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Questions tagged [hamiltonian-paths]

A path in a graph is said to be Hamiltonian if it visits each vertex exactly once.

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Hamiltonicity of k-regular graphs

It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, ...
David Eppstein's user avatar
23 votes
1 answer
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I want an easy Gadget to prove Planar Hamiltonian Cycle NP-Complete (from Hamiltonian Cycle)

It is known that Hamiltonian (Ham for short) Cycle is NP-complete and that Planar Ham Cycle is NP-Complete. The proof for Planar Ham Cycle is not from Ham Cycle. Is there a nice gadget that will, ...
Bill GASARCH's user avatar
14 votes
6 answers
510 views

Problems with Unknown Single Exponential Time Agorithms

I'm looking for examples of problems for which it is easy to get algorithms running in time $2^{O(n\log n)}$, or $2^{O(n^c)}$ for some $c>1$ but for which it is not known whether there is an ...
verifying's user avatar
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14 votes
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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 ...
Standa Zivny's user avatar
14 votes
1 answer
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Is the longest trail problem easier than the longest path problem?

The longest path problem is NP-hard. The (typical?) proof relies on a reduction of the Hamiltonian path problem (which is NP-complete). Note that here the path is taken to be (node-)simple. That is, ...
Jasper's user avatar
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13 votes
4 answers
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List of strongly NP-hard problems with numerical data

I am looking for strongly NP-hard problems for a reduction. So far I have found the following problems: 3-partition problem bin-packing problem Numerical 3-dimensional matching TSP Any NP-complete ...
sigal maon's user avatar
10 votes
1 answer
319 views

Hamilton Decomposition Decision Problem

Let $G=(V,E)$ be an undirected graph. A decomposition of $V$ into disjoint subsets $V_i$ is called a Hamilton decomposition of $G$ if the subgraph induced by each set $V_i$ is either a Hamilton graph ...
Volker Turau's user avatar
9 votes
1 answer
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What's the expected length of the shortest hamiltonian path on a randomly selected points from a planar grid?

$k$ distinct points are selected randomly from a $p\times q$ grid. (Obviously $k\leq p\times q$ and is a given constant number.) A complete weighted graph is built from these $k$ points such that ...
Javad's user avatar
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8 votes
3 answers
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Cubic graphs and hamiltonian paths

I would like to ask, if anybody knows, whether there exists a 3-regular bridgeless graph which does not have a hamiltonian path (not necessarily extended to a hamiltonian circuit). Thank you
N27's user avatar
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8 votes
1 answer
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What are some effective heuristics to find the number of Hamiltonian paths in a rectangular grid?

A particular programming problem I came across recently reduces to finding hamiltonian paths in a rectangular grid that would look something like, ...
viksit's user avatar
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4 answers
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Applications of Hamiltonian Cycle Problem

The Hamiltonian Cycle Problem and Travelling Salesman Problem are among famous NP-complete problems and has been studied extensively. I am looking for applications of the HamCycle and TSP. What are ...
user136457's user avatar
7 votes
4 answers
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Hamiltonian Cycle as an integer linear programming problem

I'm trying to do reduce Hamiltonian Cycle to integer linear programming. Here's my idea: Create variables $e_{ij}$ for every edge $(i,j)$ in the graph. Require each $$e_{ij}\in \{0,1\}$$. Create ...
Jimbob's user avatar
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7 votes
2 answers
520 views

On the class of the FNP version of the Hamiltonian Cycle problem

This post is linked to: FNP complexity class Many places say that the decision version of Hamiltonian Cycle is NP-Complete, and NP-Complete problems are those whose solution can be verified in ...
dhruvbird's user avatar
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6 votes
1 answer
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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 ...
Dave's user avatar
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3 answers
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Is there an efficient algorithm for finding edges that are part of all hamiltonian paths?

I'm wondering if there is any algorithm known for finding edges in a graph that are part of all hamiltonian paths (operating under the assumption that the graph has at least one such path). Failing ...
BenJWoodcroft's user avatar
6 votes
1 answer
676 views

Two Hamiltonian path problem variants

While formalizing the gadgets for the proposed reduction of the question Efficient algorithm for existence of permutation with differences sequence? the following problems came to my mind: Problem 1 ...
Marzio De Biasi's user avatar
5 votes
1 answer
185 views

What's the probability for a random graph with degrees greater than 1 to be Hamiltonian?

Given a random graph by the Erdős–Rényi model, if the minimal node degree is greater than 1 (or $\geq 2$), or randomly select a graph from the graphs with node degrees greater than 1 ($\geq 2$), what'...
winston's user avatar
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0 answers
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Fine-Grained Hardness for Undirected Hamiltonicity

The fastest known algorithm for detecting Hamiltonian cycles in directed graphs on $n$ nodes runs in essentially $2^n\text{poly}(n)$ time. However, for undirected graphs on $n$ nodes, there is an ...
Naysh's user avatar
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4 votes
3 answers
571 views

FNP complexity class

Where can I find more information about the FNP complexity class? The only place I did find anything on FNP was http://en.wikipedia.org/wiki/FNP_(complexity) However, that isn't sufficient for me to ...
dhruvbird's user avatar
  • 460
4 votes
1 answer
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Complexity reductions to Hamiltonian Path?

