Partitioning Problem Research Articles

Algorithmic obstructions in the random number partitioning problem

Algorithmic obstructions in the random number partitioning problem

Peak‐valley period partition and abnormal time correction for time‐of‐use tariffs under daily load curves based on improved fuzzy c‐means

AbstractPeak‐valley period partition of load curve is a key aspect of time‐of‐use (ToU) tariff to improve power load characteristics, such as shifting peak loads towards valley time periods. Fuzzy clustering algorithm is an effective and popular method commonly used to solve the peak‐valley period partition of load curves, but it still encounters the difficulty of dividing some data within the boundary regions of different time periods. Therefore, this paper presents a peak‐valley period partition and abnormal time correction scheme for ToU tariffs under typical daily load curves based on improved fuzzy C‐means (FCM) clustering algorithm. In order to improve the accuracy of peak‐valley period partition, modified fuzzy membership functions are proposed to improve the initialization of FCM clustering, and a loss function‐based method is presented for calculating the fuzzy parameters of those membership functions. To resolve the problem of abnormal time partitioning within the boundaries of different time periods, an abnormal time period recognition model and a correction model based on fuzzy subsethood are proposed to obtain the final corrected peak‐valley time period partitioning results. On the MATLAB R2020b platform, the effectiveness of the proposed method is verified through two real daily load curves with a time resolution of 5 min.

Methods for determining cycles of a specific length in undirected graphs with edge weights

In this paper, we consider the {{mathcal{N}mathcal{P}}}-hard problem of determining fixed-length cycles in undirected edge-weighted graphs. Two solution methods are proposed, one based on integer programming (IP) and one that uses bespoke local search operators. These methods are executed under a common algorithmic framework that seeks to partition problem instances into a series of smaller sub-problems. Large-scale empirical tests indicate that the local search algorithm is generally preferable to IP, even with short run times. However, it can still produce suboptimal solutions, even with relatively small graphs.

RIS Subarray Optimization With Reinforcement Learning for Green Symbiotic Communications in Internet of Things

Symbiotic communications have been deemed as a critical technology for Internet of things (IoT) communications owing to its high spectrum and energy efficiency. Reconfigurable intelligent surface (RIS), which can tune wireless transmission channels by manipulating incident waves through the corresponding electromagnetic elements, is a promising enabler of various symbiotic communications scenarios in IoT. However, when the full electromagnetic elements of RIS are activated, system capacity will be improved and energy efficiency will be reduced inevitably, also with undesirable power consumption. To address this issue, an intelligent dynamic subarray RIS framework based on deep reinforcement learning (DRL) has been proposed. The key idea is to divide RIS electromagnetic elements into several groups and optimize power amplifier factor, independent phase shifts to improve the system energy efficiency under the premise of user’s basic requirements. In particular, we formulate a hybrid optimization problem of RIS subarray partition and beamforming to maximize system energy efficiency. It can be proved that this hybrid optimization is a mixed non-convex integer programming problem. To solve this issue, we proposed a comprehensive DRL framework including two parts, i.e., (1) a Markov decision process (MDP) to model the subarray partition design, amplitude and phase shifts of RIS, and (2) an active RIS subarray optimization scheme based on deep deterministic policy gradient. Numerical results have demonstrated that, compared with the conventional fully-connected RIS, the system energy efficiency can be significantly improved.

Complexity and algorithm of setting optimal location for data sink in real-time NOMA-based IIoTs

Real-time performance is one of the most vital metrics in Non-Orthogonal Multiple Access (NOMA) based Industrial Internet of Things (IIoTs) applications. Since the relative geographic relationship between data sink and wireless sensors affects the transmission parallelism and thus the real-time performance, setting a reasonable location for the data sink is thus a feasible way to high real-time performance for applications where the locations of wireless sensors are fixed. In this paper, we consider how to find an optimal location for the data sink to minimize average access delay. We first formulate the problem and prove it to be NP-Complete by presenting an original reduction proof from classic set partition problem. Second, by tightening the NOMA decoding constraint of the original problem, a heuristic algorithm, which is designed based on Apollonian Circle Theorem and Dilworth Theorem, obtains a feasible location for the data sink. Simulation results reveal that the real-time performance loss of the heuristic algorithm is no more than 50% of the best one. Compared with the classic TDMA scheme, the real-time performance increases by more than 60% for some typical settings, and it can even reach 70% for the linear network topology, owing to the full exploitation of NOMA parallelism.

