Polynomial Running Time Research Articles

Efficient fault-tolerant scheduling on multiprocessor systems via replication and deallocation

Multiprocessor architectures have been extensively studied in the past decades. With the increasing demands on system reliability, fault-tolerant scheduling is even more important when multiprocessor system is used to run real-time applications since failure of any processor might produce disastrous results. To address this problem, we propose two fault-tolerant scheduling methods on multiprocessor systems via both active and passive backup copies. The first one uses the integer linear programming method to obtain the optimal results. The second one is a heuristic algorithm which can achieve close to optimal results within polynomial running time. In the experiments, this paper evaluates the proposed methods in terms of scheduling length for a set of DAG benchmarks. The experimental results show the effectiveness of our technique.

A fast algorithm to test the principality of ideals

Using the LLL algorithm we develop a method by which one can test the principality of ideals in some bicyclic biquadratic elds. Moreover, we will show that this algorithm run, at most, in polynomial running time.

Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed $k\ge 2$, there is a polynomial-time algorithm that, for a $1$-connected topological space $X$ given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the $k$th homotopy group $\pi_k(X)$, as well as the first $k$ stages of a Postnikov system of $X$. Combined with results of an earlier paper, this yields a polynomial-time computation of $[X,Y]$, i.e., all homotopy classes of continuous mappings $X\to Y$, under the assumption that $Y$ is $(k-1)$-connected and $\dim X\le 2k-2$. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes $X$, $Y$, where $Y$ is $(k-1)$-connected and $\dim X\le 2k-1$, plus a subspace $A\subseteq X$ and a (simplicial) map $f:A\to Y$, and the question is the extendability of $f$ to all of $X$. The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his coworkers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg--MacLane space $K(\mathbb{Z},1)$, provided in another recent paper by Krčál, Matoušek, and Sergeraert. (A corrected file is attached.)

Frequency Domain Packet Scheduling with Stability Analysis for 3GPP LTE Uplink

In this paper, we investigate the Frequency Domain Packet Scheduling (FDPS) problem for 3GPP Long Term Evolution (LTE) Uplink (UL). Instead of studying a specific scheduling policy, we provide a unified approach to tackle this issue. First, we formalize a general LTE UL FDPS problem, which is suitable for various scheduling policies. Then, we prove that the problem is MAX SNP-hard, which implies that approximation algorithms with constant approximation ratios are the best that we can hope for. Therefore, we design two approximation algorithms, both of which have polynomial runtime. The first algorithm is based on a simple greedy method. The second one is based on the Local Ratio (L-R) technique and it can approximately solve the LTE UL FDPS problem with an approximation ratio of 2. To further analyze the stability of the 2-approximation L-R algorithm, we derive a specific FDPS problem, which incorporates the queue length and channel quality information. We utilize the Lyapunov Drift to prove the L-R algorithm is stable for any $((\omega_0, \epsilon_0))$-admissible LTE UL systems. The simulation results indicate good performance of the L-R scheduler.

Runtime analysis of a multi-objective evolutionary algorithm for obtaining finite approximations of Pareto fronts

Previous theoretical analyses of evolutionary multi-objective optimization (EMO) mostly focus on obtaining ∊-approximations of Pareto fronts. However, in practical applications, an appropriate value of ∊ is critical but sometimes, for a multi-objective optimization problem (MOP) with unknown attributes, difficult to determine. In this paper, we propose a new definition for the finite representation of the Pareto front—the adaptive Pareto front, which can automatically accommodate the Pareto front. Accordingly, it is more practical to take the adaptive Pareto front, or its ∊-approximation (termed the ∊-adaptive Pareto front) as the goal of an EMO algorithm. We then perform a runtime analysis of a (μ+1) multi-objective evolutionary algorithm ((μ+1) MOEA) for three MOPs, including a discrete MOP with a polynomial Pareto front (denoted as a polynomial DMOP), a discrete MOP with an exponential Pareto front (denoted as an exponential DMOP) and a simple continuous two-objective optimization problem (SCTOP). By employing an estimator-based update strategy in the (μ+1) MOEA, we show that (1) for the polynomial DMOP, the whole Pareto front can be obtained in the expected polynomial runtime by setting the population size μ equal to the number of Pareto vectors; (2) for the exponential DMOP, the expected polynomial runtime can be obtained by keeping μ increasing in the same order as that of the problem size n; and (3) the diversity mechanism guarantees that in the expected polynomial runtime the MOEA can obtain an ∊-adaptive Pareto front of SCTOP for any given precision ∊. Theoretical studies and numerical comparisons with NSGA-II demonstrate the efficiency of the proposed MOEA and should be viewed as an important step toward understanding the mechanisms of MOEAs.

