Applications of Dynamic Programming and Heuristics to the Traveling Salesman Problem ERIC SALMON & JOSEPH SEWELL

Traveling Salesman Problem

Heuristic Algorithms for TSP A heuristic is a technique for solving a problem quicker when other methods are too slow or for finding an approximate solution to a problem.  Random Search Generate random permutation for a tour  Genetic Algorithm Mimic evolution to arrive at a tolerable tour  Simulated Annealing Find a solution by moving slowly towards a global optimum without being trapped in local optimums

Genetic Algorithm  A genetic algorithm is a search heuristic that mimics the process of natural selection.

Simulated Annealing  Name inspired from metal work  Heating and cooling an object to alter its properties  While the algorithm is ‘hot’, it is allowed to jump out of its local optimums  As the algorithm ‘cools’ it begins to hone on the global optimum

Simulated Annealing (cont.)

Dynamic Programming  A method for solving complex problems by breaking them down into simpler sub-problems  Exploits sub-problem overlap  Example: Finding Fibonacci numbers.  F(n) = F(n-2) + F(n-1)  To find F(n) you must also compute F(n-2) and F(n-1)  These values will be recomputed for each F(n) you want to find  Using Dynamic Programming, every computed value would be stored which would then be looked up before computation.

Computing Fibonacci (naïve) fib(n) if n <= 2 : f = 1 else : f = fib(n-1) + fib(n-2) return f

Dynamic Programing: Fibonacci array = {} fib(n): if n in array: return array[n] if n <= 2 : f = 1 else: f = fib(n-1) + fib(n-2) array[n] = f return f

Branch & Bound  An algorithm design for optimization problems.  Enumerations are possible solutions  Candidate partial solutions are child nodes from the root  Before enumerating child node, this branch is checked against upper/lower bounds compared to optimal solution  In the case of TSP this would be total distance up to that node  If this value is greater than the bound, discard entire branch  No added distance would ever decrease total distance  Continue enumeration through tree until solution found

Branch & Bound  is an algorithm design paradigm for discrete and combinatorial optimization problems. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm.

Branch & Bound  A branch-and-bound procedure requires two tools. The first one is a splitting procedure that, given a set S of candidates, returns two or more smaller sets S 1, S 2, … whose union covers S. Note that the minimum of f(x) over S is min{v 1, v 2, …}, where each v i is the minimum of f(x) within S i. This step is called branching, since its recursive application defines a search tree whose nodes are the subsets of S.

Branch & Bound  The second tool is a procedure that computes upper and lower bounds for the minimum value of f(x) within a given subset of S. This step is called bounding.  The key idea of the BB algorithm is: if the lower bound for some tree node (set of candidates) A is greater than the upper bound for some other node B, then A may be safely discarded from the search. This step is called pruning, and is usually implemented by maintaining a global variable m (shared among all nodes of the tree) that records the minimum upper bound seen among all sub-regions examined so far. Any node whose lower bound is greater than m can be discarded.

Branch & Bound  The recursion stops when the current candidate set S is reduced to a single element, or when the upper bound for set S matches the lower bound. Either way, any element of S will be a minimum of the function within S.

Branch & Bound on TSP  Given:

Branch & Bound on TSP