Math Models of OR: Traveling Salesman Problem

Transcription

1 Math Models of OR: Traveling Salesman Problem John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY USA November 2018 Mitchell Traveling Salesman Problem 1 / 19

2 Outline 1 Examples 2 Proving Optimality 3 Subtours 4 Reoptimizing after adding subtour elimination constraints Mitchell Traveling Salesman Problem 2 / 19

3 Examples Outline 1 Examples 2 Proving Optimality 3 Subtours 4 Reoptimizing after adding subtour elimination constraints Mitchell Traveling Salesman Problem 3 / 19

4 Examples The Traveling Salesman Problem TSP Webpage Bill Cook, Waterloo Robert Bosch Oberlin College Find shortest route that visits each city exactly once. Largest problem solved to date has more than 85,000 cities. Applications: VLSI, vehicle routing (UPS, school buses,...) Mitchell Traveling Salesman Problem 4 / 19

5 Proving Optimality Outline 1 Examples 2 Proving Optimality 3 Subtours 4 Reoptimizing after adding subtour elimination constraints Mitchell Traveling Salesman Problem 5 / 19

6 Proving Optimality How do we know we are optimal? We ve found a good tour somehow. How do we know that we can t do better? Look at relaxations. Venn diagram: set of tours relaxation of set of tours Mitchell Traveling Salesman Problem 6 / 19

7 Proving Optimality How do we know we are optimal? We ve found a good tour somehow. How do we know that we can t do better? Look at relaxations. Venn diagram: optimal tour optimal relaxation Optimal relaxation gives lower bound on length of optimal tour Mitchell Traveling Salesman Problem 6 / 19

8 Proving Optimality Constructing a Relaxation Mitchell Traveling Salesman Problem 7 / 19

9 Proving Optimality Constructing a Relaxation Use exactly two edges incident to each vertex. Mitchell Traveling Salesman Problem 7 / 19

10 Proving Optimality Optimization Formulation Use binary variables: x e = 1 if edge e is in tour 0 otherwise Let E denote the set of edges and V the set of vertices. Let c e be the cost of edge e. Let (v) denote the edges incident to vertex v. min x Pe2E c ex e subject to P e2 (v) x e = 2 for all v 2 V and x e = 0 or 1 for all e 2 E. Mitchell Traveling Salesman Problem 8 / 19

11 Subtours Outline 1 Examples 2 Proving Optimality 3 Subtours 4 Reoptimizing after adding subtour elimination constraints Mitchell Traveling Salesman Problem 9 / 19

12 May have subtours Subtours Ledge, foreach vertex s v 0 A t w r u Mitchell Traveling Salesman Problem 10 / 19

13 Subtours May have subtours s v t w r Optimal tour u Mitchell Traveling Salesman Problem 10 / 19

14 Subtours May have subtours """""" s v t w Solution to relaxation: use 2 edges at each vertex r u Mitchell Traveling Salesman Problem 10 / 19

15 Subtours May have subtours s {r,t,u}: Xrttxrutxc.us 2. v ar t w Use subtour elimination constraint to tighten relaxation: x rs + x st + x rt apple 2 u Mitchell Traveling Salesman Problem 10 / 19

16 Subtours Refinements Can add subtour elimination constraints for any subset U V, with 3 apple U apple V /2. Too many subtour elimination constraints to use them all, so add them selectively as needed. (How many possible constraints do we have?) Can also relax the restriction that x be binary: use 0 apple x e apple 1. Then have linear program, which can be solved efficiently. Extensions of this approach can solve instances of the TSP with tens of thousands of cities (and hundreds of millions of edges). Mitchell Traveling Salesman Problem 11 / 19

17 Subtours Refinements Seca, K s l U l-l. Can add subtour elimination constraints for any subset U V, with 3 apple U apple V /2. Too many subtour elimination constraints to use them all, so add them selectively as needed. (How many possible constraints do we have?) Can also relax the restriction that x be binary: use 0 apple x e apple 1. Then have linear program, which can be solved efficiently. Extensions of this approach can solve instances of the TSP with tens of thousands of cities (and hundreds of millions of edges). Mitchell Traveling Salesman Problem 11 / 19

18 Subtours Refinements Can add subtour elimination constraints for any subset U V, with 3 apple U apple V /2. Too many subtour elimination constraints to use them all, so add them selectively as needed. (How many possible constraints do we have?) Can also relax the restriction that x be binary: use 0 apple x e apple 1. Then have linear program, which can be solved efficiently. Extensions of this approach can solve instances of the TSP with tens of thousands of cities (and hundreds of millions of edges). Mitchell Traveling Salesman Problem 11 / 19

19 Subtours Refinements Can add subtour elimination constraints for any subset U V, with 3 apple U apple V /2. Too many subtour elimination constraints to use them all, so add them selectively as needed. (How many possible constraints do we have?) Can also relax the restriction that x be binary: use 0 apple x e apple 1. Then have linear program, which can be solved efficiently. Extensions of this approach can solve instances of the TSP with tens of thousands of cities (and hundreds of millions of edges). Mitchell Traveling Salesman Problem 11 / 19

20 Reoptimizing after adding subtour elimination constraints Outline 1 Examples 2 Proving Optimality 3 Subtours 4 Reoptimizing after adding subtour elimination constraints Mitchell Traveling Salesman Problem 12 / 19

21 Reoptimizing after adding subtour elimination constraints Using dual simplex to reoptimize s 5 v 1 edge lengths c e shown 1 1 t 5 w r 5 u Use two edges incident to each vertex. Eg: x rs + x rt + x ru = 2. Have 0 apple x ij apple 1 for each edge (i, j). Only impose upper bound explicitly in simplex tableau for the long edges. Mitchell Traveling Salesman Problem 13 / 19

22 Reoptimizing after adding subtour elimination constraints Initial tableau quo steal L-xrstxre + - C x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw m (= Xew + Sew Mitchell Traveling Salesman Problem 14 / 19

23 Reoptimizing after adding subtour elimination constraints Optimal tableau for relaxation 9 basic variables: columnof identitymatrix x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw i n s v \ See xru=xsv t w exe-=o r u Mitchell Traveling Salesman Problem 15 / 19

24 Reoptimizing after adding subtour elimination constraints Add subtour elimination contraint x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw x rst x rs + x rt + x st apple 2, slack variable x rst Mitchell Traveling Salesman Problem 16 / 19

25 Reoptimizing after adding subtour elimination constraints Pivot to dual canonical form x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw x rst Optimize using dual simplex. Mitchell Traveling Salesman Problem 17 / 19

26 Reoptimizing after adding subtour elimination constraints Result of first dual simplex pivot x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw x rst Optimize using dual simplex. Mitchell Traveling Salesman Problem 18 / 19

27 Reoptimizing after adding subtour elimination constraints Result of second dual simplex pivot x rs x rt x st x uv x uw x vw x ru x sv x tw s ru s sv s tw x rst Optimal in relaxation Feasible in TSP s r t w v u Optimal in TSP Mitchell Traveling Salesman Problem 19 / 19