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The Traveling Salesman - Omede Firouz
The Traveling Salesman Problem
The Traveling Salesman - Omede Firouz
Problem Statement• Motivation
– How to find the shortest tour through a set of cities. Applications in VLSI, transportation.
• Why should we care? – One of the most well-studied optimization problems.
Shows what we can do in the face of NP-Complete problems.
• Mathematically– Given a complete graph G(V,E) with some distance
function d(E), find the minimum cost Hamiltonian cycle.
The Traveling Salesman - Omede Firouz
Problem Difficulty
• A naïve approach tries all possible tours O(n!)
• Held and Karp (Berkeley) improved this to O(2nn2) in 1962, which is the best known still.
• TSP is NP-Hard, but in practice what we can do is pretty amazing.
The Traveling Salesman - Omede Firouz
Problem Difficulty Continued
Much/most of this progress is due to improved algorithms, not hardware. Moore's Law says computer speed has increased exponentially. Problem difficulty increases exponentially with size. → Therefore, hardware only improvements would imply only a linear increase in the size of solved problems (graph on the right). But historically we have improved faster.
The Traveling Salesman - Omede Firouz
Problem Difficulty Continued
• 2006: VLSI 85,900 locations.
The Traveling Salesman - Omede Firouz
Problem Classification
• Metric– Satisfies triangle inequality: d(e
ij) d(e≦
ik) + d(e
kj)
• Symmetric– d(e
ij) = d(e
ji)
• Euclidean TSP– Distance satisfies the euclidean norm. Implies both
metric and symmetric.
The Traveling Salesman - Omede Firouz
Method of Attack
• Lower Bound– A solution to an easier and relaxed problem.– In practice: Linear Programming with Branch and Cut
• Upper Bound– A feasible solution to the current problem.– In practice: Simulated Annealing, Local Search,
Genetic Algorithms, Christofides Algorithm.
• Optimality when the two bounds match.
The Traveling Salesman - Omede Firouz
Approximation Difficulty• Non-metric
– Any approximation is NP-Hard by reduction to Hamiltonian Cycle.
• Symmetric and Metric– Christofides: 3/2 approximation ratio by using a MST and
perfect matching. > 20 years old– Many believe 4/3 is possible.– 220/119 is NP-Complete (Papadimitriou)
• s,t TSP– Very recently shown to have a 1.618 approximation ratio.
The Traveling Salesman - Omede Firouz
Held-Karp Relaxation
• Integrality gap is in the range of 3/2 to 4/3.
• But how do we implement this LP?
The Traveling Salesman - Omede Firouz
Held-Karp Relaxation
• S is a subset of V, so we have 2n constraints.
• But, we only need to find and use the violated ones which we can find efficiently with network flows.
The Traveling Salesman - Omede Firouz
Branch and Cut
Branch and Cut is a very powerful technique to solve integer programs.
• Branch– Solve the LP for the optimal x*– For a non-integer result, branch into two subproblems
• x*i 0, x*≦
i 1≧
• Cut– Use a cutting plane to remove the LP optimum but not remove
any integer solutions.
The Traveling Salesman - Omede Firouz
Branch and Cut Continued
• Cutting Planes– Gomory Cuts, elegant and very generalized cutting planes.– Lift and Project Cuts– Problem Specific Cuts
• Comb, Clique Tree, Path, Wheelbarrow, Bicycle, Ladder, and Crown inequalities.
• Pool data structure– Too many cuts makes the LP slower to solve, so we
eventually drop old cuts and keep the newer ones.
The Traveling Salesman - Omede Firouz
Polyhedral Combinatorics
• Branching is naïve and computationally expensive.
• Cutting however only adds constraints to an LP.
• If we had the tightest cuts possible, facet-defining cuts, it would be easy to find integer solutions.
• How do we find the proper cuts?
The Traveling Salesman - Omede Firouz
Polyhedral Combinatorics Cont.
• Gomory and Lift and Project Cuts:– Augment any constraint matrix A with a cut.– Problem: Can take a huge number of cuts,
and not numerically stable.
• Problem Specific Cuts– Example: Knapsack – Inequalities
• At least for knapsack, guaranteed to result in the proper solution eventually
• Easy to 'separate'– What about TSP Cuts?
The Traveling Salesman - Omede Firouz
Polyhedral Combinatorics Cont.
• Comb Inequalities– Good news: They are facet defining– Bad news: There's no known polynomial
separation algorithm.– However: Comb separation hasn't been
shown to be NP-Complete
Contrast to:
• Subtour Elimination Cuts– Bad news: Exponentially many– Good news: Polynomial separation
The Traveling Salesman - Omede Firouz
Heuristics: k-Opt
• A problem is called k-Optimal if we cannot improve the tour by switching k edges.
• k-Optimality takes O(nk) time.
• Is k-Optimality enough for full optimality?– Yes if k = n (clearly). But can we do better?
→ No. Papadimitriou (Berkeley) has shown a TSP can have exponentially many local optima that are arbitrarily far from the global optima and require k = O(na) and (0 < a < 1).
The Traveling Salesman - Omede Firouz
Heuristics: k-Opt Continued
• Despite the grim picture, k-Opt does work well.
• Furthermore, O(nk) implies we look at all k groups. However, we expect a peturbation to work if it is local.
• Why don't we restrict ourselves to the p nearest neighbors to get O(pk)?
The Traveling Salesman - Omede Firouz
Heuristics: k-Opt Continued
• P Nearest Neighbors– At optimality most nodes connect to their nearest neighbors.
– However, some nodes connect to very far neighbors. i.e. the distribution has a long tail.
– Another metric of nearest neighbor is needed.
The Traveling Salesman - Omede Firouz
Heuristics: k-Opt Continued• Optimizations:
– Use alpha nearness, or distance on a minimum 1-tree.– Restrict ourselves to 2,3 - Opt moves but allow sequences of
moves.– Escape local minima with 'kicks' where we purposefully cross
edges and hope the problem relaxes lower.– First implemented by Lin and Kernighan and 'perfected' with
LKH (Keld Helsgaun).
• Results:– LKH has found every optimum and holds every record for
unknown problems (including world TSP)
The Traveling Salesman - Omede Firouz
State of the Art
• Strong heuristics speed up the relaxation, and vice versa.
• Strongest Heuristics (Upper Bounds):– Initialize a tour with Christofides Algorithm– Run LKH and possibly use genetic algorithms.
• TSP is especially suited to genetic algorithms.
– Stop when we reach lower bound.
• Strongest Relaxations (Lower Bounds):– Held-Karp LP with many problem specific cutting planes.– Use best feasible solutions to prune the search tree.
The Traveling Salesman - Omede Firouz
Conclusion
• TSP has motivated many advances in integer programming and combinatorial optimization.
• It shows how good we can do on even NP-Hard problems in practice.
• It is a simple problem with a very deep and elegant mathematical framework.
The Traveling Salesman - Omede Firouz
Conclusion
• 2011– World TSP 1,900,000 locations– 7,515,778,188 using LKH vs 7,512,218,268 from Concorde + CPLEX– .0474% Optimality Gap
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