Combinatorial Optimization Research Articles

Retraction Note: Design of extended dense coding protocol strategy based on combinatorial optimization

Retraction Note: Design of extended dense coding protocol strategy based on combinatorial optimization

COMBHelper: A Neural Approach to Reduce Search Space for Graph Combinatorial Problems

Combinatorial Optimization (CO) problems over graphs appear routinely in many applications such as in optimizing traffic, viral marketing in social networks, and matching for job allocation. Due to their combinatorial nature, these problems are often NP-hard. Existing approximation algorithms and heuristics rely on the search space to find the solutions and become time-consuming when this space is large. In this paper, we design a neural method called COMBHelper to reduce this space and thus improve the efficiency of the traditional CO algorithms based on node selection. Specifically, it employs a Graph Neural Network (GNN) to identify promising nodes for the solution set. This pruned search space is then fed to the traditional CO algorithms. COMBHelper also uses a Knowledge Distillation (KD) module and a problem-specific boosting module to bring further efficiency and efficacy. Our extensive experiments show that the traditional CO algorithms with COMBHelper are at least 2 times faster than their original versions.

EnMatch: Matchmaking for Better Player Engagement via Neural Combinatorial Optimization

Matchmaking is a core task in e-sports and online games, as it contributes to player engagement and further influences the game's lifecycle. Previous methods focus on creating fair games at all times. They divide players into different tiers based on skill levels and only select players from the same tier for each game. Though this strategy can ensure fair matchmaking, it is not always good for player engagement. In this paper, we propose a novel Engagement-oriented Matchmaking (EnMatch) framework to ensure fair games and simultaneously enhance player engagement. Two main issues need to be addressed. First, it is unclear how to measure the impact of different team compositions and confrontations on player engagement during the game considering the variety of player characteristics. Second, such a detailed consideration on every single player during matchmaking will result in an NP-hard combinatorial optimization problem with non-linear objectives. In light of these challenges, we turn to real-world data analysis to reveal engagement-related factors. The resulting insights guide the development of engagement modeling, enabling the estimation of quantified engagement before a match is completed. To handle the combinatorial optimization problem, we formulate the problem into a reinforcement learning framework, in which a neural combinatorial optimization problem is built and solved. The performance of EnMatch is finally demonstrated through the comparison with other state-of-the-art methods based on several real-world datasets and online deployments on two games.

Accelerating Cutting-Plane Algorithms via Reinforcement Learning Surrogates

Discrete optimization belongs to the set of N P-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms, which reach optimal solutions by iteratively adding inequalities known as cuts to refine a feasible set. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. In this work, we propose a method for accelerating cutting-plane algorithms via reinforcement learning. Our approach uses learned policies as surrogates for N P-hard elements of the cut generating procedure in a way that (i) accelerates convergence, and (ii) retains guarantees of optimality. We apply our method on two types of problems where cutting-plane algorithms are commonly used: stochastic optimization, and mixed-integer quadratic programming. We observe the benefits of our method when applied to Benders decomposition (stochastic optimization) and iterative loss approximation (quadratic programming), achieving up to 45% faster average convergence when compared to modern alternative algorithms.

Learning Encodings for Constructive Neural Combinatorial Optimization Needs to Regret

Deep-reinforcement-learning (DRL) based neural combinatorial optimization (NCO) methods have demonstrated efficiency without relying on the guidance of optimal solutions. As the most mainstream among them, the learning constructive heuristic (LCH) achieves high-quality solutions through a rapid autoregressive solution construction process. However, these LCH-based methods are deficient in convergency, and there is still a performance gap compared to the optimal. Intuitively, learning to regret some steps in the solution construction process is helpful to the training efficiency and network representations. This article proposes a novel regret-based mechanism for an advanced solution construction process. Our method can be applied as a plug-in to any existing LCH-based DRL-NCO method. Experimental results demonstrate the capability of our work to enhance the performance of various NCO models. Results also show that the proposed LCH-Regret outperforms the previous modification methods on several typical combinatorial optimization problems. The code and Supplementary File are available at https://github.com/SunnyR7/LCH-Regret.

