A Linear Search Strategy Using Bounds Sharlee Climer and Weixiong Zhang.

A Linear Search Strategy Using Bounds

Sharlee Climer andWeixiong Zhang

Sharlee Climer Washington University in St. Louis 2

Overview Introduction

Example problem

Results for Traveling Salesman Problem

Future Work


Introduction Linear search strategy called cut-and-solve At each step:

A chunk of the solution space is cut away and solved, providing incumbent solutions

A relaxed solution is found for remaining solution space

Iterate until relaxed solution is greater than or equal to incumbent

Cut-and-solve may not be useful for simple problem instances


Introduction

Use cut-and-solve to solve Linear Programs (LPs)

LPs are useful for modeling: Traveling Salesman Problem Constraint Satisfaction Problem Minimum cost flow problem


Introduction

Asymmetric Traveling Salesman Problem (ATSP) can be used to model: No-wait flowshop Stacker crane Tilted drilling machine Computer disk read head Robotic motion Pay phone coin collection


Introduction

Minimize Z = cij xij

s.t.: xij = 1 for j = 1,…,n

xij = 1 for i = 1,…,n

xij <= |W| - 1, for all proper non-empty subsets W of V

xij = {0,1}


Introduction NP-hard Frequently solved using search trees

Branch-and-bound Branch-and-cut

Search strategy Best-first search Depth-first search

Cut-and-solve has minimal memory requirements and no “wrong” subtrees


Introduction

ATSP solution space is a high-dimensional convex polyhedron

See paper for algorithm

Simple 2D problem for example


Example

Minimize Z = y – 4/5 x

s.t.: x >= 0y <= 3y + 13/6 x <= 9y – 5/13 x >= 1/14y + 3/5 x >= 6/5x,y integers


Example

x >= 0y <= 3y + 13/6 x <= 9y – 5/13 x >= 1/14y + 3/5 x >= 6/5x,y integers


Example


x = 0y = 3Z = 3


Example


x = 2y = 1Z = -0.6


Example


x = 3.5y = 1.4Z = -1.4


Example

Add new constraint:

y – 17/3 x >= -14


Example

Incumbent solution:

x = 3y = 2Z = -0.4


Example


x = 2.6y = 1.0Z = -1.1


Example

Add new constraint:


Example

New incumbent solution:

x = 2y = 1Z = -0.6


Example


x = 1.0y = 0.6Z = -0.2

Incumbent solution:Z = -0.6


Example

Keep new constraints and incumbent solution from one iteration to the next

Incumbent yields pruning opportunities

No children to choose between Need to decide size of cut


ATSP results

Implemented using cplex with default settings

Compared with Concorde (Applegate, Bixby, Chvatal, & Cook)

Concorde designed for STSP Used 2-node transformation


ATSP results

Testbed: All 27 TSPLIB instances 6 each: rtilt, stilt, crane, disk, coin, shop,

and super (Fischetti, Lodi, & Toth, 2002)

Cut-and-solve required 29.7% as much time as Concorde


ATSP results

rtilt class

100 cities100 trials


ATSP results

stilt class


ATSP results

crane class


ATSP results

disk class


ATSP results

coin class


ATSP results

shop class


Future work

Improve ATSP solver: Automatic determination of cut size Use specialized sparse problem solver

Clustering algorithm

A Linear Search Strategy Using Bounds Sharlee Climer and Weixiong Zhang.

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