Relevant work on SCP

Several techniques have been developed to tackle the SCP. The most notable methods in literature are based on linear programming and Lagrangian relaxations, heuristic ap- proaches and exact methods [7, 12, 29]. A survey of some dated techniques which have inspired newer methods to solve the SCP can be found in [27]. Some exact methods to solve SCP are presented in [7, 12] but they are unable to scale with larger instances of the problem where approximation and heuristic approaches are more suitable. The Greedy algorithm from [29] is one of the best approximation algorithms for SCP which finds solu- tions by iteratively selecting sets which covers the most remaining uncovered items. [79] compared a variant of the greedy algorithm with eight other algorithms which included variants of greedy, randomised algorithms and neural networks. A randomised variant of the greedy algorithm was shown to produce the best solutions with respect to cost of the final solution.

A tabu search based approach was developed in [120] and was used to solve a class of SCP problems defined from Steiner triple systems. Ant-colony based approaches to solve the SCP have been used in [108, 124, 126, 125, 33]. These approaches along with GRASP [61] construct new solutions from scratch at every iteration by employing some

random behaviour in the construction process. The solutions obtained at every iteration are feasible but the solutions tend to contain redundant sets and all of these approaches need to rely on dedicated redundancy removal operators after every iteration. Several variants of genetic algorithms have been employed to solve the SCP. These algorithms need to deal with the issue of infeasibility as well as redundancy in their solutions. Two main techniques have been employed to solve the infeasibility issue in genetic algorithms, namely penalty functions and repair. The repair mechanism converts an infeasible solution to a feasible solution by incorporating sets which cover the uncovered elements. [13, 137, 121, 9] employ repair mechanism in their GA implementations and to mitigate the problem of redundancy they use redundancy removal procedures. Penalty based approaches reduce the fitness of infeasible solutions making them less favourable than feasible solutions and driving the search towards feasible regions. [97] reduced the fitness of all infeasible solutions by an equal and high enough penalty ensuring that the fitness of any feasible solution is higher than them. The fitness of the infeasible solutions in this case does not reflect the degree of infeasibility and these solutions can no longer be compared using fitness. The genetic algorithm of [9] uses a penalty proportional to the cost it takes to cover all uncovered elements, while [150] use a penalty proportional to the number of elements not covered. An issue with all of these methods is that after few generations the chances of infeasible solutions surviving in the population are very low and therefore the infeasible region is not searched effectively.

Local search algorithms have also been utilised to solve SCP. Due to the feasibility constraint it is hard to design pure local search algorithms. [114] have proposed a local search method incorporating a tabu search like procedure with a restricted neighbourhood by using a upper bound parameter for the uni-cost version of SCP. Other local search techniques are often enhanced with methods like linear programming and Lagrangian relaxation [63, 155] and sub-gradient optimisation [84]. An AIS used to solve the SCP has been developed by [139] based on the CLONALG. They compared their AIS-SCP with the greedy algorithm and obtained better solutions. However their algorithm involves

several problem specific operators making it hard to qualify as a true meta-heuristic. All the techniques mentioned above utilise the single objective formulation of SCP and the only known work with the multi-objective formulation is the work done in [110, 130]. Like the theoretical analysis in [115], the work in [110] presents another theoretical analysis where the authors propose two island models with different topology of islands and prove that these models find good approximation for the multi-objective formulation of the SCP. The homogeneous islands model which comprises an archipelago of global simple evolutionary multi-objective optimisers is called the parallel GSEMO (PGSEMO) throughout this thesis. They provide upper bounds on the running time until an Hm

approximation for the set cover problem is reached by specifying the number of islands in the archipelago which are proportional to the probability of communication between islands and the size of the problem.

More recently [130] have introduced the constrained multi-objective artificial immune system (MAIS) based on the clonal selection principle, used to handle a real world appli- cation of find core collection for germplasms as a strategy for conservation of diversity. The problem of finding core collections is modelled as a SCP with three objectives spe- cific to the problem. They compared their algorithm with NSGA-II on a single instance of their problem of size (642 × 55) and showed that their MAIS performed better than NSGA-II on this instance based on hypervolume, extent, attainment and spacing met- rics of evaluation. Though the authors mention that parameter tuning was performed based on preliminary data, no results about the sensitivity of the algorithm towards these parameters were provided.

Even though the advantages of the multi-objective formulation of SCP are stated in [115] only one theoretical and one empirical work using this formulation is known to us at the time of this research. This presents as an opportunity to study the SCP in a multi-objective scenario as little to no work exist in this area, even when the potential advantages have been shown by theoretical studies. The empirical study by [130] utilises a multi-objective formulation that is extremely specific to the application area of germplasm

selection, which is different from the standard multi-objective formulation. Their work is also limited to a single problem instance which raises the question if the results are appli- cable to other instances. following their approach and conducting comparative analysis with their work would seriously limit the applicability of the results to the wider problem formulation of SCP and also there is the inherent problem of lack of instances to study. The work in [130] seems the most appropriate to consider for a comparative analysis with our new proposed approach, after identifying the literature gap. This is because, though their analysis is theoretical, it provides performance guarantees for the solution quality that can be achieved by their models. Therefore knowing the guarantee that their model will provide, we can make comparison with implementations of their model. This analysis will add to the scarce literature for the multi-objective formulation of SCP.