Exponential Size Research Articles

Are RNA networks scale-free?

A network is scale-free if its connectivity density function is proportional to a power-law distribution. It has been suggested that scale-free networks may provide an explanation for the robustness observed in certain physical and biological phenomena, since the presence of a few highly connected hub nodes and a large number of small-degree nodes may provide alternate paths between any two nodes on average—such robustness has been suggested in studies of metabolic networks, gene interaction networks and protein folding. A theoretical justification for why many networks appear to be scale-free has been provided by Barabási and Albert, who argue that expanding networks, in which new nodes are preferentially attached to highly connected nodes, tend to be scale-free. In this paper, we provide the first efficient algorithm to compute the connectivity density function for the ensemble of all homopolymer secondary structures of a user-specified length—a highly nontrivial result, since the exponential size of such networks precludes their enumeration. Since existent power-law fitting software, such as powerlaw, cannot be used to determine a power-law fit for our exponentially large RNA connectivity data, we also implement efficient code to compute the maximum likelihood estimate for the power-law scaling factor and associated Kolmogorov–Smirnov p value. Hypothesis tests strongly indicate that homopolymer RNA secondary structure networks are not scale-free; moreover, this appears to be the case for real (non-homopolymer) RNA networks.

Assessing how body size affects the species-time relationship in a shallow marine benthic megafauna community exposed to a strong hypoxia disturbance

The species–area relationship (SAR) and the species–time relationship (STR) are two well-established macro-ecological patterns. Species richness has also been shown to follow a humped relationship with body size, suggesting that SAR and STR may also depend on body size. We test whether the pattern of increase in the number of species with time varies with species body size. We analyzed data on carnivorous and scavenger species collected in a shallow marine benthic megafauna community in Coliumo Bay, Chile. Body size data were tabulated into exponential body size classes using two binning strategies. Species-level binning represented all individuals of each species by their average body size. Individual-level binning tabulated all data into exponential body size classes, allowing species to potentially occupy multiple size classes. We determined empirical relationships between body size and species richness, as well as estimated temporal species accumulation curves (TSAC) and nested species–time relationships (NSTR). We also tested whether the increase of species richness within body size classes follows either a power-law relationship or a logarithmic function. We find that the number of species increases with the time span studied, both for a TSAC and in the NSTR, for both species-based and individual-based data, while also showing a unimodal relationship with body size for both species and individual-based data. Thus, the STR is a general pattern, which depends on body size, despite reproductive seasonal forcing and strong hypoxia disturbance in Coliumo Bay.

Distributed Scheduling of Charging for On-Demand Electric Vehicle Fleets

Abstract Consider a fleet of N autonomous electric vehicles (EVs), where EV n has a battery state βn. Customers arrive at rate λ and each customer requests a trip that costs some battery charge b. The goal is to coordinate the pick-ups and charging of the fleet in order to minimize the discounted number of unpicked customers over an infinite time horizon. Solving the dynamic program for this scheduling problem is infeasible since the state space has an exponential size in N. We propose a distributed approach instead where each EV models the fleet as a field that picks the customer i.i.d. at random with some pick-up probability α (b). Based on this model, each EV computes its optimal charging and pick-up strategy, and reports which customer they are available to pick-up (as a function of b) or if they decided to charge. The server computes a maximal matching between customers and EVs, taking into account their reported availability. The process continues iteratively as each EV recomputes its optimal response to the new empirical α (b). We prove that the optimal local strategy is a threshold policy with respect to βn, where the threshold depends on α (b) and b. We also prove that the empirical pick-up probability α (b) converges to the true stationary probability, which explains why our algorithm converges to an equilibrium. We then study our algorithm in simulations, that show that at equilibrium, its performance is significantly better than that of an uncoordinated charging strategy. Therefore, our solution has low-complexity but is still sophisticated enough to coordinate the charging of the fleet.

Hybrid metaheuristic for generalised assignment

This paper investigates the classical generalised assignment problem (GAP), a challenging combinatorial optimisation problem that arises in numerous applications and that has attracted a great deal of research. For solving it we propose a hybrid metaheuristic combining guided search (GS), iterated local search (ILS), and very large-scale neighbourhood search (VLSN). The hybrid method is iterative. It starts with a random assignment, and in every iteration it acts in the following way: 1) The best current assignment is perturbed. 2) An exponential size neighbourhood of the perturbed assignment is constructed. It is the feasible solution set of a special GAP where only two fixed machines can execute a job. The neighbourhood construction is guided by a technique penalising poor machine/job selections. 3) The exponential neighbourhood is searched for improvement. Exploring the neighbourhood amounts to solving a monotone binary program (BP) – a monotone BP is one with two non-zero coefficients of opposite sign per column. We prove that the proposed metaheuristic runs in polynomial-time when applied to a variation of GAP. Good computational results in terms of solution quality, as well as of computation speed, are obtained with two new best values on hard instances.