Task Modules Partitioning, Scheduling and Floorplanning for Partially Dynamically Reconfigurable Systems with Heterogeneous Resources

Some field programmable gate arrays (FPGAs) can be partially dynamically reconfigurable with heterogeneous resources distributed on the chip. FPGA-based partially dynamically reconfigurable system (FPGA-PDRS) can be used to accelerate computing and improve computing flexibility. However, the traditional design of FPGA-PDRS is based on manual design. Implementing the automation of FPGA-PDRS needs to solve the problems of task modules partitioning, scheduling, and floorplanning on heterogeneous resources. Existing works only partly solve problems for the automation process of FPGA-PDRS or model homogeneous resources for FPGA-PDRS. To better solve the problems in the automation process of FPGA-PDRS and narrow the gap between algorithm and application, in this paper, we propose a complete workflow including three parts: pre-processing to generate the lists of task module candidate shapes according to the resource requirements, exploration process to search the solution of task modules partitioning, scheduling, and floorplanning, and post-optimization to improve the floorplan success rate. Experimental results show that, compared with state-of-the-art work, the pre-processing process can reduce the occupied area of task modules by 6% on average; the proposed complete workflow can improve performance by 9.6%, and reduce communication cost by 14.2% with improving the resources reuse rate of the heterogeneous resources on the chip. Based on the solution generated by the exploration process, the post-optimization process can improve the floorplan success rate by 11%.

What is a decision problem?

This paper presents a general framework about what is a decision problem. Our motivation is related to the fact that decision analysis and operational research are structured (as disciplines) around classes of methods, while instead we should first characterise the decision problems our clients present us. For this purpose, we define a decision problem as a set partitioning problem and we introduce a new framework, independent from any existing method, based upon primitives provided by (or elicited from) the client and four fundamental problem statements. We show that constructing the set to partition is a decision problem itself and that the types of primitives are finite. The result is that the number of archetypal decision problems are finite and so the archetypal decision support methods.

Reduction of Search Space for the Mean Partition Problem

The contributions of this paper are threefold. First, it conducts a formal comparison of the primary approaches to consensus clustering, using the concepts of agreement and consent. Secondly, it presents theoretical evidence justifying the preference for mean-based methods, which rely on consent, over other agreement-based procedural methods like the q-rule, which are now mostly used as quality evaluators in practice. Thirdly, the paper computes the exact reduction achieved by criteria available in existing literature to assess the quality of mean-based consensus solutions and reduce the search space’s size. Finally, it compiles the regions where consensus functions associated with well-known dissimilarity measures, such as the Mirkin metric and Variation of Information, accumulate their consensus solutions.

Quay partitioning problem

AbstractIn this paper, we introduce the quay partitioning problem (QPP), that is, a problem of partitioning quay length into berths for minimum ship waiting time. Such a problem arises when designing the terminal layout. Two schemes of quay layout are considered: with at most one ship in a berth and with at most two ships in a berth. Ship arrival times, service times, lengths, and weights are given. We show that QPP is NP‐hard. The two versions of QPP are formulated as mixed integer linear programs (MIPs). Scalability of solving QPPs as MIPs is studied. We investigate, analytically and in computational experiments, features of the QPP solutions such as (i) changes in solution quality when one long berth length is used versus choosing various berth lengths flexibly, (ii) what lengths of berths are chosen when ship lengths mixture is changing, (iii) what is the impact of congestion on the chosen berth lengths, and (iv) how much is one quay layout scheme better than the other.

Analysis and Optimization of Cache-Enabled mmWave HetNets With Integrated Access and Backhaul

In millimeter wave heterogeneous networks with integrated access and backhaul (mABHetNets), a considerable part of spectrum resources are occupied by the backhaul link, which limits the performance of the access link. In order to overcome such backhaul “ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spectrum occupancy</i> ”, we introduce cache into mABHetNets. Caching popular files at small base stations (SBSs) can offload the backhaul traffic and transfer spectrum from the backhaul link to the access link. To approach the optimal performance of cache-enabled mABHetNets, we first analyze the signal-to-interference-plus-noise ratio distribution and derive the average potential throughput (APT) expression by stochastic geometric tools. Then, based on our analytical work, we formulate a joint optimization problem of cache decision and spectrum partition to maximize the APT. Inspired by the block coordinate descent (BCD) method, we propose a joint cache decision, spectrum partition and power allocation (JCSPA) algorithm to approach the optimal solution. Numerical results show the effectiveness and enhancement of the proposed algorithm. Besides, we verify the APT under different parameters and find that the introduction of cache facilitates the transfer of backhaul spectrum to access link. Jointly deploying appropriate caching capacity at SBSs and performing specified spectrum partition can bring up about 90% APT gain in mABHetNets.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

A biased random-key genetic algorithm for the minimum quasi-clique partitioning problem

A biased random-key genetic algorithm for the minimum quasi-clique partitioning problem

SHM-NDN: A seamless hybrid mobility management scheme for named data mobile ad hoc networks

Named Data Networking (NDN), a counterpart of address-centric networking resolves consumer mobility issues to some extent, however, it confronts excessive bandwidth consumption and network congestion problems in Mobile Adhoc Networks (MANET) due to an inefficient default request re-transmission mechanism. The producer mobility in NDN-based MANET, on the other hand, poses even worse challenges owing to the continuous flooding of location advertisement messages in the network. This paper aims to address the aforementioned producer and consumer mobility issues in MANET and proposes SHM-NDN, a seamless hybrid mobility management scheme for NDN-based MANETs. The proposed scheme features new packet types and extends the functionality of wireless mobile FIB data structure to support efficient mobility management. Moreover, the SHM-NDN proposes a path recovery mechanism to deal with the network partitioning problem caused by a low signal-to-noise ratio and high congestion rate. Through an extensive simulation study, we demonstrate that SHM-NDN achieves on average 12.2% less bandwidth and energy consumption compared to CBILEM. Similarly, SHM-NDN also attains on average 25% less bandwidth and energy consumption compared to EEIF.

