april 30
backtracking
Textbook algorithm generates one child,
then examines that one; algorithm as stated is recursive.
best-first branch-and-bound
Textbook algorithm here generates all children
if node is under some bound or within constraints,
and adds them to the fringe of "live" nodes. The
one with the lowest bound (whichever level of
the tree its at) is the one considered next.
(And therefore the fringe should be implemented as a heap.)
approximations
Define an "accuracy ratio" of how close a polynomial time approximation
comes to the exact answer.
traveling salesman
Text proves that there can't be a polynomial time approximation
that does within a constant ratio of exact answer on all cases.
cities 1...N
edges E[i,j]
lower_bound = sum_i (E[i,j] + E[i,k] ) /2
for two closest cities (j,k) to i
and for a bound that includes any specific edge E[p,q],
use that edge in the sum for city p and q along with
shortest other to that city
Note that the formulation of this problem typically
assumes that you have a "complete weighted graph",
that is, you can get from any city to any other.
nearest neighbor algorithm
1. start anywhere
2. go to closest neighbor
3. keep grabbing closest unvisited (greedy)
4. until all are seen; then jump back to original
Not very good; last leg can be very long.
multifragment-heuristic algorithm
1. Sort edges by increasing weight.
2. Initialize "answer" as a set of empty edges.
3. Repeat:
Add shortest edge left if
i) it doesn't create a vertex of order 3, and
ii) it doesn't create a cycle of length less than number of cities
Better than nearest neighbor.
Can compare accuracy explicity for "Euclidean" versions of the problem.
twice around the tree algorithm
1. Find minimum spanning tree.
2. Construct "twice around" path, including duplicate over-and-back pieces.
3. Elimate duplicates by using shortcuts
Christofides algorithm
1. assumes Euclidian distances
2. similar to "twice around" but smarter
local search or iterative improvement algorithms
Examples:
- 2-opt (delete 2 edges; reconnect differently)
- 3-opt (delete 3 edges; reconnect differently)
- Lin-Kernighan (well-known, complicated version that uses both of these)
Idea is to start with any path and try to make it better
by replacing a few edges with other edges.
empirical results
This problem has been studied extensively, and is
important in many fields including circuit design.
See table of empirical results, pg 447
