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This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in t...

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... A graph is Hamiltonian when it has at least one Hamiltonian cycle, which is a cycle that visits each node of the graph exactly once. Recently, a matheuristic was developed by Jooken et al. [8], which attempts to solve this problem heuristically by using a multi-start local search algorithm. HCP can also be solved by first converting it to a travelling salesman problem (TSP) and then using a TSP solver such as Concorde or Lin-Kernighan-Helsgaun to get an exact or heuristic solution respectively. ...

... HCP can also be solved by first converting it to a travelling salesman problem (TSP) and then using a TSP solver such as Concorde or Lin-Kernighan-Helsgaun to get an exact or heuristic solution respectively. In this paper we will compare the multi-start local search algorithm (MSLS) [8] with the Concorde TSP solver (Concorde) [1]. ...

... With this knowledge, we can use existing TSP solvers such as Concorde (an exact solver) [1] or LKH (a heuristic solver) [6] to solve HCP. In the following sections we compare the performance of the multi-start local search algorithm (MSLS) [8] to the performance of Concorde on the converted instance. ...

Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is usually no standard set of instances to benchmark performance. Second, using existing graph generators results in a restricted spectrum of difficulty and the resulting graphs are usually not diverse enough to draw sound conclusions. That is why recent work proposes a new methodology to generate a diverse set of instances by using an evolutionary algorithm. We can then analyze the resulting graphs and get key insights into which attributes are most related to algorithm performance. We can also fill observed gaps in the instance space in order to generate graphs with previously unseen combinations of features. This methodology is applied to the instance space of the Hamiltonian completion problem using two different solvers, namely the Concorde TSP Solver and a multi-start local search algorithm.

Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is not always a standard set of instances to benchmark performance. Second, using existing graph generators results in a restricted spectrum of difficulty and the resulting graphs are not always diverse enough to draw sound conclusions. That is why recent work proposes a new methodology to generate a diverse set of instances by using evolutionary algorithms. We can then analyze the resulting graphs and get key insights into which attributes are most related to algorithm performance. We can also fill observed gaps in the instance space in order to generate graphs with previously unseen combinations of features. We apply this methodology to the instance space of the Hamiltonian completion problem using two different solvers, namely the Concorde TSP Solver and a multi-start local search algorithm.

The 0-1 knapsack problem is an important optimization problem, because it arises as a special case of a wide variety of optimization problems and has been generalized in several ways. Decades of research have resulted in very powerful algorithms that can solve large knapsack problem instances involving thousands of decision variables in a short amount of time. Current problem instances in the literature no longer challenge these algorithms. However, hard problem instances are important to demonstrate the strengths and weaknesses of algorithms and this knowledge can in turn be used to create better performing algorithms. In this paper, we propose a new class of hard problem instances for the 0-1 knapsack problem and provide theoretical support that helps explain why these problem instances are hard to solve to optimality. A large dataset of 3240 hard problem instances was generated and subsequently solved on a supercomputer, using approximately 810 CPU-hours. The analysis of the obtained results shows to which extent different parameters influence the hardness of the problem instances. This analysis also demonstrates that the proposed problem instances are a lot harder than the previously known hardest instances, despite being much smaller.