用 2 colors 逼近随机着色图

Question

Having trouble figuring out how to show this approximation, hoping someone could lend some advice. I'm quite new to approximation (especially with randomization) and having trouble figuring out how to narrow this down.

The problem:

Suppose we have a graph G = (V,E) , each edge with a weight w .

We want to color the graph with 2 colors, red and blue . We want to maximize the edge weight from each vertex from red to blue .

We randomly mark each vertex with either red or blue with probably 1/2 for each. The coloring is done independently of every vertex.

I need to show that this color assignment randomization algorithm is a 4-approximaton . However, not entirely sure where to start. Anyone have any ideas?

Answer 1

Even the simplest greedy algorithm will produce better approximations than randomly assigning colors.

Like this:

Mark all nodes uncolored
Mark all edges unprocessed
Sort edges into decreasing weight
LOOP until all edges processed
   Select heaviest unprocessed edge
   IF both nodes uncoloured
        color nodes on edge opposite colors
   IF one node uncolored
        color node opposite color to its partner
   mark edge processed
   ENDLOOP