Capacitated minimum spanning tree

Esau-Williams heuristic finds suboptimal CMST that are very close to the exact solutions, but on average EW produces better results than many other heuristics.

Initially, all nodes are connected to the root ${\displaystyle r}$ (star graph) and the network's cost is ${\displaystyle \displaystyle \sum _{i=0}^{n}c_{ri}}$; each of these edges is a gate. At each iteration, we seek the closest neighbor ${\displaystyle a_{j}}$ for every node in ${\displaystyle G-{r}}$ and evaluate the tradeoff function: ${\displaystyle t(a_{i})=g_{i}-c_{ij}}$. We look for the greatest ${\displaystyle t(a_{i})}$ among the positive tradeoffs and, if the resulting subtree does not violate the capacity constraints, remove the gate ${\displaystyle g_{i}}$ connecting the ${\displaystyle i}$-th subtree to ${\displaystyle a_{j}}$ by an edge ${\displaystyle c_{ij}}$. We repeat the iterations until we can not make any further improvements to the tree.

Esau-Williams heuristics for computing a suboptimal CMST:

function CMST(c,C,r):
    T = {${\displaystyle c_{1r}}$, ${\displaystyle c_{2r}}$, ..., ${\displaystyle c_{nr}}$}
    while have changes:
        for each node ${\displaystyle a_{i}}$
            ${\displaystyle a_{j}}$ = closest node in a different subtree
            ${\displaystyle t(a_{i})}$ = ${\displaystyle g_{i}}$ - ${\displaystyle c_{ij}}$
        t_max = max(${\displaystyle t(a_{i})}$)
        k = i such that ${\displaystyle t(a_{i})}$ = t_max
        if ( cost(i) + cost(j) <= c)
            T = T - ${\displaystyle g_{k}}$
            T = T union ${\displaystyle c_{kj}}$
    return T

It is easy to see that EW finds a solution in polynomial time.

Sharma's heuristic.[3]