Round Algorithm Research Articles

Approximation Algorithms for the Submodular Hitting Set Problem

In this paper, we study the submodular hitting set problem (SHSP), which is a variant of the hitting set problem. In the SHSP, we are given a supergraph H = ( V , C ) and a nonnegative submodular function on the set 2 V . The objective is to determine a vertex subset to cover all hyperedges such that the cost of submodular covering is minimized. Our main work is to present a rounding algorithm and a primal-dual algorithm respectively for the SHSP and prove that they both have the approximation ratio k , where k is the maximum number of vertices in all hyperedges.

On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Methods

This paper studies the Partial Optimal Transport (POT) problem between two unbalanced measures with at most n supports and its applications in various AI tasks such as color transfer or domain adaptation. There is hence a need for fast approximations of POT with increasingly large problem sizes in arising applications. We first theoretically and experimentally investigate the infeasibility of the state-of-the-art Sinkhorn algorithm for POT, which consequently degrades its qualitative performance in real world applications like point-cloud registration. To this end, we propose a novel rounding algorithm for POT, and then provide a feasible Sinkhorn procedure with a revised computation complexity of O(n^2/epsilon^4). Our rounding algorithm also permits the development of two first-order methods to approximate the POT problem. The first algorithm, Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD), finds an epsilon-approximate solution to the POT problem in O(n^2.5/epsilon). The second method, Dual Extrapolation, achieves the computation complexity of O(n^2/epsilon), thereby being the best in the literature. We further demonstrate the flexibility of POT compared to standard OT as well as the practicality of our algorithms on real applications where two marginal distributions are unbalanced.

Distributed half-integral matching and beyond

By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log⁎⁡n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from {0,12,1}. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree Δ=2d, and any distributed graph algorithm with round complexity T(Δ) that only depends on Δ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2d. We give a new algorithm that shows that this is also sufficient.

Exponentially Faster Massively Parallel Maximal Matching

The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, we still have a limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of n 1+Ω (1) per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm, which is a variant of an algorithm conjectured to work by Czumaj, Lacki, Madry, Mitrovic, Onak, and Sankowski [ 15 ]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among other results, leads to an O (log log Δ) round algorithm for maximal matching with O(n) space (or even mildly sublinear in n using standard techniques). As an immediate corollary, we get a 2 approximate minimum vertex cover in essentially the same rounds and space, which is the optimal approximation factor under standard assumptions. We also get an improved O (log log Δ) round algorithm for 1 + ε approximate matching. All these results can also be implemented in the congested clique model in the same number of rounds.

Practical sampling of constraint-based models: Optimized thinning boosts CHRR performance.

Thinning is a sub-sampling technique to reduce the memory footprint of Markov chain Monte Carlo. Despite being commonly used, thinning is rarely considered efficient. For sampling constraint-based models, a highly relevant use-case in systems biology, we here demonstrate that thinning boosts computational and, thereby, sampling efficiencies of the widely used Coordinate Hit-and-Run with Rounding (CHRR) algorithm. By benchmarking CHRR with thinning with simplices and genome-scale metabolic networks of up to thousands of dimensions, we find a substantial increase in computational efficiency compared to unthinned CHRR, in our examples by orders of magnitude, as measured by the effective sample size per time (ESS/t), with performance gains growing with polytope (effective network) dimension. Using a set of benchmark models we derive a ready-to-apply guideline for tuning thinning to efficient and effective use of compute resources without requiring additional coding effort. Our guideline is validated using three (out-of-sample) large-scale networks and we show that it allows sampling convex polytopes uniformly to convergence in a fraction of time, thereby unlocking the rigorous investigation of hitherto intractable models. The derivation of our guideline is explained in detail, allowing future researchers to update it as needed as new model classes and more training data becomes available. CHRR with deliberate utilization of thinning thereby paves the way to keep pace with progressing model sizes derived with the constraint-based reconstruction and analysis (COBRA) tool set. Sampling and evaluation pipelines are available at https://jugit.fz-juelich.de/IBG-1/ModSim/fluxomics/chrrt.

