The Traveling Salesman Problem (TSP) is a classic optimization challenge in computer science. It's all about finding the shortest route to visit a bunch of cities once and return home, but it's super tricky to solve perfectly.
Approximation algorithms come to the rescue when exact solutions are too hard to find. For TSP, we've got clever tricks like that give us pretty good answers without taking forever to run.
Traveling Salesman Problem
Problem Definition and Characteristics
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Traveling Salesman Problem (TSP) seeks shortest route visiting each city once and returning to start
Classified as NP-hard problem with no known polynomial-time optimal algorithm
Standard TSP assumes complete graph with weighted edges representing distances between cities
holds in standard TSP (direct path always shorter than or equal to indirect path)
Real-world applications include logistics, transportation planning, and circuit board drilling
TSP Variations
involves edge weights differing based on direction of travel
(mTSP) uses multiple salesmen starting from a depot to divide cities
requires visiting cities within specific time intervals
places cities as points in a 2D plane
Approximation Algorithms for TSP
Christofides Algorithm
Provides 3/2-approximation for
Combines minimum spanning tree with minimum-weight perfect matching
Best-known polynomial-time approximation for general metric case
Steps include:
Construct minimum spanning tree
Find minimum-weight perfect matching on odd-degree vertices
Combine tree and matching to form Eulerian circuit