Subnetwork constraints for tighter upper bounds and exact solution of the clique partitioning problem

We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.

Universal Quantum Optimization with Cold Atoms in an Optical Cavity.

Cold atoms in an optical cavity have been widely used for quantum simulations of many-body physics, where the quantum control capability has been advancing rapidly in recent years. Here, we show the atom cavity system is universal for quantum optimization with arbitrary connectivity. We consider a single-mode cavity and develop a Raman coupling scheme by which the engineered quantum Hamiltonian for atoms directly encodes number partition problems. The programmability is introduced by placing the atoms at different positions in the cavity with optical tweezers. The number partition problem solution is encoded in the ground state of atomic qubits coupled through a photonic cavity mode, which can be reached by adiabatic quantum computing. We construct an explicit mapping for the 3-SAT and vertex cover problems to be efficiently encoded by the cavity system, which costs linear overhead in the number of atomic qubits. The atom cavity encoding is further extended to quadratic unconstrained binary optimization problems. The encoding protocol is optimal in the cost of atom number scaling with the number of binary degrees of freedom of the computation problem. Our theory implies the atom cavity system is a promising quantum optimization platform searching for practical quantum advantage.

Robust transshipment problem under consistent flow constraints

AbstractIn this article, we study robust transshipment under consistent flow constraints. We consider demand uncertainty represented by a finite set of scenarios and characterize a subset of arcs as so‐called fixed arcs. In each scenario, we require an integral flow that satisfies the respective flow balance constraints. In addition, on each fixed arc, we require equal flow for all scenarios. The objective is to minimize the maximum cost occurring among all scenarios. We show that the problem is strongly ‐complete on acyclic digraphs by a reduction from the ‐Sat problem. Furthermore, we prove that the problem is weakly ‐complete on series‐parallel digraphs by a reduction from a special case of the Partition problem. If in addition the number of scenarios is constant, we observe the pseudo‐polynomial‐time solvability of the problem. We provide poly‐nomial‐time algorithms for three special cases on series‐parallel digraphs. Finally, we present a polynomial‐time algorithm for pearl digraphs.

Improved lower bound on the on-line chain partitioning of semi-orders with representation

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains in the partition. The on-line chain partitioning problem involves finding the minimal number of chains needed by an on-line algorithm. Chrobak and Ślusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most 3w−2 chains, where w is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most 2w−1 chains, over 30 years later, it remains unknown whether this is an optimal algorithm. In this paper, we improve upon previously known bounds and show that any on-line algorithm can be forced to use ⌈32w⌉ chains to partition an order presented in the form of its unit interval representation. As a consequence, we completely solve the problem for w=3. Lastly, we show that loosening the restriction from unit intervals to proper intervals in the bandwidth variant allows us to improve the lower bound by w/3.

Dynamic community partitioning for e-commerce last mile delivery with time window constraints

Community logistics (CL) is a recently proposed delivery strategy designed to deal with e-commerce last-mile delivery scheduling by dynamically assigning vehicles to designated delivery regions partitioned into “communities”. Since optimizing vehicle routes is not mandatory in the CL spectrum, the delivery solution format and optimization process can be greatly simplified. Nevertheless, abandoning vehicle routes means vehicle arrival time at each customer specified delivery destination is unknown, resulting in the inability of handling time window constraints of e-commerce orders. To expand the application scope of CL, this study introduces community time window, an aggregation of identical or adjacent order time windows. Once the community time window for a delivery community is satisfied, all orders in this community can be received within designated time windows without determining vehicle routes. With this new concept, the application range of the CL is extended to e-commerce last mile delivery contexts where order time window constraints are considered. A dynamic community partitioning problem with the time window is presented based on the Markov decision process (MDP). An efficient heuristic solution framework based on policy function approximation is proposed to solve the MDP model. Numerical results show that the CL is very effective in dealing with the time window constraints of e-commerce orders.

Topologically protected Grover's oracle for the partition problem

Topologically protected Grover's oracle for the partition problem

Minimizing properties of networks via global and local calibrations

AbstractIn this note, we prove that minimal networks enjoy minimizing properties for the length functional. A minimal network is, roughly speaking, a subset of composed of straight segments joining at triple junctions forming angles equal to ; in particular such objects are just critical points of the length functional a priori. We show that a minimal network : (i) minimizes mass among currents with coefficients in an explicit group (independent of ) having the same boundary of , (ii) identifies the interfaces of a partition of a neighborhood of solving the minimal partition problem among partitions with same boundary traces. Consequences and sharpness of such results are discussed. The proofs reduce to rather simple and direct arguments based on the exhibition of (global or local) calibrations associated to the minimal network.