Tagged Questions

The travelling salesman problem (TSP) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once.

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2
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0answers
96 views

Christofides algorithm for directed graph

Is it possible to implement the Christofides algorithm for an directed Graph? Suppose you have an undirected Graph, in which every vertex has an edges in both ways to every other in the graph (not to ...
0
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0answers
37 views

Asymmetric metric TSP when many edges have equal costs in both directions

I would like to ask whether there exists a better approximation result on a special case of the ATSP metric instances: when cost(a,b)=cost(b,a) for $O(log(|E|)$ edges, or something close/related to ...
0
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0answers
153 views

Worst case of heuristics for symmetric TSP

I have implemented the nearest neighbor heuristic for solving symmetric TSP problems. I was wondering if there is any relation between the solution found by the heuristic and the optimal solution? ...
2
votes
1answer
235 views

Heuristics for tsp without triangle inequality

Every heuristic for the traveling salesman problem that I know of (Nearest-Neighbour, Christofides, Held-Karp, ...) assumes that the triangle inequality holds. Are there heuristics to solve the tsp ...
6
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0answers
430 views

Euclidean TSP algorithms

Are there any known exact algorithms for Euclidean TSP that take advantage of the inherent structure of the problem? Do any of these algorithms have better asymptotics than $O(2^n n^2)$ of a DP ...
1
vote
0answers
255 views

Formulation of the k-TSP as an integer programming problem?

Specifically, in a complete graph, I'm trying to find the simple path with $k$ nodes that minimizes the sums of their vector edge weights. Additionally, the solution should be Pareto efficient ...
6
votes
2answers
149 views

Given a set of distances (no info regarding what points the distances correspond to) from a complete graph, is the realization of the graph unique?

There are $n$ points in $R^2$ (i.e. the 2D real space). We can think of them as a complete graph where edge weights correspond to the distance between points. Let $D$ be the distance matrix between ...
4
votes
1answer
380 views

Parameters of energy function for TSP

[This question was initially asked here. It went unanswered so I thought I should ask it in a different community.] I am reading this paper by Hopfield et al. On page six, the authors defined the ...
5
votes
1answer
620 views

Approximation for metric TSP: Worst case using nearest neighbor heuristic?

I'm looking at different heuristics that approximate solutions for a metric Traveling Salesman Problem. I was wondering if there is a worst case ratio of tours calculated by the nearest neighbor ...
0
votes
1answer
138 views

PTAS (polynomial time approximatin scheme) for euclidean TSP/Minimum-Cost k-Connected subgraph problem

Problem 1 I have read "On Approximation of the Minimum-Cost k-Connected Spanning Subgraph Problem" (by A. Czumaj, A. Lingas), and even in the abstract are 2 statements "We present a polynomial time ...
0
votes
0answers
162 views

Generate TSP instances with known optimal

Is there a known (polynomial in number of nodes) algorithm to generate TSP instances with known optimal value? The idea is to be able to generating arbitrary large instances with known optimal ...
6
votes
0answers
132 views

N shortest tours in a graph

I'm searching for papers dealing with the problem of finding not just the shortest tour in the graph (TSP) but finding N shortest tours. Somewhat surprisingly, I didn't find any mention of it, ...
8
votes
2answers
349 views

Generating interesting combinatorial optimization problems

I'm teaching a course on meta-heuristics and need to generate interesting instances of classic combinatorial problems for the term project. Let's focus on TSP. We are tackling graphs of dimension ...
4
votes
1answer
341 views

Shortest cycle with a specific number of vertices

Given an undirected graph with n nodes, I need to find the shortest cycle of involving exactly n/2 vertices (i.e. keeping the distance traveled by the cycle to a minimum). Some nodes cannot directly ...
5
votes
1answer
900 views

Guidelines to reduce general TSP to Triangle TSP

I am looking for the method / correct way to approach to reduce the traveling salesman problem to an instance of traveling salesman problem which satisfies the triangle inequality, ie: $D(a, b) \leq ...
2
votes
2answers
1k views

what is the real difference between traveling salesman problem (TSP) and vehicle routing problem (VRP)?

Both problems are well-known NP-hard problems with great similarities. In fact, I do not see the real difference between these two problems. It seems relatively easy to model TSR in the form of VRP ...
21
votes
1answer
401 views

Approximate 1d TSP with linear comparisons?

