Parameterized Mixed Graph Coloring

Abstract

Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NPcomplete. Furthermore we point out a new polynomial case where the edges form a clique.

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

“…In particular, some results that have been proven in [10,11,[27][28][29][30][31][32][33][34] for the scheduling problem GcMPT|t i = 1, [r i ]|C max and for its special cases may be interpreted as analogous results for optimal colorings c(G) of the corresponding mixed graphs G = (V, A, E). Similarly, some results that have been proven in [3][4][5]8,9,17,18,[22][23][24] for optimal mixed graph colorings c(G) may be interpreted as analogous results for general shop scheduling problems GcMPT|t i = 1, [r i ]|C max . In the other words, there are articles [3][4][5][8][9][10][11]17,18,[22][23][24][27][28][29][30][31][32][33][34] studied both of these problems without indicating that they are actually the same problems.…”

“…Articles [13,16] investigate the complexity of the job-shop scheduling problems with any fixed regular objective function. Articles [17][18][19][20][21][22][23] study different types of connections between mixed graph colorings and unit-time shop-scheduling problems. Article [24] is a survey on the mixed graph coloring problems and the equivalent unit-time shop-scheduling problems.…”

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

“…The complexity of a mixed shopsche duling problem is studied in [16; 17]. A different connection between mixed graph colourings and unittime shopscheduling problems is studied in [18][19][20][21][22][23][24]. Article [25] presents a comprehensive survey on mixed graph colourings and the equivalent unittime shopscheduling problems.…”

Mixed Graph Colorings: A Historical Review

“…Algorithms for constructing an optimal c < -coloring of the mixed graph G = (V, A, E) have been derived in [39,40]. In [40], it is shown that an optimal c < -coloring may be constructed for the mixed graph…”

“…The following polynomially solvable case for an optimal c < -coloring was discovered in [40]. In [28], an algorithm based on the mixed integer linear programming and a tabu search algorithm were developed for constricting heuristic c < -colorings of the mixed graph G = (V, A, E) and calculating upper bounds on the strict chromatic number χ < (G).…”