SAT-Based Tree Decomposition with Iterative Cascading Policy Selection

Solvers for propositional satisfiability (SAT) effectively tackle hard optimization problems. However, translating to SAT can cause a significant size increase, restricting its use to smaller instances. To mitigate this, frameworks using multiple local SAT calls for gradually improving a heuristic solution have been proposed. The performance of such algorithmic frameworks heavily relies on critical parameters, including the size of selected local instances and the time allocated per SAT call. This paper examines the automated configuration of the treewidth SAT-based local improvement method (TW-SLIM) framework, which uses multiple SAT calls for computing tree decompositions of small width, a fundamental problem in combinatorial optimization. We explore various TW-SLIM configuration methods, including offline learning and real-time adjustments, significantly outperforming default settings in multi-SAT scenarios with changing problems. Building upon insights gained from offline training and real-time configurations for TW-SLIM, we propose the iterative cascading policy—a novel hybrid technique that uniquely combines both. The iterative cascading policy employs a pool of 30 configurations obtained through clustering-based offline methods, deploying them in dynamic cascades across multiple rounds. In each round, the 30 configurations are tested according to the cascading ordering, and the best tree decomposition is retained for further improvement, with the option to adjust the following ordering of cascades. This iterative approach significantly enhances the performance of TW-SLIM beyond baseline results, even within varying global timeouts. This highlights the effectiveness of the proposed iterative cascading policy in enhancing the efficiency and efficacy of complex algorithmic frameworks like TW-SLIM.

NN-Steiner: A Mixed Neural-Algorithmic Approach for the Rectilinear Steiner Minimum Tree Problem

Recent years have witnessed rapid advances in the use of neural networks to solve combinatorial optimization problems. Nevertheless, designing the "right" neural model that can effectively handle a given optimization problem can be challenging, and often there is no theoretical understanding or justification of the resulting neural model. In this paper, we focus on the rectilinear Steiner minimum tree (RSMT) problem, which is of critical importance in IC layout design and as a result has attracted numerous heuristic approaches in the VLSI literature. Our contributions are two-fold. On the methodology front, we propose NN-Steiner which is a novel mixed neural-algorithmic framework for computing RSMTs that leverages the celebrated PTAS algorithmic framework of Arora to solve this problem (and other geometric optimization problems). Our NN-Steiner replaces key algorithmic components within Arora's PTAS by suitable neural components. In particular, NN-Steiner only needs four neural network (NN) components that are called repeatedly within an algorithmic framework. Crucially, each of the four NN components is only of bounded size independent of input size, and thus easy to train. Furthermore, as the NN component is learning a generic algorithmic step, once learned, the resulting mixed neural-algorithmic framework generalizes to much larger instances not seen in training. Our NN-Steiner, to our best knowledge, is the first neural architecture of bounded size that has capacity to approximately solve RSMT (and variants). On the empirical front, we show how NN-Steiner can be implemented and demonstrate the effectiveness of our resulting approach, especially in terms of generalization, by comparing with state-of-the-art methods (both neural and non-neural based).

Generating Diagnostic and Actionable Explanations for Fair Graph Neural Networks

A plethora of fair graph neural networks (GNNs) have been proposed to promote algorithmic fairness for high-stake real-life contexts. Meanwhile, explainability is generally proposed to help machine learning practitioners debug models by providing human-understandable explanations. However, seldom work on explainability is made to generate explanations for fairness diagnosis in GNNs. From the explainability perspective, this paper explores the problem of what subgraph patterns cause the biased behavior of GNNs, and what actions could practitioners take to rectify the bias? By answering the two questions, this paper aims to produce compact, diagnostic, and actionable explanations that are responsible for discriminatory behavior. Specifically, we formulate the problem of generating diagnostic and actionable explanations as a multi-objective combinatorial optimization problem. To solve the problem, a dedicated multi-objective evolutionary algorithm is presented to ensure GNNs' explainability and fairness in one go. In particular, an influenced nodes-based gradient approximation is developed to boost the computation efficiency of the evolutionary algorithm. We provide a theoretical analysis to illustrate the effectiveness of the proposed framework. Extensive experiments have been conducted to demonstrate the superiority of the proposed method in terms of classification performance, fairness, and interpretability.