Universal and Dynamic Locally Repairable Codes With Maximal Recoverability via Sum-Rank Codes

Locally repairable codes (LRCs) are considered with equal or unequal localities, local distances, and local field sizes. An explicit two-layer architecture with a sum-rank outer code is obtained, having disjoint local groups and achieving maximal recoverability (MR) for all families of local linear codes (MDS or not) simultaneously, up to a specified maximum locality $r $ . Furthermore, the local linear codes (thus the localities, local distances, and local fields) can be efficiently and dynamically modified without global recoding or changes in architecture or outer code, while preserving the MR property, easily adapting to new configurations in storage or new hot and cold data. In addition, local groups and file components can be added, removed or updated without global recoding. The construction requires global fields of size roughly $g^{r} $ , for $g $ local groups and maximum or specified locality $r $ . For equal localities, these global fields are smaller than those of previous MR-LRCs when $r \leq h $ (global parities). For unequal localities, they provide an exponential field size reduction on all previous best known MR-LRCs. For bounded localities and a large number of local groups, the global erasure-correction complexity of the given construction is comparable to that of Tamo–Barg codes or Reed–Solomon codes with local replication, while local repair is as efficient as for the Cartesian product of the local codes. Reed–Solomon codes with local replication and Cartesian products are recovered from the given construction when $r=1 $ and $h = 0 $ , respectively. The given construction can also be adapted to provide hierarchical MR-LRCs for all types of hierarchies and parameters. Finally, subextension subcodes and sum-rank alternant codes are introduced to obtain further exponential field size reductions, at the expense of lower information rates.

Coordinating resources in Stackelberg Security Games

In this work we formulate a Stackelberg Security Game that coordinates resources in a border patrol problem. In this security domain, resources from different precincts have to be paired to conduct patrols in the border due to logistic constraints. Given this structure, models that enumerate the pure defender strategies scale poorly. We describe the set of mixed strategies using a polynomial number of variables but exponentially many constraints that come from the matching polytope. We then include this description in a mixed integer formulation to compute the Strong Stackelberg Equilibrium efficiently with a branch and cut scheme. Since the optimal patrol solution is a probability distribution over the set of exponential size, we also introduce an efficient sampling method that can be used to deploy the security resources every shift. Our computational results evaluate the efficiency of the branch and cut scheme developed and the accuracy of the sampling method.

How to Express Hail Intensity—Modeling the Hailstone Size Distribution

AbstractThe local intensity of hail can be expressed by a variety of variables, such as hail kinetic energy, maximum hailstone size, and radar reflectivity–driven algorithms. All of these variables are connected by the hailstone size distribution. In the United States, the Community Collaborative Rain, Hail and Snow Network (CoCoRaHS) provides more than 37 000 observations that describe the diameter of the smallest, average, and largest hailstone; duration of hail; and overall hailstone number density. We use these data and the assumption of an exponential hailstone size distribution aloft to model the rate of hailstones hitting the ground per unit area, time, and hailstone size bin during the passage of a hailstorm. We found that total hail kinetic energy is proportional to the diameter of the largest hailstone to a power of less than 2. To validate these results, we compared them with hailpad observations of the largest hailstone diameters and hail kinetic energies in southwestern France. As an example application, we calculate the probability mass function for the largest hailstone observed by a hailpad given the largest hailstone occurring in its vicinity. We use this model at Risk Management Solutions, Ltd. (RMS), to calculate the vulnerability of subjects at risk as a function of the diameter of the largest hailstone.

On hard instances of non-commutative permanent

Recent developments on the complexity of the non-commutative determinant and permanent (Arvind and Srinivasan, 2010; Chien et al., 2011; Bläser, 2013; Gentry, 2014) have settled the complexity of non-commutative determinant with respect to the structure of the underlying algebra. Continuing the research further, we look to obtain more insights on hard instances of non-commutative permanent.We show that any Algebraic Branching Program (ABP) computing the Cayley permanent of a collection of disjoint directed two-cycles with distinct variables as edge labels requires exponential size on an average, where the average is taken over all possible permutations of variables. For graphs where every connected component contains at most six vertices, we show that evaluating the Cayley permanent over any algebra containing 2 × 2 matrices is #P complete.Further, we obtain efficient algorithms for computing the Cayley permanent/ determinant on graphs with bounded component size, when vertices within each component are not far apart from each other in the Cayley ordering. This gives a tight upper and lower bounds for size of ABPs computing the permanent of disjointtwo-cycles. Finally, we exhibit more families of non-commutative polynomial evaluation problems that are complete for #P.Our results demonstrate that apart from the structure of underlying algebras, relative ordering of the variables plays a crucial role in determining the complexity of non-commutative polynomials.

