Partitioning Problem Research Articles

A new multi-level algorithm for balanced partition problem on large scale directed graphs

Graph partition is a classical combinatorial optimization and graph theory problem, and it has a lot of applications, such as scientific computing, VLSI design and clustering etc. In this paper, we study the partition problem on large scale directed graphs under a new objective function, a new instance of graph partition problem. We firstly propose the modeling of this problem, then design an algorithm based on multi-level strategy and recursive partition method, and finally do a lot of simulation experiments. The experimental results verify the stability of our algorithm and show that our algorithm has the same good performance as METIS. In addition, our algorithm is better than METIS on unbalanced ratio.

Joint Task Partitioning and User Association for Latency Minimization in Mobile Edge Computing Networks

Mobile edge computing (MEC) is a promising solution to support emerging delay-sensitive mobile applications, such as self-driving, augment/virtual reality, and various Internet of Things (IoT) applications. By deploying MEC servers at network edge, e.g., close to cellular base stations (BSs), the computational tasks generated by these applications can be offloaded to edge nodes (ENs) and be quickly executed there. At the same time, with the projected large number of IoT devices, the communication and computational resources allocated to each user can be quite limited, making it challenging to provide low-latency MEC services. In this paper, we investigate the problem of task partitioning and user association in an MEC system, aiming to minimize the average latency of all users. We assume that each task can be partitioned into multiple subtasks that can be executed on local devices (e.g., vehicles), MEC servers, and/or cloud servers; each user can be associated with one of the nearby ENs. The subtasks can be independent of or dependent on each other. For each case, we formulate the joint optimization of task partitioning ratios and user association as a mixed integer programming problem. Each problem is solved by decomposing it into two subproblems. The lower-level subproblem is task partitioning under a given user association, which can be solved optimally. The higher-level subproblem is user association, we propose a dual decomposition-based approach and a matching-based approach to derive near-optimal solutions. Simulation results show that compared to benchmark schemes, the proposed schemes reduce the average latency by about 50% and 40% for the cases of independent and dependent subtasks, respectively.

Evaluating the deHoffmann‐Teller Cross‐Shock Potential at Real Collisionless Shocks

AbstractShock waves are common in the heliosphere and beyond. The collisionless nature of most astrophysical plasmas allows for the energy processed by shocks to be partitioned amongst particle sub‐populations and electromagnetic fields via physical mechanisms that are not well understood. The electrostatic potential across such shocks is frame dependent. In a frame where the incident bulk velocity is parallel to the magnetic field, the deHoffmann‐Teller frame, the potential is linked directly to the ambipolar electric field established by the electron pressure gradient. Thus measuring and understanding this potential solves the electron partition problem, and gives insight into other competing shock processes. Integrating measured electric fields in space is problematic since the measurements can have offsets that change with plasma conditions. The offsets, once integrated, can be as large or larger than the shock potential. Here we exploit the high‐quality field and plasma measurements from NASA's Magnetospheric Multiscale mission to attempt this calculation. We investigate recent adaptations of the deHoffmann‐Teller frame transformation to include time variability, and conclude that in practice these face difficulties inherent in the 3D time‐dependent nature of real shocks by comparison to 1D simulations. Potential estimates based on electron fluid and kinetic analyses provide the most robust measures of the deHoffmann‐Teller potential, but with some care direct integration of the electric fields can be made to agree. These results suggest that it will be difficult to independently assess the role of other processes, such as scattering by shock turbulence, in accounting for the electron heating.

