Sequential Optimization of Paths in Directed Graphs Relative to Different Cost Functions

Dynamic Programming Multi-Objective Combinatorial Optimization

“…Earlier, Moshkov and colleagues proposed multi-stage optimization algorithms for the following four combinatorial optimization problems: matrix chain multiplication, global sequence alignment, optimal paths in directed graphs, and binary search trees [20,21,22,23]. We were not able to find other publications related to multi-stage optimization of ten out of 11 combinatorial optimization problems considered in this dissertation.…”

“…In [20], an algorithm for multi-stage optimization of paths in a directed graph G relative to two types of cost functions was presented. Let w 1 , w 2 be weight functions In this chapter, we design a conventional circuit without repetitions describing the set of paths under consideration and define cost functions of two types for this circuit.…”

“…optimal bitonic tour [18], and segmented least squares [19]. Multi-stage optimization algorithms for the first four problems were developed earlier [20,21,22,23], although each problem was studied separately with its own problem-specific algorithms and proofs.…”

“…assigning nonnegative real weights to edges of G, and p be a directed path in G. Then the cost function ψ w 1 of the first type is equal to the sum of weights of edges in p and the cost function ϕ w 2 of the second type is equal to the minimum weight of an edge in p. The cost functions of the first type should be minimized, and the cost functions of the second type should be maximized.Two interpretations of the graph G were considered in[20]. The first interpretation of the graph G is a road network.…”

Multi-stage Optimization of Matchings in Trees with Application to Kidney Exchange

“…One of the main areas of applications for the approach is the rough set theory [10,11] in which decision trees and rules are widely used. This approach was extended also to some combinatorial optimization problems [1,[3][4][5]. Here we consider one more its application.…”