On the impact of job size variability on heterogeneity-aware load balancing

Load balancing is one of the key components in many distributed systems as it heavily impacts performance and resource utilization. We consider a heterogeneous system where each server belongs to one of K classes and the speed of the server depends on its class. Two types of load balancing strategies are considered: arriving jobs are either immediately dispatched to a server class in a randomized manner, i.e., with probability $$p_k$$ a job is assigned to class k, or are dispatched based on their size, i.e., jobs with a size in $$[T_{k-1},T_k)$$ are assigned to class k. Within each class a power of d choices rule is used to select the server that executes the job. For large systems and exponential job size durations the optimal probabilities $$p_k$$ to minimize the mean response time can be determined easily via convex optimization. In this paper we develop a mean field model (validated by simulation) to investigate how the optimal probabilities $$p_k$$ are affected by the higher moments and in particular by the variability of the job size distribution when the service discipline at each server is first-come-first-served. In addition, we make use of the cavity method to study the optimal thresholds $$T_k$$ in case the dispatching is based on the job size.

PPSZ for k ≥ 5: More Is Better

We show that for k ≥ 5, the PPSZ algorithm for k -SAT runs exponentially faster if there is an exponential number of satisfying assignments. More precisely, we show that for every k ≥ 5, there is a strictly increasing function f : [0,1] → R with f (0) = 0 that has the following property. If F is a k -CNF formula over n variables and |sat(F)| = 2 δ n solutions, then PPSZ finds a satisfying assignment with probability at least 2 − c k n − o ( n ) + f (δ) n . Here, 2 − c k n − o ( n ) is the success probability proved by Paturi et al. [11] for k -CNF formulas with a unique satisfying assignment. Our proof rests on a combinatorial lemma: given a set S ⊆ { 0,1} n , we can partition { 0,1} n into subcubes such that each subcube B contains exactly one element of S . Such a partition B induces a distribution on itself, via Pr [ B ] = |B| / 2 n for each B ∈ B . We are interested in partitions that induce a distribution of high entropy. We show that, in a certain sense, the worst case (min S : |S| = s max B H ( B )) is achieved if S is a Hamming ball. This lemma implies that every set S of exponential size allows a partition of linear entropy. This in turn leads to an exponential improvement of the success probability of PPSZ.

A branch-and-price algorithm for capacitated hypergraph vertex separation

We exactly solve the $${\mathcal {NP}}$$-hard combinatorial optimization problem of finding a minimum cardinality vertex separator with k (or arbitrarily many) capacitated shores in a hypergraph. We present an exponential size integer programming formulation which we solve by branch-and-price. The pricing problem, an interesting optimization problem on its own, has a decomposable structure that we exploit in preprocessing. We perform an extensive computational study, in particular on hypergraphs coming from the application of re-arranging a matrix into single-bordered block-diagonal form. Our experimental results show that our proposal complements the previous exact approaches in terms of applicability for larger k, and significantly outperforms them in the case $$k=\infty $$.

On the Complexity of Hazard-free Circuits

The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result, we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result, we establish the NP-completeness of several hazard detection problems.

A polyhedral model for enumeration and optimization over the set of circuits

Circuits play a fundamental role in polyhedral theory and linear programming. For instance, circuits are used as step directions in various augmentation schemes for solving linear programs or to leave degenerate vertices while running the simplex method. However, there are significant challenges when implementing these approaches: The set of circuits of a polyhedron may be of exponential size and is highly sensitive to the representation of the polyhedron.In this paper, we provide a universal framework for enumerating the set of circuits and optimizing over sets of circuits of a polyhedron in any representation—we propose a polyhedral model in which the circuits of the original polyhedron are encoded as extreme rays or vertices. Many methods in the literature and software assume that a polyhedron is in standard form; our framework is a direct generalization. We demonstrate its value by showing that the conversion of a general representation to standard form may introduce exponentially many new circuits.We then discuss the main advantages of the generalized polyhedral model. It enables the direct enumeration of useful subsets of circuits such as strictly feasible circuits or sign-compatible circuits, as well as optimization over these sets. In particular, this leads to the efficient computation of a steepest-descent circuit, which can be used in an augmentation scheme for solving linear programs or the construction of sign-compatible circuit walks with additional properties.