Approximating the minimum independent dominating set in perturbed graphs

We investigate the minimum independent dominating set in perturbed graphs g(G,p) of input graph G=(V,E), obtained by negating the existence of edges independently with a probability p>0. The minimum independent dominating set (MIDS) problem does not admit a polynomial running time approximation algorithm with worst-case performance ratio of n1−ϵ for any ϵ>0. We prove that the size of the minimum independent dominating set in g(G,p), denoted as i(g(G,p)), is asymptotically almost surely in Θ(log⁡|V|). Furthermore, we show that the probability of i(g(G,p))⩾4|V|p is no more than 2−|V|, and present a simple greedy algorithm of proven worst-case performance ratio 4|V|p and with polynomial expected running time.

Application of graph search and genetic algorithms for the single machine scheduling problem with sequence-dependent setup times and quadratic penalty function of completion times

In this paper, we consider the single machine scheduling problem with quadratic penalties and sequence-dependent (QPSD) setup times. QPSD is known to be NP-Hard. Only a few exact approaches, and to the best of our knowledge, no approximate approaches, have been reported in the literature so far. This paper discusses exact and approximate approaches for solving the problem, and presents empirical findings. We make use of a graph search algorithm, Memory-Based Depth-First Branch-and-Bound (MDFBB), and present an algorithm, QPSD_MDFBB that can optimally solve QPSD, and advances the state of the art for finding exact solutions. For finding approximate solutions to large problem instances, we make use of the idea of greedy stochastic search, and present a greedy stochastic algorithm, QPSD_GSA that provides moderately good solutions very rapidly even for large problems. The major contribution of the current paper is to apply QPSD_GSA to generate a subset of the starting solutions for a new genetic algorithm, QPSD_GEN, which is shown to provide near-optimal solutions very quickly. Owing to its polynomial running time, QPSD_GEN can be used for much larger instances than QPSD_MDFBB can handle. Experimental results have been provided to demonstrate the performances of these algorithms.

Offset polygon and annulus placement problems

The δ-annulus of a polygon P is the closed region containing all points in the plane at distance at most δ from the boundary of P. An inner (resp., outer) δ-offset polygon is the polygon defined by the inner (resp., outer) boundary of its δ-annulus.In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of δ, to cover as many points from S as possible by the δ-offset (or by the δ-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of δ such that there is a rigid transformation of the δ-offset (or the δ-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering k⩽|S| points.These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases.

A local search algorithm for binary maximum 2-path partitioning

Let G be a complete (undirected) graph with 3l vertices. Given a binary weight function on the edges of G, the binary maximum 2-path partitioning problem is to compute a set of l vertex-disjoint simple 2-edge paths with maximum total edge weight. The problem is NP-hard (Garey and Johnson 1979) [1].In this paper we propose a simple local search algorithm with polynomial running time for the problem and analyze its performance for several search depths. For depth 2, we show that the algorithm is a 0.3333-approximation, and that the bound is tight. For depth 3, we show that the algorithm is a 0.4-approximation. For depth 9, we show that the algorithm is a 0.55-approximation, improving on the best-known 0.5265 bound for the problem.We also consider the special case where G is subcubic, that is, the maximum degree in its subgraph induced by the unit-weight edges is 3. In this case we show that the algorithm is a 0.375-approximation for depth 2 and a 0.5-approximation for depth 3. In addition, we show that depth 7 is sufficient for the 0.55 bound guarantee.Finally we give, by means of bad instances, upper bounds on the performance guarantees of the algorithm. For depth 2 we show a 0.4 upper bound in the subcubic case. For depth 3 we show a 0.6 upper bound, as well as a 0.7 upper bound in the subcubic case. For the general (non-negative) weight problem we show a 0.5556 upper bound for depth 3 (for depth 2, the tight 0.3333 ratio holds for this problem as well).

The choice of the offspring population size in the ([formula omitted]) evolutionary algorithm

We extend the theory of non-elitist evolutionary algorithms (EAs) by considering the offspring population size in the (1,λ) EA. We establish a sharp threshold at λ=logee−1n≈5log10n between exponential and polynomial running times on OneMax. For any smaller value, the (1,λ) EA needs exponential time on every function that has only one global optimum. We also consider arbitrary unimodal functions and show that the threshold can shift towards larger offspring population sizes. In particular, for the function LeadingOnes there is a sharp threshold at λ=2logee−1n≈10log10n. Finally, we investigate the relationship between the offspring population size and arbitrary mutation rates on OneMax. We get sharp thresholds for λ that decrease with the mutation rate. This illustrates the balance between selection and mutation.

