OpenRaiser/Intern-Atlas huggingface dataset:
A polynomial time algorithm to approximate the mixed volume within a simply exponential factor
Venue: arXiv cs.CG 2009
Year: 2009

Let ${\bf K} = (K_1, ..., K_n)$ be an $n$-tuple of convex compact subsets in the Euclidean space $\R^n$, and let $V(\cdot)$ be the Euclidean volume in $\R^n$. The Minkowski polynomial $V_{\bf K}$ is defined as $V_{\bf K}(λ_1, ... ,λ_n) = V(λ_1 K_1 +, ..., + λ_n K_n)$ and the mixed volume $V(K_1, ..., K_n)$ as $$ V(K_1, ..., K_n) = \frac{\partial^n}{\partial λ_1...\partial λ_n} V_{\bf K}(λ_1 K_1 +, ..., + λ_n K_n). $$ Our main result is a poly-time algorithm which approximates $V(K_1, ..., K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small. Our approach is based on a particular approximation of $\log(V(K_1, ..., K_n))$ by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver-Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.
Induced Hilbert Space, Markov Chain, Diffusion Map and Fock Space in Thermophysics
Venue: arXiv cs.OH 2009
Year: 2009

In this article, we continue to explore Probability Bracket Notation (PBN), proposed in our previous article. Using both Dirac vector bracket notation (VBN) and PBN, we define induced Hilbert space and induced sample space, and propose that there exists an equivalence relation between a Hilbert space and a sample space constructed from the same base observable(s). Then we investigate Markov transition matrices and their eigenvectors to make diffusion maps with two examples: a simple graph theory example, to serve as a prototype of bidirectional transition operator; a famous text document example in IR literature, to serve as a tutorial of diffusion map in text document space. We show that the sample space of the Markov chain and the Hilbert space spanned by the eigenvectors of the transition matrix are not equivalent. At the end, we apply our PBN and equivalence proposal to Thermophysics by associating sample (phase) space with the Hilbert space of a single particle and the Fock space of many-particle systems.
WDM and Directed Star Arboricity
Venue: arXiv cs.NI 2010
Year: 2010

A digraph is $m$-labelled if every arc is labelled by an integer in $\{1, \dots,m\}$. Motivated by wavelength assignment for multicasts in optical networks, we introduce and study $n$-fibre colourings of labelled digraphs. These are colourings of the arcs of $D$ such that at each vertex $v$, and for each colour $α$, $in(v,α)+out(v,α)\leq n$ with $in(v,α)$ the number of arcs coloured $α$ entering $v$ and $out(v,α)$ the number of labels $l$ such that there is at least one arc of label $l$ leaving $v$ and coloured with $α$. The problem is to find the minimum number of colours $λ_n(D)$ such that the $m$-labelled digraph $D$ has an $n$-fibre colouring. In the particular case when $D$ is $1$-labelled, $λ_1(D)$ is called the directed star arboricity of $D$, and is denoted by $dst(D)$. We first show that $dst(D)\leq 2Δ^-(D)+1$, and conjecture that if $Δ^-(D)\geq 2$, then $dst(D)\leq 2Δ^-(D)$. We also prove that for a subcubic digraph $D$, then $dst(D)\leq 3$, and that if $Δ^+(D), Δ^-(D)\leq 2$, then $dst(D)\leq 4$. Finally, we study $λ_n(m,k)=\max\{λ_n(D) \tq D \mbox{is $m$-labelled} \et Δ^-(D)\leq k\}$. We show that if $m\geq n$, then $\ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq λ_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n}$ for some constant $C$. We conjecture that the lower bound should be the right value of $λ_n(m,k)$.
Detection of Gauss-Markov Random Fields with Nearest-Neighbor Dependency
Venue: arXiv cs.IT 2010
Year: 2010

