A Min-Max Theorem on Feedback Vertex Sets

Abstract

We establish a necessary and sufficient condition for the linear system {x : Hx ≥ e, x ≥ 0} associated with a bipartite tournament to be totally dual integral, where H is the cycle-vertex incidence matrix and e is the all-one vector. The consequence is a min-max relation on packing and covering cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem on the corresponding bipartite tournaments. In addition, we show that the feedback vertex set problem on general bipartite tournaments is NP-complete and approximable within 3.5 based on the min-max theorem.

An Efficient Algorithm for Finding Maximum Cycle Packings in Reducible Flow Graphs

“…vertex) set. See [3]- [5] for a complete characterization of all tournaments and bipartite tournaments and [7] and [8] for the description of all undirected graphs.…”

“…. , α, and for each arc (a, b) of G mm (v) its capacity (5) c (a, b) is set to be c(a, b) if (a, b) is an arc in G mm (v) and to be α j=1: (a,b)∈Q j y j if (a, b) is outside G mm (v). Figure 5 illustrates the construction of G mm (v 1 ) for the network depicted in Figure 4.…”

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

“…It is also one of the classical NP-complete problems from Karp's list [22]. Thus not surprisingly, for several decades, many different algorithmic approaches were tried on this problem including approximation algorithms [1,2,12,23], linear programming [9], local search [4], polyhedral combinatorics [7,20], probabilistic algorithms [26], and parameterized complexity [10,11,21].…”

Feedback arc set in bipartite tournaments is NP-complete

“…A bipartite tournament is an orientation of a complete bipartite graph. Cai et al [5] showed that FVS in bipartite tournaments is NP-complete. They have also established a min-max theorem for FVS in bipartite tournaments.…”

“…However, the observations used in his proof do not hold for bipartite tournaments, thus requiring a fairly different reduction. Interestingly, Cai et al [5] also observe in their NPcompleteness proof for FVS in bipartite tournaments that it seems to be a formidable (if not impossible) task to directly adapt a reduction for the non-bipartite case to the more complicated bipartite case. Note that our reduction also shows that FAS is NP-complete for cpartite tournaments for any fixed c 2.…”