On the Relation between Star-Topology Decoupling and Petri Net Unfolding

Petri net unfolding expands concurrent sub-threads of a transition system separately. In AI Planning, star-topology decoupling (STD) finds a partitioning of state variables into components whose dependencies take a star shape, and expands leafcomponent state spaces separately. Thus both techniques rely on the separate expansion of state-space composites. How do they relate? We show that, provided compatible search orderings, STD state space size dominates that of unfolding if every component contains a single state variable, and unfolding dominates STD in the absence of prevail conditions (nondeleted action preconditions). In all other cases, exponential state space size advantages are possible on either side. Thus the sources of exponential advantages of STD are exactly a) state space size in the presence of prevail conditions (our results), and b) decidability of reachability in time linear in state space size vs. NP-hard for unfolding (known results).

Leading exponential finite size corrections for non-diagonal form factors

We derive the leading exponential finite volume corrections in two dimensional integrable models for non-diagonal form factors in diagonally scattering theories. These formulas are expressed in terms of the infinite volume form factors and scattering matrices. If the particles are bound states then the leading exponential finite-size corrections (μ-terms) are related to virtual processes in which the particles disintegrate into their constituents. For non-bound state particles the leading exponential finite-size corrections (F-terms) come from virtual particles traveling around the finite world. In these F-terms a specifically regulated infinite volume form factor is integrated for the momenta of the virtual particles. The F-term is also present for bound states and the μ-term can be obtained by taking an appropriate residue of the F-term integral. We check our results numerically in the Lee-Yang and sinh-Gordon models based on newly developed Hamiltonian truncations.

Event Size Distribution of Shallow Tectonic Tremor in the Nankai Trough

AbstractSlow earthquake size distributions have been found to follow either a power law or an exponential law, depending on the data set. We investigated the size distribution of shallow tectonic tremor events along the Nankai trough, southwestern Japan, observed in 2009, 2015, 2016, and 2018 and estimated the seismic energy radiated during the events as a measure of event size. The obtained size distributions mostly followed a power law with a b value of about 1, but an exponential law also fits well the data of the largest events; this result indicates that the size distribution follows the Gutenberg‐Richter law with an exponential taper. If only the largest events were observed, an exponential size distribution would be obtained. This exponential taper could be attributed to a source fault limitation. Because the maximum event size differed among activity periods, the source fault size varied among them, although the source areas largely overlap.

Connecting Knowledge Compilation Classes Width Parameters

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs).

THE FLUTED FRAGMENT REVISITED

AbstractWe study the fluted fragment, a decidable fragment of first-order logic with an unbounded number of variables, motivated by the work of W. V. Quine. We show that the satisfiability problem for this fragment has nonelementary complexity, thus refuting an earlier published claim by W. C. Purdy that it is in NExpTime. More precisely, we consider ${\cal F}{{\cal L}^m}$, the intersection of the fluted fragment and the m-variable fragment of first-order logic, for all $m \ge 1$. We show that, for $m \ge 2$, this subfragment forces $\left\lfloor {m/2} \right\rfloor$-tuply exponentially large models, and that its satisfiability problem is $\left\lfloor {m/2} \right\rfloor$-NExpTime-hard. We further establish that, for $m \ge 3$, any satisfiable ${\cal F}{{\cal L}^m}$-formula has a model of at most ($m - 2$)-tuply exponential size, whence the satisfiability (= finite satisfiability) problem for this fragment is in ($m - 2$)-NExpTime. Together with other, known, complexity results, this provides tight complexity bounds for ${\cal F}{{\cal L}^m}$ for all $m \le 4$.

Spruce forest stands in a stationary state

We present stationarity criteria for forest stands, and establish embodiments using a Norwegian empirical stand development model. The natural stationary states only slightly differ from the outcome of long-term simulations previously implemented using the same empirical model. Human interference in terms of diameter-limit cutting is introduced. Consequently, stationary states differing from the natural one appear. Standing volume, growth and monetary value appear low but the financial return rate may be significant. Volume yield and financial return clearly contradict each other, the former arising from harvesting large trees, the latter from frequent removal of small trees. An exponential tree size distribution does not appear to comply with the stationarity criterion.

Sub-optimal investment for insurers

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.