2022 IEEE NIGERCON 978-1-6654-7978-3/22/$31.00 ©2022 IEEE Integrating Local Search Methods in Metaheuristic Algorithms for Combinatorial Optimization: The Traveling Salesman Problem and its Variants Abstract—Metaheuristic Algorithms (MAs) have demonstrated exceptional competence in solving Combinatorial Optimization Problems (COP) such as the Minimum Spanning Tree Problem (MSTP), the Features Selection Problem (FSP) with the most popular and oldest being the Traveling Salesman Problem (TSP). However, this class of algorithms many a time suffers from local optima stagnation leading to sub-optimal performance. Thus, there is a need for such algorithms to be supported with specific Local Search (LS) procedures either as an inner component or as a post-processing mechanism to enhance the search process for better performance. This paper presents a comprehensive review of the integration of LS methods in metaheuristic algorithms for solving single, multi, and many-objective COP with a focus on the TSP and its variants. The LS methods reviewed in this study were classified into one-way and two-way based on their mode of operation. In addition, practical suggestions were discussed and possible future directions were pointed out. Keywords—combinatorial optimization; metaheuristic algorithms; traveling salesman problem; local search; single, multi and many-objective I. INTRODUCTION Combinatorial Optimization (CO) is a concept that has found widespread use in practical sectors such as routing, scheduling, planning, transportation, and telecommunication, among others, where there are often not only a single but multi or many objectives involved [1]. Several existing strategies for addressing COP, such as deterministic exhaustive search, have significant limitations, including high computing cost, no guarantee of an optimal solution, and the requirement for a domain expert to develop algorithmic rules for each problem [1]. COP in its basic sense involves finding an optimal object from a collection of a bounded set of objects. Most times, the set of achievable solutions is discrete or reducible to a discrete set [2]. In a more formal sense, COP is a collection of instances say  = {, } , where  is a collection or set of achievable solutions while  is a cost function: ∶  → ℝ. This can be expressed as: from the set of feasible solutions, find the best solution. Also, find the fitness of the best solution [3]. Typical examples of COP include the Traveling Salesman Problem (TSP), the Cutting Stock Problem (CSP), the Minimum Spanning Tree Problem (MSTP), the Features Selection Problem (FSP), the Packing Problems (PP), the Subset Sum Problem (SSP), the Partition Problem (PP), the Graph Coloring Problem (GCP) among others, with the most popular and oldest being the TSP [1]. The TSP is a well- known problem for evaluating and improving a variety of optimization algorithms, and it is classified as an NP-hard problem [4]. Many real-world problems necessitate working with multiple objectives that must be optimized at the same time [5]. Problems with more than one objective are referred to as Multi-Objective Optimization Problems (MOOPs) or Many-Objective Optimization Problems (MaOPs) when more than three objectives are involved [6]. Because it does not require an aftermath decision, COP with only one objective is often simple and less complex. [7]. However, with an increasing number of objectives, it becomes demanding i.e., it is likely that most of the solutions will gradually become non- dominated and distance metrics become less discriminatory [5]. Exhaustive search-based methods for solving COP are frequently guided by heuristics that have proven to be inefficient over time, with no assurance of delivering an optimal solution [8]. Taking this constraint into account, more resilient and efficient machines, such as Metaheuristic Algorithms (MAs), were developed specifically for solving COP. MAs are based on the idea of real-world natural phenomena and are designed to overcome the limitations of heuristic methods in addressing NP-hard problems. Prominent and early MAs such as the Genetic Algorithm (GA) [9] and the Ant Colony Optimization (ACO) [10] are among the most applied in solving COP in literature. These algorithms have so far been classified into diverse classes based on their mode of operation with one of the notable classes being the population- based and trajectory-based MAs [11]. The population-based MAs deal with a collection of candidate solutions that, over numerous generations, grow into better solutions through communication and learning. An example is the ACO. On the other hand, trajectory-based MAs (also known as local search methods) deal with a single solution and evolve it across numerous generations into a better one. An example is the Simulated Annealing (SA). Fig. 1 presents the general taxonomy of CO techniques. Isuwa Jeremiah Computer Science Department, Federal University of Kashere, Gombe-Nigeria isuwajeremiah@gmail.com Mohammed Abdullahi Computer Science Department, Ahmadu Bello University, Zaria-Nigeria. abdullahilwafu@abu.edu.ng Sahabi Ali Yusuf Computer Science Department, Ahmadu Bello University, Zaria-Nigeria sahabiali@yahoo.com Muhammad Nuruddeen Idris Computer Science Department, Federal University Gusau, Zamfara-Nigeria minuruddeen66@gmail.com Baffa Shuaibu Garko Computer Science Department, Ahmadu Bello University, Zaria-Nigeria baffancy1@gmail.com Muhammad Yusuf Haruna Computer Science Department, Ahmadu Bello University, Zaria-Nigeria muhdyusufharuna@gmail.com