Almost universally optimal distributed Laplacian solvers via low-congestion shortcuts

In this paper, we refine the (almost) existentially optimal distributed Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS ‘21) into an (almost) universally optimal distributed Laplacian solver. Specifically, when the topology is known (i.e., the Supported-CONGEST model), we show that any Laplacian system on an n-node graph with shortcut qualitytextrm{SQ}(G) can be solved after n^{o(1)} text {SQ}(G) log (1/epsilon ) rounds, where epsilon >0 is the required accuracy. This almost matches our lower bound that guarantees that any correct algorithm on G requires widetilde{Omega }(textrm{SQ}(G)) rounds, even for a crude solution with epsilon le 1/2. Several important implications hold in the unknown-topology (i.e., standard CONGEST) case: for excluded-minor graphs we get an almost universally optimal algorithm that terminates in D cdot n^{o(1)} log (1/epsilon ) rounds, where D is the hop-diameter of the network; as well as n^{o(1)} log (1/epsilon )-round algorithms for the case of textrm{SQ}(G) le n^{o(1)}, which holds for most networks of interest. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique model. In this model, we show the existence of a Laplacian solver with round complexity n^{o(1)} log (1/epsilon ). The unifying thread of these results, and our main technical contribution, is the development of near-optimal algorithms for a novel rho -congested generalization of the standard part-wise aggregation problem, which could be of independent interest.

Reveal: Robustness-aware VNF placement and request scheduling in edge clouds

In the edge cloud network, service providers place virtual network functions (VNFs) in edge clouds to serve users’ requests. Thus, it is essential to consider VNF placement and request scheduling in edge clouds. Existing works often focus on minimizing request completion time or maximizing network throughput to utilize network resources and ensure users’ QoS efficiently. However, they ignore two practical factors: malicious users and failed VNFs, leading to poor network robustness. To this end, this paper studies robustness-aware VNF placement and request scheduling, named Reveal. Specifically, we limit the number of VNFs each user can access and the number of users each VNF can serve to control the influence scope of malicious users and VNF failures. Since placing VNFs is time-consuming and requests arrive dynamically, we solve this problem through two phases: robust VNF placement and assignment, and online request scheduling. For the first phase, we design an efficient knapsack-based rounding algorithm with bounded approximation factors. For online request scheduling, we propose a primal–dual based algorithm with a competitive ratio of 1−ϵ,O(log1/ϵ) where ϵ∈(0,1). Experiment and simulation results show that Reveal can achieve better performance and robustness than other alternatives.

Space Information Network With Joint Virtual Network Function Deployment and Flow Routing Strategy With QoS Constraints

Space information network (SIN) can provide global coverage in 6G network. Furthermore, SIN with network function virtualization (NFV) can achieve flexible deployment of network functions and improve the utilization of resources. In SIN with NFV, network functions can be virtualized into virtual network functions (VNFs). However, in SIN with NFV, the mission flow must satisfy the service function chain (SFC) constraint, i.e., the mission flow must be processed by all VNFs in the predefined order. Furthermore, each VNF can be deployed on multiple physical nodes. Moreover, different kinds of services may have the diverse quality of service (QoS) requirements. In this paper, we investigate the joint VNFs deployment and flow routing strategy (VNF-R) to maximize the number of completed missions with the guaranteed end-to-end latency under SFC constraints in time-varying SINs. Specifically, the problem can be formulated as a mixed integer linear programming (MILP) problem, which is proved to be NP-hard. In order to effectively solve the problem, we propose a novel low-complexity near-optimal penalty successive upper bound minimization rounding LP relaxation iterative rounding (PSUM-R-LRIR) algorithm. The simulation results show that the PSUM-R-LRIR algorithm can achieve near-optimal performance, and our proposed VNF-R scheme significantly outperforms the fixed VNF deployment scheme.

Enhanced AES-256 cipher round algorithm for IoT applications

Objectives: Networks have become a significant mode of communication in recent years. As a result, internet security has become a critical requirement for secure information exchange. Cryptography is used to securely send passwords over large networks. Cryptographic algorithms are sequences of processes used to encipher and decipher messages in a cryptographic system. One of those is the Advanced Encryption Standard (AES), which is a standard for data encryption in hardware and software to hide sensitive and vital information. The main objective is to design an AES system with modifications by the addition of primitive operations which can withstand several attacks and is more efficient.Method: AES works with three different key lengths: 128-bit keys, 192- bit keys, and 256-bit keys. The early rounds of AES have a poor diffusion rate. Better diffusion properties can be brought about by putting in additional operations in the cipher round and key generation algorithm of the conventional AES.Findings: The diffusion characteristics of the conventional AES and the proposed methodology are compared using the avalanche effect. The proposed AES algorithm shows an increased avalanche effect, which proves it to be more secure than the conventional AES. The proposed algorithm is executed on Vivado 2016.2 ISE Design Suite and the results are targeted on Zybo–Zynq Z-7010 AP SoC development board.Novelty: In addition, this paper also proposes an improved AES algorithm that was accomplished by altering the sub-bytes operation. This change was made to make it more reliant on round keys. This algorithm was also extended to a higher key length of 256 bits which makes the algorithm less vulnerable to attacks.

