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The minimum feedback arc set and maximum acyclic subgraph are equivalent for the purposes of exact optimization, as one is the complement set of the other. However, for parameterized complexity and approximation, they differ, because the analysis used for those kinds of algorithms depends on the size of the solution and not just on the size of the input graph, and the minimum feedback arc set ...
The NP-complete problem Minimum feedback arc set reduces to Min-ULR[≥], with exactly one 1 and one -1 in each constraint, and all right-hand sides equal to 1. [ 6 ] Min-ULR[=,>,≥] are polynomial if the number of variables n is constant: they can be solved polynomially using an algorithm of Greer [ 7 ] in time O ( n ⋅ m n / 2 n − 1 ...
NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. [3]: ND2 Feedback vertex set [2] [3]: GT7 Feedback arc set [2] [3]: GT8 Graph coloring [2] [3]: GT4
The problem of finding a smallest feedback edge set is equivalent to finding a spanning forest, which can be done in polynomial time. The analogous concept in a directed graph is the feedback arc set (FAS) - a set of directed arcs whose removal makes the graph acyclic. Finding a smallest FAS is an NP-hard problem.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
Two examples of such algorithms are the Karger–Stein algorithm [1] and the Monte Carlo algorithm for minimum feedback arc set. [2] The name refers to the Monte Carlo casino in the Principality of Monaco, which is well-known around the world as an icon of gambling. The term "Monte Carlo" was first introduced in 1947 by Nicholas Metropolis. [3]
If the input graph is not already a directed acyclic graph, a set of edges is identified the reversal of which will make it acyclic. Finding the smallest possible set of edges is the NP-complete feedback arc set problem, so often greedy heuristics are used here in place of exact optimization algorithms.
A feedback arc set is chosen, and the edges of this set reversed, in order to convert the input into a directed acyclic graph with (if possible) few reversed edges. The vertices of the graph are given integer y -coordinates in such a way that, for each edge, the starting vertex of the edge has a higher coordinate than the ending vertex, with at ...