OpenRaiser/Intern-Atlas huggingface dataset:
Approximating the volume of unions and intersections of high-dimensional geometric objects
Venue: arXiv cs.CG 2010
Year: 2010

We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time $2^{d^{1-ε}}$-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time $ε$-approximation.
A computer verified, monadic, functional implementation of the integral
Venue: arXiv cs.LO 2010
Year: 2010

We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O'Connor, this may be seen as the beginning of the realization of Bishop's vision to use constructive mathematics as a programming language for exact analysis.
Bi-directional half-duplex protocols with multiple relays
Venue: arXiv cs.IT 2010
Year: 2010

In a bi-directional relay channel, two nodes wish to exchange independent messages over a shared wireless half-duplex channel with the help of relays. Recent work has considered information theoretic limits of the bi-directional relay channel with a single relay. In this work we consider bi-directional relaying with multiple relays. We derive achievable rate regions and outer bounds for half-duplex protocols with multiple decode and forward relays and compare these to the same protocols with amplify and forward relays in an additive white Gaussian noise channel. We consider three novel classes of half-duplex protocols: the (m,2) 2 phase protocol with m relays, the (m,3) 3 phase protocol with m relays, and general (m, t) Multiple Hops and Multiple Relays (MHMR) protocols, where m is the total number of relays and 3<t< m+3 is the number of temporal phases in the protocol. The (m,2) and (m,3) protocols extend previous bi-directional relaying protocols for a single m=1 relay, while the new (m,t) protocol efficiently combines multi-hop routing with message-level network coding. Finally, we provide a comprehensive treatment of the MHMR protocols with decode and forward relaying and amplify and forward relaying in the Gaussian noise, obtaining their respective achievable rate regions, outer bounds and relative performance under different SNRs and relay geometries, including an analytical comparison on the protocols at low and high SNR.
Relating Web pages to enable information-gathering tasks
Venue: arXiv cs.IR 2010
Year: 2010

We argue that relationships between Web pages are functions of the user's intent. We identify a class of Web tasks - information-gathering - that can be facilitated by a search engine that provides links to pages which are related to the page the user is currently viewing. We define three kinds of intentional relationships that correspond to whether the user is a) seeking sources of information, b) reading pages which provide information, or c) surfing through pages as part of an extended information-gathering process. We show that these three relationships can be productively mined using a combination of textual and link information and provide three scoring mechanisms that correspond to them: {\em SeekRel}, {\em FactRel} and {\em SurfRel}. These scoring mechanisms incorporate both textual and link information. We build a set of capacitated subnetworks - each corresponding to a particular keyword - that mirror the interconnection structure of the World Wide Web. The scores are computed by computing flows on these subnetworks. The capacities of the links are derived from the {\em hub} and {\em authority} values of the nodes they connect, following the work of Kleinberg (1998) on assigning authority to pages in hyperlinked environments. We evaluated our scoring mechanism by running experiments on four data sets taken from the Web. We present user evaluations of the relevance of the top results returned by our scoring mechanisms and compare those to the top results returned by Google's Similar Pages feature, and the {\em Companion} algorithm proposed by Dean and Henzinger (1999).
Model checking memoryful linear-time logics over one-counter automata
Venue: arXiv cs.LO 2010
Year: 2010

We study complexity of the model-checking problems for LTL with registers (also known as freeze LTL) and for first-order logic with data equality tests over one-counter automata. We consider several classes of one-counter automata (mainly deterministic vs. nondeterministic) and several logical fragments (restriction on the number of registers or variables and on the use of propositional variables for control locations). The logics have the ability to store a counter value and to test it later against the current counter value. We show that model checking over deterministic one-counter automata is PSPACE-complete with infinite and finite accepting runs. By constrast, we prove that model checking freeze LTL in which the until operator is restricted to the eventually operator over nondeterministic one-counter automata is undecidable even if only one register is used and with no propositional variable. As a corollary of our proof, this also holds for first-order logic with data equality tests restricted to two variables. This makes a difference with the facts that several verification problems for one-counter automata are known to be decidable with relatively low complexity, and that finitary satisfiability for the two logics are decidable. Our results pave the way for model-checking memoryful (linear-time) logics over other classes of operational models, such as reversal-bounded counter machines.
On the Decoder Error Probability of Rank Metric Codes and Constant-Dimension Codes
Venue: arXiv cs.IT 2010
Year: 2010

