Connectedness of efficient solutions in multiple criteria combinatorial optimization

Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization

“…However, Ehrgott and Klamroth (1997) demonstrated the incorrectness of the connectedness conjecture for MOSP and MOST problems by a counterexample and, thus, disproved Martins (1984). Ehrgott and Klamroth (1997) showed that any graph can be extended in such a way that the adjacency graph (of MOSP and MOST) for the problem on the extended graph is not connected.…”

“…Instead of the topological connectedness, a graph theoretical definition can be used for MOCO problems as described, for example, in Ehrgott and Klamroth (1997) and Paquete and Stützle (2006). Definition 1.1 For a given MOCO problem the adjacency graph of efficient solutions G = (V, A) of the MOCO problem is defined as follows: V consists of all efficient solutions of the given MOCO problem.…”

Connectedness and Local Search for Bicriteria Knapsack Problems

“…Despite of the negative results reported in the literature for similar problems [3,5], the experimental analysis for the problems investigated in this paper are quite promising. For both bicriteria versions of the unconstrained optimization problem and knapsack problem with a bounded cardinality constraint, the experiments suggest that small-sized neighborhood structures give rise to connected efficient sets quite Table 3.…”

“…the efficient set ) [3] is analyzed from an experimental point of view, and related to the performance of a particular class of multicriteria local search algorithms [9]. For a given efficient set of a MCO problem instance, a graph can be constructed such that each node represents an optimal solution and an edge connects two nodes if the corresponding optimal solutions are neighbors for a given neighborhood structure.…”

“…If the efficient set is connected and the neighborhood structure is tractable from a computational point of view, local search algorithms would be able to find the efficient set in a very effective manner [9], by starting from at least one efficient solution. However, worst-case results have shown that the efficient set for many MCO problems is not connected in general with respect to different neighborhood structures [3,5], except for very few particular cases [6,11]. Moreover, some recent results indicate that approximate solutions that are obtained from independent metaheuristic runs on the bicriteria traveling salesman problem are strongly clustered with respect to small-sized neighborhood structures [10].…”