A heuristic scheme for multivariate set partitioning problems with application to classifying heterogeneous populations for multiple binary attributes

We provide a novel heuristic approach to solve a class of multivariate set partitioning problems in which each item is characterized by three attribute values. The scheme first identifies a series of orderings of the items and then solves a corresponding sequence of shortest path problems. We provide theoretical findings on the structure of an optimal solution that motivate the design of the proposed heuristic scheme. The proposed algorithm runs in polynomial-time and is independent of the number of groups in the partition, making it more efficient than existing algorithms. To measure the performance of our solutions, we construct bounds for special instances which allow us to provide optimality gaps. We conduct an extensive numerical experiment in which we solve a large number of problem instances and show that our proposed approach converges to the global optimal solution in the vast majority of cases and in the case it does not, it yields very low optimality gaps. We demonstrate our findings with an application in the context of classifying a large heterogeneous population as positive or negative for multiple binary attributes as efficiently as possible. We conduct a case study on the screening of three of the most prevalent sexually transmitted diseases in the United States. The resulting solutions are shown to be within 2.6% of optimality and lead to a 26% cost saving over current screening practices.

Stability aware spatial cut of metapopulations ecological networks

Ecological complex networks are common in the study of patched ecological systems where evolving populations interact within and among the patches. The loss of the dispersal connections between patches due to reasons such as erosion of migration corridors and road construction can cause an undesirable partitioning of such networks resulting in instability or negative impact on the metapopulations. A partitioning or spatial cut that is aware of the stability of the dynamics in the resulting daughter sub-networks can be an effective tool in dealing with the situation like proposing road alignment through a metapopulations network. This paper provides some mathematical conditions along with an heuristic graph partitioning algorithm that can help in finding ecologically suitable partitions of the metapopulations networks. Our study noted the crucial role of network connectivity (measured by Fiedler value) in stabilizing the metapopulations. That is, a sufficiently connected metapopulations network along with constrained internal patch dynamics has stable dynamics around its homogeneous co-existential equilibrium solution. With the considered mathematical model in this paper, network partitioning does not alter the internal patch dynamics around its homogeneous equilibrium point, but it can change the connectivity levels in the partitioned subnetworks. Thus, the proposed partitioning problem for an already stable metapopulations network is reduced to finding its subnetworks with desirable connectivity levels.

Decomposing subcubic graphs into claws, paths or triangles

AbstractLet be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph into graphs taken from any nonempty . The problem is known to be ‐complete for any possible choice of in general graphs. In this paper, we assume that the input graph is subcubic (i.e., all its vertices have degree at most 3), and study the computational complexity of the problem of partitioning its edge set for any choice of . We identify all polynomial and ‐complete problems in that setting.

Multi-Attribute Community Detection in International Trade Network

Understanding the structure of communities in a network has a great importance in the economic analysis. Communities are indeed characterized by specific properties, that are different from those of both the individual nodes and the whole network, and they can affect various processes on the network. In the International Trade Network, community detection aims to search sets of countries (or of trade sectors) which have a high intra-cluster connectivity and a low inter-cluster connectivity. In general, exchanges among countries occur according to preferential economic relationships ranging over different sectors. In this paper, we combine community detection with specific topological indicators, such as centrality measures. As a result, a new weighted network is constructed from the original one, in which weights are determined taking into account all the topological indicators in a multi-criteria approach. To solve the resulting Clique Partitioning Problem and find homogeneous group of nations, we use a new fast algorithm, based on quick descents to a local optimal solution. The analysis allows to cluster countries by interconnections, economic power and intensity of trade, giving an important overview on the international trade patterns.

An improved approximation algorithm for the minimum common integer partition problem

Given a collection of multisets {X1,X2,…,Xk} (k≥2) of positive integers, a multiset S is a common integer partition for them if S is an integer partition of every multiset Xi,1≤i≤k. The minimum common integer partition (k-MCIP) problem is defined as to find a CIP for {X1,X2,…,Xk} with the minimum cardinality. We present a 65-approximation algorithm for the 2-MCIP problem, improving the previous best algorithm of performance ratio 54 designed by Chen et al. in 2006. We then extend it to obtain an absolute 0.6k-approximation algorithm for k-MCIP when k is even (when k is odd, the approximation ratio is 0.6k+0.4).