Rank metric codes and constant-dimension codes (CDCs) have been considered for error control in random network coding. Since decoder errors are more detrimental to system performance than decoder failures, in this paper we investigate the decoder error probability (DEP) of bounded distance decoders (BDDs) for rank metric codes and CDCs. For rank metric codes, we consider a channel motivated by network coding, where errors with the same row space are equiprobable. Over such channels, we establish upper bounds on the DEPs of BDDs, determine the exact DEP of BDDs for maximum rank distance (MRD) codes, and show that MRD codes have the greatest DEPs up to a scalar. To evaluate the DEPs of BDDs for CDCs, we first establish some fundamental geometric properties of the projective space. Using these geometric properties, we then consider BDDs in both subspace and injection metrics and derive analytical expressions of their DEPs for CDCs, over a symmetric operator channel, as functions of their distance distributions. Finally, we focus on CDCs obtained by lifting rank metric codes and establish two important results: First, we derive asymptotically tight upper bounds on the DEPs of BDDs in both metrics; Second, we show that the DEPs for KK codes are the greatest up to a scalar among all CDCs obtained by lifting rank metric codes.
Efficient Interpolation in the Guruswami-Sudan Algorithm
Venue: arXiv cs.IT 2010
Year: 2010

A novel algorithm is proposed for the interpolation step of the Guruswami-Sudan list decoding algorithm. The proposed method is based on the binary exponentiation algorithm, and can be considered as an extension of the Lee-O'Sullivan algorithm. The algorithm is shown to achieve both asymptotical and practical performance gain compared to the case of iterative interpolation algorithm. Further complexity reduction is achieved by integrating the proposed method with re-encoding. The key contribution of the paper, which enables the complexity reduction, is a novel randomized ideal multiplication algorithm.
Detecting lacunary perfect powers and computing their roots
Venue: arXiv cs.SC 2010
Year: 2010

We consider solutions to the equation f = h^r for polynomials f and h and integer r > 1. Given a polynomial f in the lacunary (also called sparse or super-sparse) representation, we first show how to determine if f can be written as h^r and, if so, to find such an r. This is a Monte Carlo randomized algorithm whose cost is polynomial in the number of non-zero terms of f and in log(deg f), i.e., polynomial in the size of the lacunary representation, and it works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give two deterministic algorithms to compute the perfect root h given f and r. The first is output-sensitive (based on the sparsity of h) and works only over Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second approach to computing h, which is extremely efficient and works over both GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of these algorithms are unconditionally polynomial-time in the lacunary size of the input polynomial f. Finally, we demonstrate the efficiency of the randomized detection algorithm and the latter perfect root computation algorithm with an implementation in the C++ library NTL.
Feature Hashing for Large Scale Multitask Learning
Venue: arXiv cs.AI 2010
Year: 2010

Empirical evidence suggests that hashing is an effective strategy for dimensionality reduction and practical nonparametric estimation. In this paper we provide exponential tail bounds for feature hashing and show that the interaction between random subspaces is negligible with high probability. We demonstrate the feasibility of this approach with experimental results for a new use case -- multitask learning with hundreds of thousands of tasks.
Inseparability and Strong Hypotheses for Disjoint NP Pairs
Venue: arXiv cs.CC 2010
Year: 2010

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k))-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs.
On the (semi)lattices induced by continuous reducibilities
Venue: arXiv cs.LO 2010
Year: 2010

Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most important definitions, and especially introduce suprema for them. The suprema are shown to commutate with several characteristic numbers.
Kalman Filtering with Intermittent Observations: Weak Convergence to a Stationary Distribution
Venue: arXiv cs.IT 2010
Year: 2010

The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RARE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. In particular, this weak convergence holds under broad conditions and even when the observations arrival rate is below the critical probability for mean stability. We apply the weak-Feller property of the Markov process governing the RARE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure ergodicity of the sample paths to easily compute the moments of the invariant distribution. A one dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.
Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference
Venue: arXiv cs.AI 2010
Year: 2010

In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified message-passing algorithm architecture. We generalize the Belief Propagation (BP) algorithms of sum-product and max-product and tree-rewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on "convex-free-energy" and Linear-Programming (LP) relaxation as a zero-temprature of a convex-free-energy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of $f + \sum_i h_i$ where the function $f$ is an extended-valued, strictly convex but non-smooth and the functions $h_i$ are extended-valued functions (not necessarily convex). We use tools from convex duality to present the "primal-dual ascent" algorithm which is an extension of the Bregman successive projection scheme and is designed to handle optimization of the general type $f + \sum_i h_i$. Mapping the fractional-free-energy variational principle to this framework introduces the "norm-product" message-passing. Special cases include sum-product and max-product (BP algorithms) and the TRBP algorithms. When the fractional-free-energy is set to be convex (convex-free-energy) the norm-product is globally convergent for estimating of marginal probabilities and for approximating the LP-relaxation. We also introduce another branch of the norm-product, the "convex-max-product". The convex-max-product is convergent (unlike max-product) and aims at solving the LP-relaxation.
Optimization and Analysis of Distributed Averaging with Short Node Memory
Venue: arXiv cs.DC 2010
Year: 2010

In this paper, we demonstrate, both theoretically and by numerical examples, that adding a local prediction component to the update rule can significantly improve the convergence rate of distributed averaging algorithms. We focus on the case where the local predictor is a linear combination of the node's two previous values (i.e., two memory taps), and our update rule computes a combination of the predictor and the usual weighted linear combination of values received from neighbouring nodes. We derive the optimal mixing parameter for combining the predictor with the neighbors' values, and carry out a theoretical analysis of the improvement in convergence rate that can be obtained using this acceleration methodology. For a chain topology on n nodes, this leads to a factor of n improvement over the one-step algorithm, and for a two-dimensional grid, our approach achieves a factor of n^1/2 improvement, in terms of the number of iterations required to reach a prescribed level of accuracy.
A System of Interaction and Structure IV: The Exponentials and Decomposition
Venue: arXiv cs.LO 2010
Year: 2010

We study a system, called NEL, which is the mixed commutative/non-commutative linear logic BV augmented with linear logic's exponentials. Equivalently, NEL is MELL augmented with the non-commutative self-dual connective seq. In this paper, we show a basic compositionality property of NEL, which we call decomposition. This result leads to a cut-elimination theorem, which is proved in the next paper of this series. To control the induction measure for the theorem, we rely on a novel technique that extracts from NEL proofs the structure of exponentials, into what we call !-?-Flow-Graphs.
Approximability of Sparse Integer Programs
Venue: arXiv cs.DS 2010
Year: 2010

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx: Ax >= b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >= 2 and eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax <= b, 0 <= x <= d} where A has at most k nonzeroes per column, we give a (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.
Seidel Minor, Permutation Graphs and Combinatorial Properties
Venue: arXiv cs.DM 2010
Year: 2010

A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at one of it vertex $v$ consists to complement the edges between the neighborhood and the non-neighborhood of $v$. Two graphs are Seidel complement equivalent if one can be obtained from the other by a successive application of Seidel complementation. In this paper we introduce the new concept of Seidel complementation and Seidel minor, we then show that this operation preserves cographs and the structure of modular decomposition. The main contribution of this paper is to provide a new and succinct characterization of permutation graphs i.e. A graph is a permutation graph \Iff it does not contain the following graphs: $C_5$, $C_7$, $XF_{6}^{2}$, $XF_{5}^{2n+3}$, $C_{2n}, n\geqslant6$ and their complement as Seidel minor. In addition we provide a $O(n+m)$-time algorithm to output one of the forbidden Seidel minor if the graph is not a permutation graph.
What does Newcomb's paradox teach us?
Venue: arXiv cs.GT 2010
Year: 2010