The one-dimensional traveling salesperson path problem is, obviously, the same thing as sorting, and so can be solved exactly by comparisons in $O(n\log n)$ time, but it is formulated in such a way ...
2
votes
0answers
219 views

TSP with multiple visits

Can you please suggest possible approaches for the following problem: find a path through graph verticies so that the distance (sum of edges weights) between two vertex i occurrences would be no more ...
9
votes
3answers
445 views

Special cases of Graphic TSP

In Graphic TSP, you are given an unweighted undirected graph $G$ and the goal is to find a shortest tour in $G$ that visits every vertex at least once. Note that this is NOT same as finding a ...
3
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0answers
196 views

Ordered routing problem which is NP-hard

All the np-hard routing problems I know are of the form, minimize some quantity while visiting the verticies in an unordered way. Are there problems which are still np-hard, if one has to visit the ...
18
votes
3answers
842 views

Solving Superstring Exactly

What is known about exact complexity of the shortest superstring problem? Can it be solved faster than $O^*(2^n)$? Are there known algorithms that solve shortest superstring without reducing to TSP? ...
0
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0answers
322 views

Algorithm to maximize profit: ways to solve/approach? (Advanced NP-Complete)

This one's hard, so all help really appreciated! I know it is NP-Complete and thus cannot be solved in polynomial time, but looking for help in analysis, i.e. what type of NP-Complete problem it ...
34
votes
4answers
4k views

Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
6
votes
2answers
386 views

Travelling Salesman and Planar Travel - Generalized TSP

Our beloved Travelling Salesman just bought the Manual of the Planes and wants to make some use of it. He is not a great adventurer though, so he will restrain his travels in the Parallel and ...
0
votes
0answers
181 views

Is there a series of algorithms for approximating TSP polynomially?

I've began studying some CS recently, and I've faced the TSP. The decision problem version of the TSP is NP-complete, right? I've noticed (and elaborated myself) that there exists several polynomial ...
1
vote
3answers
191 views

Simple spatial ordering or TSP algorithms?

I'm not sure if this is the right place to ask, but I suppose you'll tell me. I'm writing a program that produces a series of points on a map, and I need to put the points in some linear order so ...
2
votes
1answer
165 views

Question on the Prize-Collecting TSP's ratio related to inapprox. of general TSP

The Prize-Collecting TSP (PCTSP) is defined as the ordinary TSP with the difference that penalties are added to nodes; so we may avoid visiting a node paying its penalty, which is added to the overall ...
4
votes
2answers
557 views

Why is Metric TSP's best possible achieved approximation ratio believed to be 4/3?

Is it just that integrality gaps (LP/IP) for specific instances do not give more than 4/3? Thanks in priori.
7
votes
2answers
270 views

Is there a local variant of TSP?

I'm a traveling salesman and I have n days to sell, I can start anywhere, I can sell once per city. I want to know where to start and what route to take. It's likely NP-hard, I was just wondering if ...
2
votes
1answer
368 views

ATSP with direction restrictions

I'm trying to find any material on this problem. It extends the Asymmetric Travelling Salesman Problem (ATSP) in that it requires for some destinations that they are approached in the specified ...
2
votes
1answer
677 views

Ant colony optimization for traveling salesman problem with changing graph-nodes/vertices

Are there any publications focusing on solving TSP with ant colony optimization that consider small changes in the graph's nodes or vertices? So what I have is: a traveling salesman problem (TSP) ...
6
votes
3answers
1k views

A simple approximation algorithm for the TSP

Consider the following extremely simple approximation algorithm for the TSP. Input: A complete weighted graph $G=(V,E).$ Take any three vertices $a,b,c\in V$ and let $H:=(a,b,c,a).$ While there ...
13
votes
3answers
692 views

Classes of graphs with easy Hamiltonian cycle but NP-hard TSP

The Hamiltonian Cycle Problem (HC) consists in finding a cycle that goes through all vertices in a given undirected graph. The Travelling Salesman Problem (TSP) consists in finding a cycle that goes ...
9
votes
1answer
3k views

Time complexity of Held-Karp algorithm for TSP

When I looked through "A Dynamic Programming Approach to Sequencing Problems" by Michael Held and Richard M. Karp, I came up with the following question: why the complexity of their algorithm for TSP ...
9
votes
1answer
457 views

Any SAT/SMT formulations of the VRP/VRPTW (TSP, Job-Shop-Scheduling)?

i wonder if they are any approaches formulating a Vehicle-Routing-Problem with Time-Windows (VRPTW) (as a decision problem) as a SAT/SMT instance? (alternative: TSP) For example: "Is there a valid ...
18
votes
4answers
742 views

DNA-algorithms and NP-completeness

What is the relationship between DNA-algorithms and the complexity classes defined using Turing machines? What do the complexity measures like time and space correspond to in DNA-algorithms? Can they ...