Approximation Algorithms for the Traveling Salesman Problem Shayan Oveis Gharan

Traveling Salesman Problem (TSP) 2 Seattle What is the fastest route?

Problem Formulation 3 May represent time, gas usage, …

Applications 4

Methods of Attack Naïve Approach: Try all permutations! 5 ….. #permutations of 75 cities >> #atoms in the universe

Methods of Attack Naïve Approach: Try all permutations! Optimistic Approach: Practical instances are easy TSP on the 2,000,000 cities in the whole world 6

Methods of Attack Naïve Approach: Try all permutations! Optimistic Approach: Practical instances are easy 7 Bad scenarios happen in practice! Theory of Computing Approach: Find good solutions efficiently in the worst case.

NP Completeness 8

Approximation Algorithms 9 Run in time n or n 2 or n 3

Motivations for Worst case Approximation 10

Approximation Algorithms for TSP 11

General Approach 12 Discrete Optimization Problem Near Optimal Solution Very difficult: Hard to characterize optimum Rounding Linear Program Relaxation Integer Program Formulation Optimal Fractional solution LP-Solving

Formulation of the Optimum 13

Integer Program 14 Hard to solve Optimally Cost of the solution Exit whenever Enter Enter every subset of vertices

General Approach 15 Discrete Optimization Problem Integer Program Linear Program Optimal Fractional solution Near Optimal Solution Very difficult: Hard to characterize optimum Relaxation Rounding Formulation LP-Solving

LP Relaxation Proposed by Dantzig, Fulkerson, Johnson 1954 and Held, Karp Optimum remains feasible!

General Approach 17 Discrete Optimization Problem Integer Program Linear Program Optimal Fractional solution Near Optimal Solution Very difficult: Hard to characterize optimum Relaxation Rounding Formulation LP-Solving

Easy with LP Solvers matlab, cplex, mosek, gorubi, … 18 x i,j = 0.5 for all dashed edges

General Approach 19 Discrete Optimization Problem Integer Program Linear Program Optimal Fractional solution Near Optimal Solution Very difficult: Hard to characterize optimum Relaxation Rounding Formulation LP-Solving

Rounding (A Geometric View) 20 In higher dimensions rounding is more complicated. IP LP Rounding LP solution is not necessarily integral

Rounding Can be quite complicated in the worst case. Easy in typical instances, because frac sols are sparse. 21

Eulerian Graphs A graph G is Eulerian, if it is “connected” and the indegree of each vertex is equal to its outdegree. Any Eulerian graph has a walk that visits each edge exactly once. By triangle inequality, we can extract a TSP tour from an Eulerian walk of a smaller cost. 22

Rounding Can be quite complicated in the worst case. Easy in typical instances, because frac sols are sparse approximation x2 x i,j = 0.5 for all dashed edges 2 ≥ 2 =

Our Contribution [AGMOS] [FGM] us Approximation Factor Time

Conclusion 25