Superfast coloring in CONGEST via efficient color sampling

We present a procedure for efficiently sampling colors in the CONGEST model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to Θ(log⁡n) semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with O(log⁎⁡Δ) complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an O(log⁎⁡Δ)-round CONGEST algorithm for (1+ϵ)Δ-edge coloring when Δ=Ω(log1+1/log⁎⁡n⁡n), and a poly(log⁡log⁡n)-round algorithm for (2Δ−1)-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.

Array Scheduling With Power and Bandwidth Allocation for Simultaneous Multibeam Tracking Low-Angle Targets in a VHF-MIMO Radar

The very high frequency (VHF) radar suffers from the multipath interference and false alarms when tracking low-angle targets. The co-located multiple-input multiple-output (MIMO) radar technique can mitigate these adverse effects with waveform diversity. However, the limited resources restrict the potential of the VHF-MIMO radar. In this paper, an array scheduling with power and bandwidth allocation (ASPBA) strategy is proposed for the VHF-MIMO radar tracking low-angle targets. The predicted posterior Cramér-Rao lower bound (PCRLB) under the multipath interference and measurement origin uncertainty (MOU) effects is derived to serve as the optimization metric. The measurement amplitude information (AI) is incorporated into the PCRLB to facilitate target detection and tracking. It is shown that the ASPBA problem is a mixed integer programming and NP-hard problem, where the array, power, and bandwidth allocation variables are both coupled in the objective and constraints. An efficient three-stage-based solution is proposed for problem-solving. The approximated relationship among three variables is derived using Hölder's inequality, and an auxiliary variable is introduced to describe the array contribution. Thereby, the array scheduling is determined by a heuristic rounding algorithm, and the power and bandwidth allocation are achieved using the approximated relationship. Simulation results confirm the effectiveness and efficiency of the proposed ASPBA strategy, compared with state-of-the-art algorithms. It is also shown that the target height is a main factor that influences the resource allocation results in the low-angle tracking scenario.

On the firefighter problem with spreading vaccination for maximizing the number of saved nodes: the IP model and LP rounding algorithms.

When an infectious disease spreads, how to quickly vaccinate with a limited budget per time step to reduce the impact of the virus is very important. Specifically, vaccination will be carried out in every time step, and vaccinated nodes will no longer be infected. Meanwhile, the protection from vaccination can spread to the neighbors of a vaccinated node. Our goal is to efficiently find optimal and approximation solutions to our problem with various algorithms. In this paper, we first design an integer linear program to solve this problem. We then propose approximation algorithms of (1) Linear programming (LP) deterministic threshold rounding, (2) LP dependent randomized rounding, and (3) LP independent randomized rounding. We prove that the LP independent randomized rounding algorithm has a high probability of finding a feasible solution that gives an approximation ratio of , where a small constant between 0 and 1 reduces the lower bound on the feasibility probability. We also provide experimental results for three different rounding algorithms to show that they perform numerically well in terms of approximation ratios. These analytical and numerical studies allow each individual to adopt the most appropriate approximation algorithm to efficiently resolve the vaccination problem when her reliance on commercial optimization solvers is costly.

Distributed coloring and the local structure of unit-disk graphs

Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph G with at most 4ω(G) colors, where ω(G) is the clique number of G. This improves upon a classical 3-approximation algorithm for this problem, for all unit-disk graphs whose chromatic number significantly exceeds their clique number. When nodes do not know their coordinates in the plane, we give a distributed algorithm in the LOCAL model that colors every unit-disk graph G with at most 5.68ω(G)+1 colors in O(log⁎⁡n) rounds. This algorithm is based on a study of the local structure of unit-disk graphs, which is of independent interest. We conjecture that every unit-disk graph G has average degree at most 4ω(G), which would imply the existence of a O(log⁡n) round algorithm coloring any unit-disk graph G with (approximately) 4ω(G) colors in the LOCAL model. We provide partial results towards this conjecture using Fourier-analytical tools.

Received Signal Strength-Based Wireless Source Localization With Inaccurate Anchor Positions

Received signal strength (RSS)-based wireless localization is easy to implement at low cost. In practice, exact positions of anchors may not be available. This article focuses on determining the location of a source in the presence of inaccurate positions of anchors based on RSS directly. We first use Taylor expansion and a min–max approach to get a maximum likelihood estimator of the coordinates of the source. Then we propose a relaxed semidefinite programming model to circumvent the nonconvexity. We also propose a rounding algorithm considering both inaccurate source locations and inaccurate anchor locations. Simulation results together with analysis are presented to validate the proposed method.

