Local search for minimum spanning trees. Consider the set of all spanning trees (not just minimum ones) of a weighted, connected, undirected graph
G = (V, E).
Recall from Section 5.1 that adding an edge e to a spanning tree T creates an unique cycle, and subsequently removing any other edge e' ≠ e from this cycle gives back a different spanning tree T'. We will say that T and T' differ by a single edge swap (e, e') and that they are neighbors.
(a) Show that it is possible to move from any spanning tree T to any other spanning tree T' by performing a series of edge-swaps, that is, by moving from neighbor to neighbor. At most how many edge-swaps are needed?
(b) Show that if T' is an MST, then it is possible to choose these swaps so that the costs of the spanning trees encountered along the way are nonincreasing. In other words, if the sequence of spanning trees encountered is
T = T0→ T1 →....→ T2 → Tk = T′
then cost(Ti+1)
(c) Consider the following local search algorithm which is given as input an undirected graph with distinct edge weights.
Let T be any spanning tree of G
while there is an edge-swap (e, e') which reduces
cost(T):
T→ T + e – e′
return T
Show that this procedure always returns a minimum spanning tree. At most how many iterations does it take?
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