A top-down approach for approximate data anonymisation

Data sharing in today's information society poses a threat to individual privacy and organisational confidentiality. k-anonymity is a widely adopted model to prevent the owner of a record being re-identified. By generalising and/or suppressing certain portions of the released dataset, it guarantees that no records can be uniquely distinguished from at least other k−1 records. A key requirement for the k-anonymity problem is to minimise the information loss resulting from data modifications. This article proposes a top-down approach to solve this problem. It first considers each record as a vertex and the similarity between two records as the edge weight to construct a complete weighted graph. Then, an edge cutting algorithm is designed to divide the complete graph into multiple trees/components. The Large Components with size bigger than 2k−1 are subsequently split to guarantee that each resulting component has the vertex number between k and 2k−1. Finally, the generalisation operation is applied on the vertices in each component (i.e. equivalence class) to make sure all the records inside have identical quasi-identifier values. We prove that the proposed approach has polynomial running time and theoretical performance guarantee O(k). The empirical experiments show that our approach results in substantial improvements over the baseline heuristic algorithms, as well as the bottom-up approach with the same approximate bound O(k). Comparing to the baseline bottom-up O(logk)-approximation algorithm, when the required k is smaller than 50, the adopted top-down strategy makes our approach achieve similar performance in terms of information loss while spending much less computing time. It demonstrates that our approach would be a best choice for the k-anonymity problem when both the data utility and runtime need to be considered, especially when k is set to certain value smaller than 50 and the record set is big enough to make the runtime have to be taken into account.

Graph Isomorphisms and Automorphisms via Spectral Signatures

An isomorphism between two graphs is a connectivity preserving bijective mapping between their sets of vertices. Finding isomorphisms between graphs, or between a graph and itself (automorphisms), is of great importance in applied sciences. The inherent computational complexity of this problem is as yet unknown. Here, we introduce an efficient method to compute such mappings using heat kernels associated with the graph Laplacian. While the problem is combinatorial in nature, in practice we experience polynomial runtime in the number of vertices. As we demonstrate, the proposed method can handle a variety of graphs and is competitive with state-of-the-art packages on various important examples.

Improved approximation algorithms for the Min–Max Selecting Items problem

We give a simple deterministic O(logK/loglogK) approximation algorithm for the Min–Max Selecting Items problem, where K is the number of scenarios. While our main goal is simplicity, this result also improves over the previous best approximation ratio of O(logK) due to Kasperski, Kurpisz, and Zieliński (2013) [4]. Despite using the method of pessimistic estimators, the algorithm has a polynomial runtime also in the RAM model of computation. We also show that the LP formulation for this problem by Kasperski and Zieliński (2009) [6], which is the basis for the previous work and ours, has an integrality gap of at least Ω(logK/loglogK).

Minimizing accumulative memory load cost on multi-core DSPs with multi-level memory

In multi-core Digital Signal Processing (DSP) Systems, the processor-memory gap remains the primary obstacle in improving system performance. This paper addresses this bottleneck by combining task scheduling and memory accesses so that the system architecture and memory modules of a multi-core DSP can be utilized as efficiently as possible. To improve the system and memory utilization, the key is to take advantage of locality as much as possible and integrate it into task scheduling. Two algorithms are proposed to optimize memory accesses while scheduling tasks with timing and resource constraints. The first one uses Integer Linear Programming (ILP) to produce a schedule with the most efficient memory access sequence while satisfying the constraints. The second one is a heuristic algorithm which can produce a near optimal schedule with polynomial running time. The experimental results show that the memory access cost can be reduced up to 60% while the schedule length is also shortened.

Frequency Domain Packet Scheduling with MIMO for 3GPP LTE Downlink

In this paper, we formalize a general Frequency Domain Packet Scheduling (FDPS) problem for 3GPP LTE Downlink (DL). The DL FDPS problem incorporates the Single-User Multiple Input Multiple Output (SU-MIMO) technique, and can express various scheduling policies, including the Proportional-Fair metric, the MaxWeight scheduling, etc. For LTE DL SU-MIMO, the constraint of selecting only one MIMO mode (transmit diversity or spatial multiplexing) per user in each transmission time interval (TTI) increases the hardness of the FDPS problem. We prove the problem is MAX SNP-hard, which implies approximation algorithms with constant approximation ratios are the best we can expect. Subsequently, we propose an approximation algorithm of polynomial runtime. The solution is based on a greedy method for maximizing a non-decreasing submodular function over a matroid. The algorithm can solve the general DL FDPS problem with an approximation ratio of 4. We implement the proposed algorithm and compare its performance with other well-known schedulers.