Shortest Path Method for Hardware/Software Partitioning Problems

Smart systems are becoming more present in every aspect of our daily lives. The main component of such systems is an embedded system; this latter assures the collection, the treatment, and the transmission of the accurate information in the right time and for the right component. Modern embedded systems are facing several challenges; the objective is to design a system with high performance and to decrease the cost and the development time. Consequently, some robust methodologies like the Codesign were developed to fulfill those requirements. The most important step of the Codesign is the partitioning of the systems' functionalities between a hardware set and a software set. This article deals with this problem and uses a heuristic approach based on shortest path optimizations to solve the problem. The aim is to minimize the total hardware area and to respect a constraint on the overall execution time of the system. Experiments results demonstrate that the proposed method is very fast and gives better results compared to the genetic algorithm.

Model abstraction for discrete-event systems by binary linear programming with applications to manufacturing systems.

Model abstraction for finite state automata is helpful for decreasing computational complexity and improving comprehensibility for the verification and control synthesis of discrete-event systems (DES). Supremal quasi-congruence equivalence is an effective method for reducing the state space of DES and its effective algorithms based on graph theory have been developed. In this paper, a new method is proposed to convert the supremal quasi-congruence computation into a binary linear programming problem which can be solved by many powerful integer linear programming and satisfiability (SAT) solvers. Partitioning states to cosets is considered as allocating states to an unknown number of cosets and the requirement of finding the coarsest quasi-congruence is equivalent to using the least number of cosets. The novelty of this paper is to solve the optimal partitioning problem as an optimal state-to-coset allocation problem. The task of finding the coarsest quasi-congruence is equivalent to the objective of finding the least number of cosets. Then the problem can be solved by optimization methods, which are respectively implemented by mixed integer linear programming (MILP) in MATLAB and binary linear programming (BLP) in CPLEX. To reduce the computation time, the translation process is first optimized by introducing fewer decision variables and simplifying constraints in the programming problem. Second, the translation process formulates a few techniques of converting logic constraints on finite automata into binary linear constraints. These techniques will be helpful for other researchers exploiting integer linear programming and SAT solvers for solving partitioning or grouping problems. Third, the computational efficiency and correctness of the proposed method are verified by two different solvers. The proposed model abstraction approach is applied to simplify the large-scale supervisor model of a manufacturing system with five automated guided vehicles. The proposed method is not only a new solution for the coarsest quasi-congruence computation, but also provides us a more intuitive understanding of the quasi-congruence relation in the supervisory control theory. A future research direction is to apply more computationally efficient solvers to compute the optimal state-to-coset allocation problem.

Fog radio access network optimization for 5G leveraging user mobility and traffic data

The surging data traffic and dynamic user mobility in 5G networks have posed significant demands for mobile operators to increase data processing capacity and ensure user handover quality. Specifically, a cost-effective and quality-aware radio access network (RAN) is in great necessity. With the emergence of fog-computing-based RAN architecture (Fog-RAN), the data processing units (BBUs) can be separated from base stations (RRHs) and hosted in distributed fog servers, where each server accommodates a community of RRHs to handle data traffic and user handover. The key problem in Fog-RAN optimization is how to cluster complementary RRHs into communities and allocate adequate numbers of BBUs for the fog servers, since real-world traffic and mobility patterns are highly dynamic to model, and it is not trivial to find an optimal RRH clustering and BBU allocation scheme from potentially enormous numbers of candidates. In this work, we propose a data-driven framework for cost-effective and quality-aware Fog-RAN optimization. In the RRH clustering phase, we build a weighted graph model to characterize user mobility patterns across RRHs, and propose a size-constrained community detection (SCUD) algorithm to cluster RRHs into communities with frequent internal handover events. In the BBU allocation phase, we formulate BBU allocation in each community fog server as a set partitioning problem, and propose a column-reduced integer programming (CLIP) algorithm to find optimal BBU allocation schemes that maximize BBU utilization rate. Evaluations using two large-scale real-world datasets collected from Ivory Coast and Senegal show that compared to the traditional RAN architecture, our framework effectively reduces the average handover overhead to 12.8% and 27.3%, and increases the average BBU utilization rate to 49.7% and 52.3% in both cities, respectively, which consistently outperforms the state-of-the-art baseline methods.

