graph/coloring: warm start optimisation of exact coloring algorithm #1705
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The graphs were obtained from https://mat.tepper.cmu.edu/COLOR/instances.html and converted to graph6 using the following code: package main import ( "bufio" "fmt" "os" "strconv" "strings" "gonum.org/v1/gonum/graph/encoding/graph6" "gonum.org/v1/gonum/graph/simple" ) func main() { g := simple.NewUndirectedGraph() sc := bufio.NewScanner(os.Stdin) for sc.Scan() { line := sc.Text() if len(line) == 0 || line[0] != 'e' { continue } f := strings.Fields(line) g.SetEdge(simple.Edge{F: node(f[1]), T: node(f[2])}) } fmt.Printf("%q\n", graph6.Encode(g)) } func node(s string) simple.Node { i, err := strconv.Atoi(s) if err != nil { panic(err) } return simple.Node(i) } To convert graphs (ASCII col format only), use `go run main.go < <input>.col'.
There is an optimisation that is available for the Dsatur recursive search that takes advantage of the trivial observation that the colouring of a maximum clique in the graph must be consistent with the colouring of the graph and that a maximum clique will therefore constrain the colouring of the graph. This means that if the maximum clique is first trivially coloured the search can continue from there. However, if there are multiple co-equal maximum cliques it is possible for a clique to be chosen that will force the search down a pessimal search path, so we choose the clique to start from as the clique with the least constraining effect on the rest of the graph. The optimisation has the effect of excluding large sections of the search tree with a small amount of initial work to assess the maximum cliques. This change chooses the maximum clique with the lowest degree into the rest of the graph and assigns colours to all its nodes and progresses from there.
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Codecov Report
@@ Coverage Diff @@
## master #1705 +/- ##
==========================================
+ Coverage 73.22% 73.36% +0.13%
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Files 502 511 +9
Lines 59536 60462 +926
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+ Hits 43598 44355 +757
- Misses 13487 13605 +118
- Partials 2451 2502 +51
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DO NOT MERGE
This is here to make the optimisation public. It can be used to demonstrate the improvement in the number of recursive calls, but it not yet complete since the underlying implementation of Dsatur is quite inefficient wrt allocations. This is something that I want to improve before I complete this addition so that the comparison between the optimisation and naive implementation can be made with respect to time as well.
The optimisation is in the final two commits, the first three being from #1615.