Distributed Submodular Minimization and Motion Planning Over Discrete State Space

Submodular function minimization is a polynomially solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have high-order polynomial time complexity. In many applications the objective function is locally defined in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every valued constraint satisfaction problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally defined submodular functions. From a theoretical point of view, this brings to light a previously unknown connection between submodularity and soft arc consistency.

We consider a ground set E and a submodular function acting on it. We first propose a “set multicovering” problem in which the value (price) of any is . We show that the linear program (LP) of this problem can be directly solved by applying a submodular function minimization (SFM) algorithm on the dual LP. However, the main results of this study concern prize‐collecting multicovering with submodular pricing, that is, a more general and more difficult “multicovering” problem in which elements can be left uncovered by paying a penalty. We formulate it as a large‐scale LP (with variables representing all subsets of E) that could be tackled by column generation (CG; for a CG algorithm for “set‐covering” problems with submodular pricing). However, we do not solve this large‐scale LP by CG, but we solve it in polynomial time by exploiting its integrality properties. More exactly, after appropriate restructuring, the dual LP can be transformed into an LP that has been thoroughly studied (as a primal) in the SFM theory. Solving this LP reduces to optimizing a strong map of submodular functions. For this, we use the Fleischer–Iwata framework that optimizes all these functions within the same asymptotic running time as solving a single SFM, that is, in , where and γ is the complexity of one submodular evaluation. Besides solving the problem, the proposed approach can be useful to (a) simultaneously find the best solution of up to functions under strong map relations in time, (b) perform sensitivity analysis in very short time (even at no extra cost), and (c) reveal combinatorial insight into the primal–dual optimal solutions. We present several potential applications throughout the paper, from production planning to combinatorial auctions.

Bisubmodular functions are a natural "directed", or "signed", extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata's fully combinatorial version of IFF to BSFM.

A novel approach using transient chaotic neural network (TCNN) and self-organizing feature map (SOFM) process to solve large-scale combinatorial optimization problems has been proposed. With the clustering function of self-organizing feature map, the computational cost of a large-scale combinatorial optimization problem solved by TCNN is reduced. Numerical simulation of TSP shows that the proposed method is effective to solve large-scale optimization problems.

In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2−ϵ), (1−1/e−ϵ) of approximation ratios and O(nϵlog(logBϵ)logk), O(nϵlogB), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.

This article critically investigates the limitations of the simulated annealing algorithm using probabilistic bits (pSA) in solving large-scale combinatorial optimization problems. The study begins with an in-depth analysis of the pSA process, focusing on the issues resulting from unexpected oscillations among p-bits. These oscillations hinder the energy reduction of the Ising model and thus obstruct the successful execution of pSA in complex tasks. Through detailed simulations, we unravel the root cause of this energy stagnation, identifying the feedback mechanism inherent to the pSA operation as the primary contributor to these disruptive oscillations. To address this challenge, we propose two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA). These algorithms are designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems. On the 16 benchmarks from 800 to 5000 nodes, the proposed methods improve the normalized cut value from 0.8 to 98.4% on average in comparison with the conventional pSA.

Earthquake rupture forecasting is a critical component of seismic hazard analysis and requires identifying physically plausible rupture paths across complex fault networks. At its core, this task could be formulated as a large-scale combinatorial optimization problem, which involves selecting an optimal subset of fault segments from a set of candidates. Such problems pose significant challenges for traditional algorithms, as the number of admissible multi-fault ruptures grows combinatorially. Current operational workflows rely on locally applied plausibility filters. While computationally efficient, such greedy local heuristics risk excluding globally competitive rupture scenarios, particularly in regions with dense fault connectivity and competing rupture pathways. This work investigates whether quantum annealing hardware can serve as a sampling accelerator for exploring the ensemble of physically plausible ruptures beyond what is currently accessible to classical approaches. Designing a problem formulation that remains scientifically meaningful while respecting the constraints of current quantum hardware (size, noise, etc.) is nontrivial, and naïve encodings often collapse under these limitations. We encode rupture plausibility modeling as low energy solutions to a quadratic unconstrained binary optimization (QUBO) problem defined on a fault-network middle graph. Binary variables represent activated fault segments, while local interaction terms encode Coulomb stress transfer (as a proxy for pairwise rupture compatibility), continuity preferences, and branching behavior. Here, we adopt a classical-quantum hybrid workflow: the quantum annealer is used to sample globally competitive rupture candidates, while classical post-processing implements physical constraints to filter out non-physical ruptures. The workflow is demonstrated on a subset of the fault network used in the 2023 USGS National Seismic Hazard Model including over 200 fault segments, using both simulated thermal and quantum annealing on D-Wave hardware. Results show that our hardware-aware formulation and conditioning enable robust sampling on comparatively large fault-network instances. Views on quantum computing are polarized: some overstate its power, while others dismiss it as impractical. Our results help bridge this by establishing a concrete, testable pathway for integrating quantum annealing into rupture-modeling workflows on existing purpose-built quantum devices.—---This work is supported by the ICSC National Research Centre for High Performance Computing, Big Data and Quantum Computing (CN00000013, CUP D53C22001300005) within the European Union-NextGenerationEU program (National Recovery and Resilience Plan (PNRR) - Mission 4 Component 2 Investment 1.4.)