Computational study of a branching algorithm for the maximum [formula omitted]-cut problem

This work considers the graph partitioning problem known as maximum k-cut. It focuses on investigating features of a branch-and-bound method to obtain global solutions. An exhaustive experimental study is carried out for the two main components of a branch-and-bound algorithm: Computing bounds and branching strategies. In particular, we propose the use of a variable neighborhood search metaheuristic to compute good feasible solutions, the k-chotomic strategy to split the problem, and a branching rule based on edge weights to select variables. Moreover, we analyze a linear relaxation strengthened by semidefinite-based constraints, a cutting plane algorithm, and node selection strategies. Computational results show that the resulting method outperforms the state-of-the-art approach and discovers the solution of several instances, especially for problems with k≥5.

Modelling of Microstructure Evolution during Laser Processing of Intermetallic Containing Ni-Al Alloys

There is a growing interest in laser melting processes, e.g., for metal additive manufacturing. Modelling and numerical simulation can help to understand and control microstructure evolution in these processes. However, standard methods of microstructure simulation are generally not suited to model the kinetic effects associated with rapid solidification in laser processing, especially for material systems that contain intermetallic phases. In this paper, we present and employ a tailored phase-field model to demonstrate unique features of microstructure evolution in such systems. Initially, the problem of anomalous partitioning during rapid solidification of intermetallics is revisited using the tailored phase-field model, and the model predictions are assessed against the existing experimental data for the B2 phase in the Ni-Al binary system. The model is subsequently combined with a Potts model of grain growth to simulate laser processing of polycrystalline alloys containing intermetallic phases. Examples of simulations are presented for laser processing of a nickel-rich Ni-Al alloy, to demonstrate the application of the method in studying the effect of processing conditions on various microstructural features, such as distribution of intermetallic phases in the melt pool and the heat-affected zone. The computational framework used in this study is envisaged to provide additional insight into the evolution of microstructure in laser processing of industrially relevant materials, e.g., in laser welding or additive manufacturing of Ni-based superalloys.

Fixed-parameter tractability for the Tree Assembly problem

Calculating the “distance” between two given objects with respect to a designated “editing” operation is a hot research area in bioinformatics, where the “distance” is always defined as the minimum number of the “editing” operations required to transform one object into the other one. One of the famous problems in the area is the Minimum Common String Partition problem, which is the simplified variant of the Minimum Tree Cut/Paste Distance problem. Within the paper, we consider another simplified variant of the Minimum Tree Cut/Paste Distance problem, named Tree Assembly problem, of which the edge-deletion operations are specified. More specifically, the Tree Assembly problem aims to transform a given forest into a given tree by edge-addition operations only. In our investigations, we present a fixed-parameter algorithm with runtime 2O(klog⁡k)nO(1) for the Tree Assembly problem, where k is the number of trees in the given forest, and n is the number of nodes in the given tree and forest. Additionally, we give a polynomial time algorithm for a restricted variant of the problem.

