TY - JOUR

T1 - Coloring graphs using two colors while avoiding monochromatic cycles

AU - Talla Nobibon, F.

AU - Hurkens, C.A.J.

AU - Leus, R.

AU - Spieksma, F.C.R.

PY - 2012

Y1 - 2012

N2 - We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs. Applications of this problem include testing rationality of collective consumption behavior, a subject in micro-economics. We prove that the problem is NP-complete even for oriented graphs and argue that the existence of a constant-factor approximation algorithm is unlikely for an optimization version which maximizes the number of vertices that can be colored using two colors while avoiding monochromatic cycles. We present three exact algorithms, namely an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm. We compare these three logorithms both on real-life instances and on randomly generated graphs. We find that for the latter set of graphs, every algorithm solves instances of considerable size within few seconds; however, the CPU time of the integer-programming algorithm increases with the number of vertices in the graph more clearly than the the CPU time of the two other procedures. For real-life instances, the integer-programming algorithm solves the largest instance in about a half hour while the branch-and-check algorithm takes about ten minutes and the backtracking algorithm less than five minutes. Finally, for every algorithm, we also study empirically the transition from a high to a low probability of a YES answer as function of the number of arcs divided by the number of vertices.

AB - We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs. Applications of this problem include testing rationality of collective consumption behavior, a subject in micro-economics. We prove that the problem is NP-complete even for oriented graphs and argue that the existence of a constant-factor approximation algorithm is unlikely for an optimization version which maximizes the number of vertices that can be colored using two colors while avoiding monochromatic cycles. We present three exact algorithms, namely an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm. We compare these three logorithms both on real-life instances and on randomly generated graphs. We find that for the latter set of graphs, every algorithm solves instances of considerable size within few seconds; however, the CPU time of the integer-programming algorithm increases with the number of vertices in the graph more clearly than the the CPU time of the two other procedures. For real-life instances, the integer-programming algorithm solves the largest instance in about a half hour while the branch-and-check algorithm takes about ten minutes and the backtracking algorithm less than five minutes. Finally, for every algorithm, we also study empirically the transition from a high to a low probability of a YES answer as function of the number of arcs divided by the number of vertices.

U2 - 10.1287/ijoc.1110.0466

DO - 10.1287/ijoc.1110.0466

M3 - Article

VL - 24

SP - 485

EP - 499

JO - INFORMS Journal on Computing

JF - INFORMS Journal on Computing

SN - 1091-9856

IS - 3

ER -