Algorithm Design

688
Chapter 12 Local Search
this difference? The only possible change in the assignment penalties could
come from nodes in V~: for each i ~ V~, the change is ci(f*(i))- ciff(i))-
The separation penalties differ between the two labelings only ’in edges (i, j)
that have at least one end in V~. The following inequality accounts for these
differences.
(12.10) For a labeling f and its neighbor fa, roe have
E
d~(fa) -- ~(f) _ 0 for
all a ~ L. We use (12.10) to bound a2(fa) - ~(f) and then add the resulting
inequalities for all labels to obtain the following:
12.6 Classification via Local Search
We will rearrange the inequality by grouping the positive terms on the lefihand
side and the negative terms on the right-hand side. On the lefi-hand
side, we get Q(f*(i)) for all nodes i, which is exactly the assignment penalty
of f*. In addition, we get the term pi] twice for each of the edges separated by
f* (once for each of the two labels f*(i) and f*(j)).
On the right-hand side, we get Q(f(i)) for each node i, which is exactly the
assignment penalty of f. In addition, we get the terms pij for edges separated
by f. We get each.such separation penalty at least once, and possibly twice if
it is also separated by f*. -
In summary, we get the following.
2qb(f*) > E
a~L
proving the claimed bound.
E q(f*(i)) + E P~i~
_i~V~ (i,]) leaving V~
E Q(f(i)) + E P~J] >i~
V~ (i,j) in or leaving V~
We proved that all local optima are good approximations to the labeling
with minimum penalty. There is one more issue to consider: How fast does
the algorithm find a local optimum? Recall that in the case of the Maximum-
Cut Problem, we had to resort to a variant of the algorithm that accepts only
big improvements, as repeated local improvements may not run in polynomial
time. The same is also true here. Let ~ > 0 be a constant. For a given labeling f,
we will consider a neighboring labeling f’ a significant improvement if ~ (f’) <
(1 - ¢/3k) a~ (f). To make sure the algorithm runs in polynomial time, we should
only accept significant improvements, and terminate when no significant
improvements are possible. After at most ~-lk significant improvements, the
penalty decreases by a constant factor; hence the algorithm will terminate in
polynomial time. It is not hard to adapt the proof of (12.11) to establish the
following.
(12.12) For any fixed ~ > O, the version of the local search algorithm that only
accepts significant improvements terminates in polynomial time and results in
a labeling f such that q)(f)