The problem of hypothesis testing against independence for a Gauss-Markov random field (GMRF) is analyzed. Assuming an acyclic dependency graph, an expression for the log-likelihood ratio of detection is derived. Assuming random placement of nodes over a large region according to the Poisson or uniform distribution and nearest-neighbor dependency graph, the error exponent of the Neyman-Pearson detector is derived using large-deviations theory. The error exponent is expressed as a dependency-graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals. The exponent is analyzed for different values of the variance ratio and correlation. It is found that a more correlated GMRF has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio.
Untangling planar graphs from a specified vertex position - Hard cases
Venue: arXiv cs.DM 2010
Year: 2010

Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $π$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $π$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))\sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $π$ exists even if it is presupposed that the vertices occupy any prescribed set of points in the plane. Similar questions are discussed for other families of planar graphs.
Cryptanalysis of Two McEliece Cryptosystems Based on Quasi-Cyclic Codes
Venue: arXiv cs.CR 2010
Year: 2010

We cryptanalyse here two variants of the McEliece cryptosystem based on quasi-cyclic codes. Both aim at reducing the key size by restricting the public and secret generator matrices to be in quasi-cyclic form. The first variant considers subcodes of a primitive BCH code. We prove that this variant is not secure by finding and solving a linear system satisfied by the entries of the secret permutation matrix. The other variant uses quasi-cyclic low density parity-check codes. This scheme was devised to be immune against general attacks working for McEliece type cryptosystems based on low density parity-check codes by choosing in the McEliece scheme more general one-to-one mappings than permutation matrices. We suggest here a structural attack exploiting the quasi-cyclic structure of the code and a certain weakness in the choice of the linear transformations that hide the generator matrix of the code. Our analysis shows that with high probability a parity-check matrix of a punctured version of the secret code can be recovered in cubic time complexity in its length. The complete reconstruction of the secret parity-check matrix of the quasi-cyclic low density parity-check codes requires the search of codewords of low weight which can be done with about $2^{37}$ operations for the specific parameters proposed.
Mechanizing the Metatheory of LF
Venue: arXiv cs.LO 2010
Year: 2010

LF is a dependent type theory in which many other formal systems can be conveniently embedded. However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's judgments. Although detailed informal proofs of these properties have been published, they have not been formally verified in a theorem prover. We have formalized these properties within Isabelle/HOL using the Nominal Datatype Package, closely following a recent article by Harper and Pfenning. In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others. We also formally derive a version of the type checking algorithm from which Isabelle/HOL can generate executable code. Besides its intrinsic interest, our formalization provides a foundation for studying the adequacy of LF encodings, the correctness of Twelf-style metatheoretic reasoning, and the metatheory of extensions to LF.
Towards a stable definition of Kolmogorov-Chaitin complexity
Venue: arXiv cs.IT 2010
Year: 2010

Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the Kolmogorov-Chaitin complexity of a string s. Some attempts have been made to arrive at a framework stable enough for a concrete definition of K, independent of any constant under a programming language, by appealing to the "naturalness" of the language in question. The aim of this paper is to present an approach to overcome the problem by looking at a set of models of computation converging in output probability distribution such that that "naturalness" can be inferred, thereby providing a framework for a stable definition of K under the set of convergent models of computation.
Submodular approximation: sampling-based algorithms and lower bounds
Venue: arXiv cs.DS 2010
Year: 2010

We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees for most of our algorithms are of the order of sqrt(n/ln n). We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.
Stabilizing Tiny Interaction Protocols
Venue: arXiv cs.DC 2010
Year: 2010

In this paper we present the self-stabilizing implementation of a class of token based algorithms. In the current work we only consider interactions between weak nodes. They are uniform, they do not have unique identifiers, are static and their interactions are restricted to a subset of nodes called neighbours. While interacting, a pair of neighbouring nodes may create mobile agents (that materialize in the current work the token abstraction) that perform traversals of the network and accelerate the system stabilization. In this work we only explore the power of oblivious stateless agents. Our work shows that the agent paradigm is an elegant distributed tool for achieving self-stabilization in Tiny Interaction Protocols (TIP). Nevertheless, in order to reach the full power of classical self-stabilizing algorithms more complex classes of agents have to be considered (e.g. agents with memory, identifiers or communication skills). Interestingly, our work proposes for the first time a model that unifies the recent studies in mobile robots(agents) that evolve in a discrete space and the already established population protocols paradigm.
The Heap Lambda Machine
Venue: arXiv cs.LO 2010
Year: 2010

