both RIP and MIP ; and this is true even the utility gap is
required to be constant. Furthermore the full power of each MRIP class can be
achieved using only two provers and three rounds. (A preliminary version of
this paper appeared at ITCS 2016. This is the full version that contains new
results.)Comment: Proceedings of the 2016 ACM Conference on Innovations in Theoretical
Computer Science. ACM, 201
Doctor of Philosophy
dissertationThe contributions of this dissertation are centered around designing new algorithms in the general area of sublinear algorithms such as streaming, core sets and sublinear verification, with a special interest in problems arising from data analysis including data summarization, clustering, matrix problems and massive graphs. In the first part, we focus on summaries and coresets, which are among the main techniques for designing sublinear algorithms for massive data sets. We initiate the study of coresets for uncertain data and study coresets for various types of range counting queries on uncertain data. We focus mainly on the indecisive model of locational uncertainty since it comes up frequently in real-world applications when multiple readings of the same object are made. In this model, each uncertain point has a probability density describing its location, defined as distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by examining only this subset. For each type of query we provide coreset constructions with approximation-size trade-offs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancy-based techniques on axis-aligned range queries. In the second part, we focus on designing sublinear-space algorithms for approximate computations on massive graphs. In particular, we consider graph MAXCUT and correlation clustering problems and develop sampling based approaches to construct truly sublinear () sized coresets for graphs that have polynomial (i.e., for any ) average degree. Our technique is based on analyzing properties of random induced subprograms of the linear program formulations of the problems. We demonstrate this technique with two examples. Firstly, we present a sublinear sized core set to approximate the value of the MAX CUT in a graph to a factor. To the best of our knowledge, all the known methods in this regime rely crucially on near-regularity assumptions. Secondly, we apply the same framework to construct a sublinear-sized coreset for correlation clustering. Our coreset construction also suggests 2-pass streaming algorithms for computing the MAX CUT and correlation clustering objective values which are left as future work at the time of writing this dissertation. Finally, we focus on streaming verification algorithms as another model for designing sublinear algorithms. We give the first polylog space and sublinear (in number of edges) communication protocols for any streaming verification problems in graphs. We present efficient streaming interactive proofs that can verify maximum matching exactly. Our results cover all flavors of matchings (bipartite/ nonbipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP and exact triangle counting, as well as for graph primitives such as the number of connected components, bipartiteness, minimum spanning tree and connectivity. In particular, these are the first results for weighted matchings and for metric TSP in any streaming verification model. Our streaming verifiers use only polylogarithmic space while exchanging only polylogarithmic communication with the prover in addition to the output size of the relevant solution. We also initiate a study of streaming interactive proofs (SIPs) for problems in data analysis and present efficient SIPs for some fundamental problems. We present protocols for clustering and shape fitting including minimum enclosing ball (MEB), width of a point set, -centers and -slab problem. We also present protocols for fundamental matrix analysis problems: We provide an improved protocol for rectangular matrix problems, which in turn can be used to verify (approximate) eigenvectors of an integer matrix . In general our solutions use polylogarithmic rounds of communication and polylogarithmic total communication and verifier space