In Newcomb's paradox you choose to receive either the contents of a particular closed box, or the contents of both that closed box and another one. Before you choose though, an antagonist uses a prediction algorithm to deduce your choice, and fills the two boxes based on that deduction. Newcomb's paradox is that game theory's expected utility and dominance principles appear to provide conflicting recommendations for what you should choose. A recent extension of game theory provides a powerful tool for resolving paradoxes concerning human choice, which formulates such paradoxes in terms of Bayes nets. Here we apply this to ol to Newcomb's scenario. We show that the conflicting recommendations in Newcomb's scenario use different Bayes nets to relate your choice and the algorithm's prediction. These two Bayes nets are incompatible. This resolves the paradox: the reason there appears to be two conflicting recommendations is that the specification of the underlying Bayes net is open to two, conflicting interpretations. We then show that the accuracy of the prediction algorithm in Newcomb's paradox, the focus of much previous work, is irrelevant. We similarly show that the utility functions of you and the antagonist are irrelevant. We end by showing that Newcomb's paradox is time-reversal invariant; both the paradox and its resolution are unchanged if the algorithm makes its `prediction' \emph{after} you make your choice rather than before.
Guarded resolution for answer set programming
Venue: arXiv cs.AI 2010
Year: 2010

We describe a variant of resolution rule of proof and show that it is complete for stable semantics of logic programs. We show applications of this result.
Multihomogeneous Resultant Formulae for Systems with Scaled Support
Venue: arXiv cs.SC 2010
Year: 2010

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.
Fundamentals of the Backoff Process in 802.11: Dichotomy of the Aggregation
Venue: arXiv cs.NI 2010
Year: 2010

This paper discovers fundamental principles of the backoff process that governs the performance of IEEE 802.11. A simplistic principle founded upon regular variation theory is that the backoff time has a truncated Pareto-type tail distribution with an exponent of $(\log γ)/\log m$ ($m$ is the multiplicative factor and $γ$ is the collision probability). This reveals that the per-node backoff process is heavy-tailed in the strict sense for $γ>1/m^2$, and paves the way for the following unifying result. The state-of-the-art theory on the superposition of the heavy-tailed processes is applied to establish a dichotomy exhibited by the aggregate backoff process, putting emphasis on the importance of time-scale on which we view the backoff processes. While the aggregation on normal time-scales leads to a Poisson process, it is approximated by a new limiting process possessing long-range dependence (LRD) on coarse time-scales. This dichotomy turns out to be instrumental in formulating short-term fairness, extending existing formulas to arbitrary population, and to elucidate the absence of LRD in practical situations. A refined wavelet analysis is conducted to strengthen this argument.
On-the-fly erasure coding for real-time video applications
Venue: arXiv cs.NI 2010
Year: 2010

This paper introduces a robust point-to-point transmission scheme: Tetrys, that relies on a novel on-the-fly erasure coding concept which reduces the delay for recovering lost data at the receiver side. In current erasure coding schemes, the packets that are not rebuilt at the receiver side are either lost or delayed by at least one RTT before transmission to the application. The present contribution aims at demonstrating that Tetrys coding scheme can fill the gap between real-time applications requirements and full reliability. Indeed, we show that in several cases, Tetrys can recover lost packets below one RTT over lossy and best-effort networks. We also show that Tetrys allows to enable full reliability without delay compromise and as a result: significantly improves the performance of time constrained applications. For instance, our evaluations present that video-conferencing applications obtain a PSNR gain up to 7dB compared to classic block-based erasure codes.