Three-quarter approximation for the number of unused colors in graph coloring

Conjugate conflict continuation graphs for multi-layer constrained via minimization

“…As can be seen, three vias are eliminated. A feasible solution to 2-layer CVM can be also obtained by coloring the graph in two colors such that the two vertices of any conflict edge are marked with different colors [5,18,29]. Note that the broken lines in an original layout should not be confused with those in a graph model.…”

Differential Ratio Approximation

“…Sub-sequent papers ( [23,24,40,36,37,57,32]) have improved its differential approximation ratio from 1/2 to 289/360. This problem is also a typical example of a problem that behaves in completely different ways when dealing with the standard or the differential paradigms.…”

z-Approximations

“…Very simple algorithms with factors of 1 2 and 1 3 are described in [24], improvements are given in [22] (see also [36]), and finally a factor of 59 72 is given in [16]. These algorithms approximate the color saving problem, which amounts to maximizing the saving in colors over the trivial solution which assigns a different color to each vertex.…”