The time‐consistent dial‐a‐ride problem

AbstractIn the context of door‐to‐door transportation of people with disabilities, service quality considerations such as maximum ride time and service time consistency are critical requirements. To identify a good trade‐off between these considerations and economic objectives, we define a new variant of the multiperiod dial‐a‐ride problem called the time‐consistent dial‐a‐ride problem. A transportation planning is supposed to be time consistent if for each passenger, the same service time is used all along the planning horizon. However, considering the numerous variations in transportation demands over a week, designing consistent plans for all passengers can be too expensive. It is therefore necessary to find a compromise solution between costs and time‐consistency objectives. The time‐consistent dial‐a‐ride problem is solved using an epsilon‐constraint approach to illustrate the trade‐off between these two objectives. It computes an approximation of the Pareto front, using a matheuristic framework that combines a large neighbourhood search with the solution of set partitioning problems. This approach is benchmarked on time‐consistent vehicle routing problem literature instances. Experiments are also conducted in the context of door‐to‐door transportation for people with disabilities, using real data. These experiments support managerial insights regarding the inter‐relatedness of costs and quality of service.

The biclique partitioning polytope

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.

A branch-and-price procedure for clustering data that are graph connected

This paper studies the Graph-Connected Clique-Partitioning Problem (GCCP), a clustering optimization model in which units are characterized by both individual and relational data. This problem, introduced by Benati, Puerto, and Rodríguez-Chía (2017) under the name of Connected Partitioning Problem, shows that the combination of the two data types improves the clustering quality in comparison with other methodologies. Nevertheless, the resulting optimization problem is difficult to solve; only small-sized instances can be solved exactly, large-sized instances require the application of heuristic algorithms. In this paper we improve the exact and the heuristic algorithms previously proposed. Here, we provide a new Integer Linear Programming (ILP) formulation, that solves larger instances, but at the cost of using an exponential number of variables. In order to limit the number of variables necessary to calculate the optimum, the new ILP formulation is solved implementing a branch-and-price (B&P) algorithm. The resulting pricing problem is itself a new combinatorial model: the Maximum-weighted Graph-Connected Single-Clique problem (MGCSC), that we solve testing various Mixed Integer Linear Programming (MILP) formulations and proposing a new fast “random shrink” heuristic. In this way, we are able to improve the previous algorithms: The B&P method outperforms the computational times of the previous MILP algorithms and the new random shrink heuristic, when applied to GCCP, is both faster and more accurate than the previous heuristic methods. Moreover, the combination of column generation and random shrink is itself a new MILP-relaxed matheuristic that can be applied to large instances too. Its main advantage is that all heuristic local optima are combined together in a restricted MILP, consisting in the application of the exact B&P method but solving heuristically the pricing problem.

Solving the minimum convex partition of point sets with integer programming

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map – which represents, say, a geographical area – into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp ). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.

Group Testing on a Network

Group testing---where multiple samples are tested together using a single test kit and individual tests are performed only for samples in positive groups---is a popular strategy to optimize the use of testing resources. We investigate how to effectively group samples for testing based on a transmission network. We formalize the group assembling problem as a graph partitioning problem, where the goal is to minimize the expected number of tests needed to screen the entire network. The problem is shown to be computationally hard and thus we focus on designing effective heuristics for it. Using realistic epidemic models on real contact networks, we show that our approaches save up to 33\% of resources---compared to the best baseline---at 4\% prevalence, are still effective at higher prevalence, and are robust to missing transmission data.

Enhancing Balanced Graph Edge Partition with Effective Local Search

Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in power-law graphs than vertex partition and thereby has been more widely adopted as the default partition strategy by existing graph systems. The graph edge partition problem, which is to split the edge set into multiple balanced parts with the objective of minimizing the total number of copied vertices, has been widely studied from the view of optimization and algorithms. In this paper, we study local search algorithms for this problem to further improve the partition results from existing methods. More specifically, we propose two novel concepts, namely adjustable edges and blocks. Based on these, we develop a greedy heuristic as well as an improved search algorithm utilizing the property of max-flow model. To evaluate the performance of our algorithms, we first provide adequate theoretical analysis in terms of approximation quality. We significantly improve the previous known approximation ratio for this problem. Then we conduct extensive experiments on a large number of benchmark datasets and state-of-the-art edge partition strategies. The results show that our proposed local search framework can further improve the quality of graph partition by a wide margin.