Network Optimization Problems: Models and Algorithms In this handout: Approximation Algorithms Traveling Salesman Problem

Classes of discrete optimization problems Class 1 problems have polynomial-time algorithms for solving the problems optimally. Ex.: Min. Spanning Tree problem For Class 2 problems (NP-hard problems) No polynomial-time algorithm is known; And more likely there is no one. Ex.: Traveling Salesman Problem

Three main directions to solve NP-hard discrete optimization problems: Integer programming techniques Heuristics Approximation algorithms We gave examples of the first two methods for TSP. In this handout, an approximation algorithm for TSP.

Definition of Approximation Algorithms Definition: An α-approximation algorithm is a polynomial-time algorithm which always produces a solution of value within α times the value of an optimal solution. That is, for any instance of the problem Zalgo / Zopt  α , (for a minimization problem) where Zalgo is the cost of the algorithm output, Zopt is the cost of an optimal solution. α is called the approximation guarantee (or factor) of the algorithm.

Some Characteristics of Approximation Algorithms Time-efficient (sometimes not as efficient as heuristics) Don’t guarantee optimal solution Guarantee good solution within some factor of the optimum Rigorous mathematical analysis to prove the approximation guarantee Often use algorithms for related problems as subroutines Next we will give an approximation algorithm for TSP.

An approximation algorithm for TSP Given an instance for TSP problem, Find a minimum spanning tree (MST) for that instance. (using the algorithm of the previous handout) To get a tour, start from any node and traverse the arcs of MST by taking shortcuts when necessary. Example: Stage 1 Stage 2 red bold arcs form a tour start from this node

Approximation guarantee for the algorithm In many situations, it is reasonable to assume that triangle inequality holds for the cost function c: E  R defined on the arcs of network G=(V,E) : cuw  cuv + cvw for any u, v, w V Theorem: If the cost function satisfies the triangle ineqality, then the algorithm for TSP is a 2-approximation algorithm. v w u

Approximation guarantee for the algorithm (proof) First let’s compare the optimal solutions of MST and TSP for any problem instance G=(V,E), c: E  R . Idea: Get a tour from Minimum spanning tree without increasing its cost too much (at most twice in our case). Optimal TSP sol-n A tree obtained from the tour Optimal MST sol-n (*) Cost (Opt. TSP sol-n) ≥ Cost (of this tree) ≥ Cost (Opt. MST sol-n)

Approximation guarantee for the algorithm (proof) red bold arcs form a tour The algorithm takes a minimum spanning tree starts from any node traverse the MST arcs by taking shortcuts when necessary to get a tour. What is the cost of the tour compared to the cost of MST? Each tour (bold) arc e is a shortcut for a set of tree (thin) arcs f1, …, fk (or simply coincides with a tree arc) 1 2 3 4 5 6 start from this node

Approximation guarantee for the algorithm (proof) Based on triangle inequality, c(e)  c(f1)+…+c(fk) E.g, c15  c13 + c35 c23  c23 But each tree (thin) arc is shortcut exactly twice. (**) E.g., tree arc 3-5 is shortcut by tour arcs 1-5 and 5-6 . The following chain of inequalities concludes the proof, by using the facts we obtained so far: red bold arcs form a tour 1 2 3 4 5 6 start from this node

Performance of TSP algorithms in practice A more sophisticated algorithm (which again uses the MST algorithm as a subroutine) guarantees a solution within factor of 1.5 of the optimum (Christofides). For many discrete optimization problems, there are benchmarks of instances on which algorithms are tested. For TSP, such a benchmark is TSPLIB. On TSPLIB instances, the Christofides’ algorithm outputs solutions which are on average 1.09 times the optimum. For comparison, the nearest neighbor algorithm outputs solutions which are on average 1.26 times the optimum. A good approximation factor often leads to good performance in practice.