Thompson Sampling for Real-Valued Combinatorial Pure Exploration of Multi-Armed Bandit

We study the real-valued combinatorial pure exploration of the multi-armed bandit (R-CPE-MAB) problem. In R-CPE-MAB, a player is given stochastic arms, and the reward of each arm follows an unknown distribution. In each time step, a player pulls a single arm and observes its reward. The player's goal is to identify the optimal action from a finite-sized real-valued action set with as few arm pulls as possible. Previous methods in the R-CPE-MAB require enumerating all of the feasible actions of the combinatorial optimization problem one is considering. In general, since the size of the action set grows exponentially large with respect to the number of arms, this is almost practically impossible when the number of arms is large. We introduce an algorithm named the Generalized Thompson Sampling Explore (GenTS-Explore) algorithm, which is the first algorithm that can work even when the size of the action set is exponentially large with respect to the number of arms. We also introduce a novel problem-dependent sample complexity lower bound of the R-CPE-MAB problem, and show that the GenTS-Explore algorithm achieves the optimal sample complexity up to a problem-dependent constant factor.

Comparison of New and Old Optimization Algorithms for Traveling Salesman Problem on Small, Medium, and Large-scale Benchmark Instances

The Traveling Salesman Problem (TSP), a prominent combinatorial optimization issue, is the subject of this study's evaluation of the performance of new and old optimization techniques. This paper seeks to expand knowledge of optimization techniques and how they might be applied to solve TSP challenges. The goal of the research is to compare various algorithms' scalability, convergence, and computation times on benchmark instances of several sizes. To achieve this goal, this paper carried out extensive testing using the Artificial Bee Colony (ABC), Grey Wolf Optimization (GWO), and Salp Swarm Algorithm (SSA) as new optimization algorithms and the Genetic Algorithm (GA), Ant Colony Optimization (ACO), and Simulated Annealing (SA) as old optimization algorithms. On small, medium, and large-scale benchmark cases, these algorithms were examined. The findings of this investigation show that the new optimization techniques are more convergent and scalable than the old ones, especially for medium-scale scenarios. They perform better performance in terms of solution quality by applying objective function values. The new methods also exhibit improved scalability, successfully adjusting to medium-scale instances. However, there were no discernible changes between the smaller and larger instances. This study makes an impact by offering insightful information about how well optimization methods perform while solving the TSP. Each algorithm's strengths and downsides have been reported, and these details offer useful guidance for choosing an algorithm for a certain scenario. The results also show the practical ramifications of applying novel optimization techniques, especially in medium-scale instances..

CEGAR-Based Approach for Solving Combinatorial Optimization Modulo Quantified Linear Arithmetics Problems

Bioinformatics has always been a prolific domain for generating complex satisfiability and optimization problems. For instance, the synthesis of multi-scale models of biological networks has recently been associated with the resolution of optimization problems mixing Boolean logic and universally quantified linear constraints (OPT+qLP), which can be benchmarked on real-world models. In this paper, we introduce a Counter-Example-Guided Abstraction Refinement (CEGAR) to solve such problems efficiently. Our CEGAR exploits monotone properties inherent to linear optimization in order to generalize counter-examples of Boolean relaxations. We implemented our approach by extending Answer Set Programming (ASP) solver Clingo with a quantified linear constraints propagator. Our prototype enables exploiting independence of sub-formulas to further exploit the generalization of counter-examples. We evaluate the impact of refinement and partitioning on two sets of OPT+qLP problems inspired by system biology. Additionally, we conducted a comparison with the state-of-the-art ASP solver Clingo[lpx] that handles non-quantified linear constraints, showing the advantage of our CEGAR approach for solving large problems.