This paper introduces a new machine architecture for evaluating lambda expressions using the normal-order reduction, which guarantees that every lambda expression will be evaluated if the expression has its normal form and the system has enough memory. The architecture considered here operates using heap memory only. Lambda expressions are represented as graphs, and all algorithms used in the processing unit of this machine are non-recursive.
Fair Scheduling in Networks Through Packet Election
Venue: arXiv cs.IT 2010
Year: 2010

We consider the problem of designing a fair scheduling algorithm for discrete-time constrained queuing networks. Each queue has dedicated exogenous packet arrivals. There are constraints on which queues can be served simultaneously. This model effectively describes important special instances like network switches, interference in wireless networks, bandwidth sharing for congestion control and traffic scheduling in road roundabouts. Fair scheduling is required because it provides isolation to different traffic flows; isolation makes the system more robust and enables providing quality of service. Existing work on fairness for constrained networks concentrates on flow based fairness. As a main result, we describe a notion of packet based fairness by establishing an analogy with the ranked election problem: packets are voters, schedules are candidates and each packet ranks the schedules based on its priorities. We then obtain a scheduling algorithm that achieves the described notion of fairness by drawing upon the seminal work of Goodman and Markowitz (1952). This yields the familiar Maximum Weight (MW) style algorithm. As another important result we prove that algorithm obtained is throughput optimal. There is no reason a priori why this should be true, and the proof requires non-traditional methods.
Solving the apparent diversity-accuracy dilemma of recommender systems
Venue: arXiv cs.IR 2010
Year: 2010

Recommender systems use data on past user preferences to predict possible future likes and interests. A key challenge is that while the most useful individual recommendations are to be found among diverse niche objects, the most reliably accurate results are obtained by methods that recommend objects based on user or object similarity. In this paper we introduce a new algorithm specifically to address the challenge of diversity and show how it can be used to resolve this apparent dilemma when combined in an elegant hybrid with an accuracy-focused algorithm. By tuning the hybrid appropriately we are able to obtain, without relying on any semantic or context-specific information, simultaneous gains in both accuracy and diversity of recommendations.
The Complexity of Reasoning for Fragments of Default Logic
Venue: arXiv cs.CC 2010
Year: 2010

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\SigmaPtwo$-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.
An overview of the transmission capacity of wireless networks
Venue: arXiv cs.IT 2010
Year: 2010

This paper surveys and unifies a number of recent contributions that have collectively developed a metric for decentralized wireless network analysis known as transmission capacity. Although it is notoriously difficult to derive general end-to-end capacity results for multi-terminal or \adhoc networks, the transmission capacity (TC) framework allows for quantification of achievable single-hop rates by focusing on a simplified physical/MAC-layer model. By using stochastic geometry to quantify the multi-user interference in the network, the relationship between the optimal spatial density and success probability of transmissions in the network can be determined, and expressed -- often fairly simply -- in terms of the key network parameters. The basic model and analytical tools are first discussed and applied to a simple network with path loss only and we present tight upper and lower bounds on transmission capacity (via lower and upper bounds on outage probability). We then introduce random channels (fading/shadowing) and give TC and outage approximations for an arbitrary channel distribution, as well as exact results for the special cases of Rayleigh and Nakagami fading. We then apply these results to show how TC can be used to better understand scheduling, power control, and the deployment of multiple antennas in a decentralized network. The paper closes by discussing shortcomings in the model as well as future research directions.
A local construction of the Smith normal form of a matrix polynomial
Venue: arXiv cs.SC 2010
Year: 2010

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the original matrix. The performance of the algorithm in exact arithmetic is reported for several test cases.
A Minimum Relative Entropy Principle for Learning and Acting
Venue: arXiv cs.AI 2010
Year: 2010