A Water-Filling Primal-Dual Algorithm for Approximating NonLinear Covering Problems

Obtaining strong linear relaxations of capacitated covering problems constitute a major technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities achieves an integrality gap of 2. These inequalities have been exploited in more general environments, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a new general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a $(2+\varepsilon)$-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm, which effectively balances the difficulties given by having different slopes in the cost functions: when some superior portion of an item presents a low slope, it helps to increase the priority with which the inferior portions may be taken. We also present a rounding algorithm with an approximation guarantee of 2. We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting where items can be taken fractionally. For this problem we obtain a $(4+\varepsilon)$-approximation algorithm in polynomial time, almost matching the $4$-approximation known for the classical setting.

A simple optimal contention resolution scheme for uniform matroids

Contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak and Zenklusen, are a class of randomized rounding algorithms for converting a fractional solution to a relaxation for a down-closed constraint family into an integer solution. A CR scheme takes a fractional point x in a relaxation polytope, rounds each coordinate xi independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the constraints. Intuitively, a CR scheme is c-balanced if every element i is selected with probability at least c⋅xi.It is known that general matroids admit a (1−1/e)-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple and explicit monotone CR scheme for uniform matroids of rank k on n elements with a balancedness of 1−(nk)(1−kn)n+1−k(kn)k, and show that this is optimal. As n grows, this expression converges from above to 1−e−kkk/k!. While this asymptotic bound can be obtained by combining previously known results, these require defining an exponential-sized linear program, as well as using random sampling and the ellipsoid algorithm. Our procedure, on the other hand, has the advantage of being simple and explicit. This scheme extends naturally into an optimal CR scheme for partition matroids.

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [ 31 ] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut , for many others, e.g., Max-SAT and Max-DiCut , the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut , Max-2SAT , and Max-DiCut , and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

Exponentially Faster Shortest Paths in the Congested Clique

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain poly(log log n )-round algorithms for the following problems in unweighted undirected n -vertex graphs: ( 1 + ϵ )-approximation of multi-source shortest paths (MSSP) from O (√ n ) sources. (2 + ϵ )-approximation of all pairs shortest paths (APSP). (1 + ϵ , β)-approximation of APSP where β = O (log log n / ϵ ) log log n . These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold t = O ( β / ϵ ) where β = O (log log n / ϵ ) log log n . Handling the long distances is done by devising a new algorithm for computing a sparse (1 + ϵ , β ) emulator with O ( n log log n ) edges. For the short distances, we provide distance-sensitive variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius t , their round complexities get improved from poly (log n ) to poly (log t ). Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.

Optimal Configuration of Array Elements for Hybrid Distributed PA-MIMO Radar System Based on Target Detection

This paper establishes a hybrid distributed phased array multiple-input multiple-output (PA-MIMO) radar system model to improve the target detection performance by combining coherent processing gain and spatial diversity gain. First, the radar system signal model and array space configuration model for the PA-MIMO radar are established. Then, a novel likelihood ratio test (LRT) detector is derived based on the Neyman–Pearson (NP) criterion in a fixed noise background. It can jointly optimize the coherent processing gain and spatial diversity gain of the system by implementing subarray level and array element level optimal configuration at both receiver and transmitter ends in a uniform blocking manner. On this basis, three typical optimization problems are discussed from three aspects, i.e., the detection probability, the effective radar range, and the radar system equipment volume. The approximate closed-form solutions of them are constructed and solved by the proposed quantum particle swarm optimization-based stochastic rounding (SR-QPSO) algorithm. Through the simulations, it is verified that the proposed optimal configuration of the hybrid distributed PA-MIMO radar system offers substantial improvements compared to the other typical radar systems, detection probability of 0.98, and an effective range of 1166.3 km, which significantly improves the detection performance.

Multimedia System Design and Data Storage Optimization Based on Machine Learning Algorithm.

With the advancement of science and technology, digital technology and Internet of Things network technology have been developed rapidly, and multimedia technology has also been widely used. Multimedia formats such as digital TV and elevator posters are shaking up traditional media. At the same time, many media operation models and multimedia technologies are combined to plan operational strategies, determine operational goals, and change the traditional media structure to achieve commercial profits and society benefit. However, due to limitations in the existing operating model or unreasonable technical solutions, it is not easy to maximize the value of multimedia technology. The XML-based database has been submitted, and it will carry out the business requirements of the transaction network and the business platform of the transaction network. Integrated management mechanism is analyzed and applied. The framework design includes parallel quota processing module, update processing module, result processing module, and storage library and database connection management module. The department runs multiple parts of the system together and completes the database. The development of cloud database is based on cloud computing. It can effectively fill the shortcomings and gaps of traditional database storage and processing, and it can also provide high-reciprocity databases to provide storage and management services. It has high reliability. Cloud servers use fair weighted rounding algorithms to achieve load balancing and use the in-memory database Redis to realize terminal data caching. After a comprehensive test of the system, the system can perform all functions normally, and it has good performance and stable operation.