A Linear Search Strategy Using Bounds

A Linear Search Strategy Using Bounds Sharlee Climer and Weixiong Zhang

Overview • Introduction • Example problem • Results for Traveling Salesman Problem • Future Work Washington University in St. Louis

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 Washington University in St. Louis

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 Washington University in St. Louis

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 Washington University in St. Louis

Introduction Minimize Z = SScij xij s.t.: Sxij = 1 for j = 1,…,n Sxij = 1 for i = 1,…,n SSxij <= |W| - 1, for all proper non-empty subsets W of V xij = {0,1} Washington University in St. Louis

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 Washington University in St. Louis

Introduction • ATSP solution space is a high-dimensional convex polyhedron • See paper for algorithm • Simple 2D problem for example Washington University in St. Louis

Example Minimize Z = y – 4/5 x s.t.: x >= 0 y <= 3 y + 13/6 x <= 9 y – 5/13 x >= 1/14 y + 3/5 x >= 6/5 x,y integers Washington University in St. Louis

Example x >= 0 y <= 3 y + 13/6 x <= 9 y – 5/13 x >= 1/14 y + 3/5 x >= 6/5 x,y integers Washington University in St. Louis

Example Minimize Z = y – 4/5 x x = 0 y = 3 Z = 3 Washington University in St. Louis

Example Minimize Z = y – 4/5 x x = 2 y = 1 Z = -0.6 Washington University in St. Louis

Example Minimize Z = y – 4/5 x x = 3.5 y = 1.4 Z = -1.4 Washington University in St. Louis

Example Add new constraint: y – 17/3 x >= -14 Washington University in St. Louis

Example Incumbent solution: x = 3 y = 2 Z = -0.4 Washington University in St. Louis

Example Minimize Z = y – 4/5 x x = 2.6 y = 1.0 Z = -1.1 Washington University in St. Louis

Example Add new constraint: Washington University in St. Louis

Example New incumbent solution: x = 2 y = 1 Z = -0.6 Washington University in St. Louis

Example Minimize Z = y – 4/5 x x = 1.0 y = 0.6 Z = -0.2 Incumbent solution: Z = -0.6 Washington University in St. Louis

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 Washington University in St. Louis

ATSP results • Implemented using cplex with default settings • Compared with Concorde (Applegate, Bixby, Chvatal, & Cook) • Concorde designed for STSP • Used 2-node transformation Washington University in St. Louis

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 Washington University in St. Louis

ATSP results rtilt class 100 cities 100 trials Washington University in St. Louis

ATSP results stilt class Washington University in St. Louis

ATSP results crane class Washington University in St. Louis

ATSP results disk class Washington University in St. Louis

ATSP results coin class Washington University in St. Louis

ATSP results shop class Washington University in St. Louis

Future work • Improve ATSP solver: • Automatic determination of cut size • Use specialized sparse problem solver • Clustering algorithm Washington University in St. Louis

A Linear Search Strategy Using Bounds