This paper proposes a method to construct an adaptive agent that is universal with respect to a given class of experts, where each expert is an agent that has been designed specifically for a particular environment. This adaptive control problem is formalized as the problem of minimizing the relative entropy of the adaptive agent from the expert that is most suitable for the unknown environment. If the agent is a passive observer, then the optimal solution is the well-known Bayesian predictor. However, if the agent is active, then its past actions need to be treated as causal interventions on the I/O stream rather than normal probability conditions. Here it is shown that the solution to this new variational problem is given by a stochastic controller called the Bayesian control rule, which implements adaptive behavior as a mixture of experts. Furthermore, it is shown that under mild assumptions, the Bayesian control rule converges to the control law of the most suitable expert.
Best-effort Group Service in Dynamic Networks
Venue: arXiv cs.DC 2010
Year: 2010

We propose a group membership service for dynamic ad hoc networks. It maintains as long as possible the existing groups and ensures that each group diameter is always smaller than a constant, fixed according to the application using the groups. The proposed protocol is self-stabilizing and works in dynamic distributed systems. Moreover, it ensures a kind of continuity in the service offer to the application while the system is converging, except if too strong topology changes happen. Such a best effort behavior allows applications to rely on the groups while the stabilization has not been reached, which is very useful in dynamic ad hoc networks.
A complexity dichotomy for hypergraph partition functions
Venue: arXiv cs.CC 2010
Year: 2010

We consider the complexity of counting homomorphisms from an $r$-uniform hypergraph $G$ to a symmetric $r$-ary relation $H$. We give a dichotomy theorem for $r>2$, showing for which $H$ this problem is in FP and for which $H$ it is #P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the case $r=2$, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation $H$ is weighted, and the goal is to compute the \emph{partition function}, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of $r$ sites interact symmetrically. In the weighted case, our dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs, where $r=2$. When $r=2$, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all $r>2$ the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.
Improved Capacity Scaling in Wireless Networks With Infrastructure
Venue: arXiv cs.IT 2010
Year: 2010

This paper analyzes the impact and benefits of infrastructure support in improving the throughput scaling in networks of $n$ randomly located wireless nodes. The infrastructure uses multi-antenna base stations (BSs), in which the number of BSs and the number of antennas at each BS can scale at arbitrary rates relative to $n$. Under the model, capacity scaling laws are analyzed for both dense and extended networks. Two BS-based routing schemes are first introduced in this study: an infrastructure-supported single-hop (ISH) routing protocol with multiple-access uplink and broadcast downlink and an infrastructure-supported multi-hop (IMH) routing protocol. Then, their achievable throughput scalings are analyzed. These schemes are compared against two conventional schemes without BSs: the multi-hop (MH) transmission and hierarchical cooperation (HC) schemes. It is shown that a linear throughput scaling is achieved in dense networks, as in the case without help of BSs. In contrast, the proposed BS-based routing schemes can, under realistic network conditions, improve the throughput scaling significantly in extended networks. The gain comes from the following advantages of these BS-based protocols. First, more nodes can transmit simultaneously in the proposed scheme than in the MH scheme if the number of BSs and the number of antennas are large enough. Second, by improving the long-distance signal-to-noise ratio (SNR), the received signal power can be larger than that of the HC, enabling a better throughput scaling under extended networks. Furthermore, by deriving the corresponding information-theoretic cut-set upper bounds, it is shown under extended networks that a combination of four schemes IMH, ISH, MH, and HC is order-optimal in all operating regimes.
EcoTRADE - a multi player network game of a tradable permit market for biodiversity credits
Venue: arXiv cs.GT 2010
Year: 2010

EcoTRADE is a multi player network game of a virtual biodiversity credit market. Each player controls the land use of a certain amount of parcels on a virtual landscape. The biodiversity credits of a particular parcel depend on neighboring parcels, which may be owned by other players. The game can be used to study the strategies of players in experiments or classroom games and also as a communication tool for stakeholders participating in credit markets that include spatially interdependent credits.