Limited Query Graph Connectivity Test

We propose a combinatorial optimisation model called Limited Query Graph Connectivity Test. We consider a graph whose edges have two possible states (On/Off). The edges' states are hidden initially. We could query an edge to reveal its state. Given a source s and a destination t, we aim to test s−t connectivity by identifying either a path (consisting of only On edges) or a cut (consisting of only Off edges). We are limited to B queries, after which we stop regardless of whether graph connectivity is established. We aim to design a query policy that minimizes the expected number of queries. Our model is mainly motivated by a cyber security use case where we need to establish whether attack paths exist in a given network, between a source (i.e., a compromised user node) and a destination (i.e., a high-privilege admin node). Edge query is resolved by manual effort from the IT admin, which is the motivation behind query minimization. Our model is highly related to Stochastic Boolean Function Evaluation (SBFE). There are two existing exact algorithms for SBFE that are prohibitively expensive. We propose a signifcantly more scalable exact algorithm. While previous exact algorithms only scale for trivial graphs (i.e., past works experimented on at most 20 edges), we empirically demonstrate that our algorithm is scalable for a wide range of much larger practical graphs (i.e., graphs representing Windows domain networks with tens of thousands of edges). We also propose three heuristics. Our best-performing heuristic is via limiting the planning horizon of the exact algorithm. The other two are via reinforcement learning (RL) and Monte Carlo tree search (MCTS). We also derive an algorithm for computing the performance lower bound. Experimentally, we show that all our heuristics are near optimal. The heuristic building on the exact algorithm outperforms all other heuristics, surpassing RL, MCTS and eight existing heuristics ported from SBFE and related literature.

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

Parallel Beam Search Algorithms for Domain-Independent Dynamic Programming

Domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming, has shown promising performance on multiple combinatorial optimization problems compared with mixed integer programming (MIP) and constraint programming (CP). The current DIDP solvers are based on heuristic search, and the state-of-the-art solver, complete anytime beam search (CABS), uses beam search. However, the current DIDP solvers cannot utilize multiple threads, unlike state-of-the-art MIP and CP solvers. In this paper, we propose three parallel beam search algorithms and develop multi-thread implementations of CABS. With 32 threads, our multi-thread DIDP solvers achieve 9 to 39 times speedup on average and significant performance improvement over the sequential solver, finding the new best solutions for two instances of the traveling salesperson problem with time windows. In addition, our solvers outperform multi-thread MIP and CP solvers in four of the six combinatorial optimization problems evaluated.

Agile earth observation satellite constellations scheduling for large area target imaging using heuristic search

Demand for imaging of large area targets by space-based assets such as constellations of Earth Observation Satellites (EOS) is on the rise for different applications. The area targets coverage cannot be achieved by a single satellite in a single observation opportunity, as the region for imaging spreads wider than the coverage swaths of imaging sensors. Hence, the imaging satellites are scheduled in a cooperative way to achieve maximum coverage of the area in a given planning time horizon. The scheduling algorithms also aim to maximize or minimize some specific objectives of interest while simultaneously complying with all resource constraints for an efficient use of precious space resources. As the current generation satellites are highly agile, there are many ways of choosing an imaging location in the target area for each single observation opportunity of a satellite. Even though the possible imaging region in continuous space is discretized into a finite number of strips for each observation, the huge combinatorial search space formed with all satellites’ imaging opportunities makes the problem intractable. Thus, the area target imaging scheduling is a complex combinatorial NP-hard optimization problem. Many exact and heuristic methods were evolved to provide a solution to this scheduling problem. Even though the exact methods solve the problem to optimality, they are limited to smaller problem instances because of the high computational complexities involved in those methods. More focus is seen in the literature on heuristic approaches as they guarantee scheduling performance with feasible solutions within an acceptable computational complexity. This paper presents our efficient heuristic approach proposed for the scheduling problem for imaging a large area target using a constellation of multiple satellites. Our approach is an extension of a widely used greedy approach in combination with insights from dynamic programming. In this paper, first we addressed the scheduling problem by describing it in detail and formulating it into a non-linear integer programming problem, considering the multi-objectives of achieving maximum coverage and minimum imaging resolution. The solution to the problem comprises two phases, namely the area decomposition phase and the scheduling phase. In the area decomposition phase, the area target is divided into strips dynamically for each observation opportunity of a satellite. Then exact observation period of each strip is computed using a simplified semi-analytical computational method. In the scheduling phase, we applied our heuristic search strategy, namely Forward and Backward Heuristic Search (FBHS) to obtain a near optimal solution within an acceptable computational time. Through the extensive simulations conducted under various scenarios, the effectiveness of the proposed method is verified in comparison with the baseline greedy approach.