Systematic parameter estimation in data-rich environments for cell signalling dynamics

Motivation: Computational models of biological signalling networks, based on ordinary differential equations (ODEs), have generated many insights into cellular dynamics, but the model-building process typically requires estimating rate parameters based on experimentally observed concentrations. New proteomic methods can measure concentrations for all molecular species in a pathway; this creates a new opportunity to decompose the optimization of rate parameters.Results: In contrast with conventional parameter estimation methods that minimize the disagreement between simulated and observed concentrations, the SPEDRE method fits spline curves through observed concentration points, estimates derivatives and then matches the derivatives to the production and consumption of each species. This reformulation of the problem permits an extreme decomposition of the high-dimensional optimization into a product of low-dimensional factors, each factor enforcing the equality of one ODE at one time slice. Coarsely discretized solutions to the factors can be computed systematically. Then the discrete solutions are combined using loopy belief propagation, and refined using local optimization. SPEDRE has unique asymptotic behaviour with runtime polynomial in the number of molecules and timepoints, but exponential in the degree of the biochemical network. SPEDRE performance is comparatively evaluated on a novel model of Akt activation dynamics including redox-mediated inactivation of PTEN (phosphatase and tensin homologue).Availability and implementation: Web service, software and supplementary information are available at www.LtkLab.org/SPEDRESupplementary information: Supplementary data are available at Bioinformatics online.Contact: LisaTK@nus.edu.sg

The knapsack problem with a minimum filling constraint

AbstractWe study a knapsack problem with an additional minimum filling constraint, such that the total weight of selected items cannot be less than a given threshold. The problem has several applications in shipping, e‐commerce, and transportation service procurement. When the threshold equals the knapsack capacity, even finding a feasible solution to the problem is NP‐hard. Therefore, we consider the case when the ratio α of threshold to capacity is less than 1. For this case, we develop an approximation scheme that returns a feasible solution with a total profit not less than (1 ‐ ε) times the total profit of an optimal solution for any ε > 0, and with a running time polynomial in the number of items, 1/ε, and 1/(1‐α). © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013

Computing Borcherds products

AbstractWe present an algorithm for computing Borcherds products, which has polynomial runtime. It deals efficiently with the bounds on Fourier expansion indices originating in Weyl chambers. Naive multiplication has exponential runtime due to inefficient handling of these bounds. An implementation of the new algorithm shows that it is also much faster in practice.

Mining functional subgraphs from cancer protein-protein interaction networks

BackgroundProtein-protein interaction (PPI) networks carry vital information about proteins' functions. Analysis of PPI networks associated with specific disease systems including cancer helps us in the understanding of the complex biology of diseases. Specifically, identification of similar and frequently occurring patterns (network motifs) across PPI networks will provide useful clues to better understand the biology of the diseases.ResultsIn this study, we developed a novel pattern-mining algorithm that detects cancer associated functional subgraphs occurring in multiple cancer PPI networks. We constructed nine cancer PPI networks using differentially expressed genes from the Oncomine dataset. From these networks we discovered frequent patterns that occur in all networks and at different size levels. Patterns are abstracted subgraphs with their nodes replaced by node cluster IDs. By using effective canonical labeling and adopting weighted adjacency matrices, we are able to perform graph isomorphism test in polynomial running time. We use a bottom-up pattern growth approach to search for patterns, which allows us to effectively reduce the search space as pattern sizes grow. Validation of the frequent common patterns using GO semantic similarity showed that the discovered subgraphs scored consistently higher than the randomly generated subgraphs at each size level. We further investigated the cancer relevance of a select set of subgraphs using literature-based evidences.ConclusionFrequent common patterns exist in cancer PPI networks, which can be found through effective pattern mining algorithms. We believe that this work would allow us to identify functionally relevant and coherent subgraphs in cancer networks, which can be advanced to experimental validation to further our understanding of the complex biology of cancer.

FlipCut Supertrees: Towards Matrix Representation Accuracy in Polynomial Time

In computational phylogenetics, supertree methods provide a way to reconstruct larger clades of the Tree of Life. The supertree problem can be formalized in different ways, to cope with contradictory information in the input. In particular, there exist methods based on encoding the input trees in a matrix, and methods based on finding minimum cuts in some graph. Matrix representation methods compute supertrees of superior quality, but the underlying optimization problems are computationally hard. In contrast, graph-based methods have polynomial running time, but supertrees are inferior in quality. In this paper, we present a novel approach for the computation of supertrees called FlipCut supertree. Our method combines the computation of minimum cuts from graph-based methods with a matrix representation method, namely Minimum Flip Supertrees. Here, the input trees are encoded in a 0/1/?-matrix. We present a heuristic to search for a minimum set of 0/1-flips such that the resulting matrix admits a directed perfect phylogeny. We then extend our approach by using edge weights to weight the columns of the 0/1/?-matrix. In our evaluation, we show that our method is extremely swift in practice, and orders of magnitude faster than the runner up. Concerning supertree quality, our method is sometimes on par with the “gold standard” Matrix Representation with Parsimony.