An interactive method for the selection of design criteria and the formulation of optimization problems in computer aided optimal design

On submodular prophet inequalities and correlation gap

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound [Formula: see text]. Second, we exhibit a general combinatorial black box approach to turn [Formula: see text]-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an [Formula: see text] algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their [Formula: see text] algorithm.

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Recently, the large-scale optimization problems have become a common research topic in the field of evolutionary computation. It is hard to find optimal solutions when solving large-scale multi-objective optimization problems (LSMOPs), due to the ineffectiveness of existing operators. In the other word, the search ability of most existing MOEAs on solving LSMOPs is still weak. To address this issue, an efficient competitive swarm optimizer with a strong exploration ability, denoted as E-CSO, is presented in this paper, which designs a novel three-particle-based particle updating strategy to improve the search efficiency. The experimental results validate the high efficiency and effectiveness of our proposed approach when solving various LSMOPs.

Distribution network reconfiguration is a very complex and large-scale combinatorial optimization problem. In network reconfiguration, whether an effective solution can be obtained is a key issue. Aiming at the problems in network reconstruction by traditional algorithm, such as long time required, more times of power flow calculation and high network loss, a network optimization design algorithm based on improved ant colony algorithm for high voltage power distribution network is proposed. After analyzing the operating characteristics of the high voltage power distribution network, the network topology of the high voltage power distribution network is described by constructing a hierarchical variable-structure distribution network model. A mathematical model of distribution network reconstruction considering the opportunity constraint with the minimum network loss as the objective function is established. The power flow distribution is calculated by using the pre-push back-generation method combined with the hierarchical structure of the distribution network. The maximum and minimum ant colony algorithm is introduced to improve the pheromone updating method of the traditional ant colony algorithm, and the search range is expanded, so that the algorithm can jump out of the local optimization trap to realize the accurate solution of the power distribution network reconstruction model. The experimental results show that compared with the current network reconstruction algorithm, the proposed algorithm requires less time for convergence, less power flow calculation, and lower network loss.

In this paper, we propose an implementation of a maintenance scheduler for cloud infrastructures. Live migration associated with maintenance work is important to ensure service continuity for all virtual machines in an infrastructure. However, executing the migration process when the machines are under heavy load negatively affects cloud users' businesses, such as by degrading performance and extending downtime. We can avoid this by finding an appropriate time period for live migration and performing the migration then. This idea is convenient for cloud users but inconvenient for cloud providers, that is, maintenance work should be completed as soon as possible for security reasons. To satisfy both the users' convenience and providers' requirements, we designed a maintenance scheduling problem to find the appropriate time period and to shorten the maintenance work period. Since it is a large-scale combinatorial optimization problem with complex constraints on maintenance requirements, we described the constraints by using answer set programming and implemented a maintenance scheduler on the basis of a divide-and-conquer approach to reduce the computational complexity exponentially. We evaluated our scheduler by using information on a real configuration of a commercial cloud infrastructure. While a naive approach to solving the maintenance scheduling problem could not find any feasible solutions within a realistic amount of time and memory, our implementation generated the best maintenance schedule for 1032 physical machines and 14208 virtual machines in 206 s with a memory usage of 1086 MB.

Large neighborhood search is a popular hybrid metaheuristic which results from the use of a complete technique—such as dynamic programming, constraint programming or MIP solvers—for finding the best neighbor within a large neighborhood of the incumbent solution. In this work we present an application of large neighborhood search to a strategic supply chain management problem from the Chemical industry, namely the configuration of a three-echelon hydrogen network for vehicle use with the goal of minimizing the total cost. Traditionally, these large-scale combinatorial optimization problems have been solved by means of mathematical programming techniques. Our experimental results show that large neighborhood search has the potential to be a viable alternative, especially when the complexity of the problem grows.