Erratum: Progress toward favorable landscapes in quantum combinatorial optimization [Phys. Rev. A 104 , 032401 (2021)

Erratum: Progress toward favorable landscapes in quantum combinatorial optimization [Phys. Rev. A <b>104</b> , 032401 (2021)

AquaNutriOpt: Optimizing nutrients for controlling harmful algal blooms in Python—A case study of Lake Okeechobee

We present AquaNutriOpt, a user-friendly Python package designed to tackle a complex combinatorial optimization problem aimed at optimizing nutrient management for the control of harmful algal blooms. This optimization process involves the identification of optimal Best Management Practices (BMPs) and Treatment Technologies (TTs). AquaNutriOpt is constructed based on a novel integer programming model, which we present in this paper. The package can accommodate various user inputs, automatically transforming them into an optimization model, and then solving it using a free solver. To demonstrate AquaNutriOpt’s efficacy, we conduct a series of experiments on two watersheds around Lake Okeechobee in Florida, USA. These experiments illustrate that the optimal BMPs/TTs obtained by AquaNutriOpt can significantly reduce Phosphorus loads into the lake across various budget scenarios. We validate the results by running simulations with the process-based Watershed Assessment Model (WAM), confirming that the estimated percentage reductions closely align with the reports from WAM.

Guided quantum walk

We utilize the theory of local amplitude transfer (LAT) to gain insights into quantum walks (QWs) and quantum annealing (QA) beyond the adiabatic theorem. By representing the eigenspace of the problem Hamiltonian as a hypercube graph, we demonstrate that probability amplitude traverses the search space through a series of local Rabi oscillations. We argue that the amplitude movement can be systematically guided towards the ground state using a time-dependent hopping rate based solely on the problem's energy spectrum. Building upon these insights, we extend the concept of multistage QW by introducing the guided quantum walk (GQW) as a bridge between QW-like and QA-like procedures. We assess the performance of the GQW on exact cover, traveling salesperson, and garden optimization problems with 9 to 30 qubits. Our results provide evidence for the existence of optimal annealing schedules, beyond the requirement of adiabatic time evolutions. These schedules might be capable of solving large-scale combinatorial optimization problems within evolution times that scale linearly in the problem size. Published by the American Physical Society 2024

Prompt Optimization in Large Language Models

Prompt optimization is a crucial task for improving the performance of large language models for downstream tasks. In this paper, a prompt is a sequence of n-grams selected from a vocabulary. Consequently, the aim is to select the optimal prompt concerning a certain performance metric. Prompt optimization can be considered as a combinatorial optimization problem, with the number of possible prompts (i.e., the combinatorial search space) given by the size of the vocabulary (i.e., all the possible n-grams) raised to the power of the length of the prompt. Exhaustive search is impractical; thus, an efficient search strategy is needed. We propose a Bayesian Optimization method performed over a continuous relaxation of the combinatorial search space. Bayesian Optimization is the dominant approach in black-box optimization for its sample efficiency, along with its modular structure and versatility. We use BoTorch, a library for Bayesian Optimization research built on top of PyTorch. Specifically, we focus on Hard Prompt Tuning, which directly searches for an optimal prompt to be added to the text input without requiring access to the Large Language Model, using it as a black-box (such as for GPT-4 which is available as a Model as a Service). Albeit preliminary and based on “vanilla” Bayesian Optimization algorithms, our experiments with RoBERTa as a large language model, on six benchmark datasets, show good performances when compared against other state-of-the-art black-box prompt optimization methods and enable an analysis of the trade-off between the size of the search space, accuracy, and wall-clock time.

Ranking species in complex ecosystems through nestedness maximization

Identifying the rank of species in a complex ecosystem is a difficult task, since the rank of each species invariably depends on the interactions stipulated with other species through the adjacency matrix of the network. A common ranking method in economic and ecological networks is to sort the nodes such that the layout of the reordered adjacency matrix looks maximally nested with all nonzero entries packed in the upper left corner, called Nestedness Maximization Problem (NMP). Here we solve this problem by defining a suitable cost-energy function for the NMP which reveals the equivalence between the NMP and the Quadratic Assignment Problem, one of the most important combinatorial optimization problems, and use statistical physics techniques to derive a set of self-consistent equations whose fixed point represents the optimal nodes’ rankings in an arbitrary bipartite mutualistic network. Concurrently, we present an efficient algorithm to solve the NMP that outperforms state-of-the-art network-based metrics and genetic algorithms. Eventually, our theoretical framework may be easily generalized to study the relationship between ranking and network structure beyond pairwise interactions, e.g. in higher-order networks.