I am looking for a NP-hardness reduction from an arbitrary problem to the Hamiltonian Path problem such that the reduced no-instances of Hamiltonian path are "far" from having a Hamiltonian path. Do ...
jbensmai's user avatar
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4 votes
1 answer
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Complexity of finding a path visiting all leaves on a tree while respecting a distance bound

I am interested in the complexity of a specific variant of the Hamiltonian path problem where we want to visit all leaves of a tree while respecting a distance bound. Formally, given an (undirected, ...
a3nm's user avatar
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4 votes
1 answer
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Best Hamiltonian Cycle Problem solver

What is the best Hamiltonian Cycle Problem (HCP) solvers available in the market? Googling so far shows that there is one created by Flinders University that can solve at most 5000 node instances. I ...
Izzah Leari's user avatar
4 votes
0 answers
78 views

Name for a cyclic path in a graph that visits every vertex while minimizing the maximum number of times a given vertex is revisited?

Me and my colleague are interested in whether anyone has looked into a generalization of Hamiltonian cycles where vertices can be revisited, but we want to minimize the maximum number of times a given ...
aellab's user avatar
  • 439
3 votes
1 answer
219 views

Minimum offset while measuring TSP paths

I have Euclidean graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices. I am trying to solve TSP with brute algorithm, and I want to ...
Ilya Gazman's user avatar
3 votes
1 answer
299 views

A decision problem related to the problem of counting Hamiltonian cycles

Define a decision problem H as follows. The input of H is a pair (G1,G2) of graphs, and the problem is to verify whether the number of Hamiltonian cycles in G1 is greater than the number of ...
Marcelo's user avatar
  • 31
3 votes
2 answers
417 views

Complexity of the Hamiltonian Subcycle problem

The problem is as follows: Given a graph $G$, find a (vertex) disjoint set of cycles $C$ on $G$ such that every vertex is visited by a cycle exactly once. My question is then: what is the ...
Alex ten Brink's user avatar
3 votes
2 answers
531 views

Hamiltonian cycle on a subset of 2D points, constrained by maximum total length

We are given a list of 2d coordinates, each coordinate representing a node in a graph, and a scalar D, which is a constraint on total length of the cycle. The task is to find a Hamiltonian cycle on a ...
Jan Hadáček's user avatar
3 votes
1 answer
287 views

Finding a hamiltonian cycle in $G'$ given a hamiltonian cycle in $G$

Say I have an undirected, weighted graph $G=(V,E)$ and I know a hamiltonian cycle of minimum weight in that graph. Can I use that information to efficiently find a hamiltonian cycle in $G'=(V',E')$ ...
mikeazo's user avatar
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3 votes
1 answer
482 views

Is there a known extension of Dirac's / Ghoulia-Houri's theorems for $k$-path existence?

In the well studied problem of Hamiltonicity, several papers/theorems gave sufficient "degree conditions" for the existence of Hamiltonian path in a graph. These include: Dirac's theorem , 1952, ...
R B's user avatar
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3 votes
0 answers
52 views

Partial Hamiltonian Path Optimization Problem

Let $G = (V,E)$ be a directed graph. Define the optimization problem in which the goal is to find a subset of edges in $G$ of maximum cardinality, such that (i) the in-degree and out-degree of each ...
John's user avatar
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3 votes
0 answers
75 views

Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles

Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles. Question: Is there some constant $k$ so that the $...
Elle Najt's user avatar
  • 1,469
2 votes
3 answers
1k views

Best bounds for the longest path optimization problem in cubic Hamiltonian graph?

optimization problem Input: cubic Hamiltonian graph feasible solution: A simple path measure to optimize: length of the simple path Design a polynomial-time algorithm that outputs the longest path ...
Mohammad Al-Turkistany's user avatar
2 votes
1 answer
182 views

Is balanced Hamiltonian cycle NP complete on maximal plane graphs?

I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs. If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
Elle Najt's user avatar
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2 votes
1 answer
245 views

Searching for name of equivalence property in hamiltonian paths

This one has been bugging me for a while. A long time ago in undergrad, I noticed this while learning about TSP. Nobody recognized it and I basically gave up. Given a hamiltonian path, any subpath ...
user3266's user avatar
2 votes
0 answers
102 views

Finding Hamilton cycles in random graphs

For a random graph $G$ of minimum degree 3, can we find a Hamilton cycle in linear time (with high probability for every edge density)? If this is an open problem, I will also accept an empirically ...
Dmytro Taranovsky's user avatar
1 vote
1 answer
267 views

On the paper "Quantum Computing Hamiltonian cycles"

The paper Quantum Computing Hamiltonian cycles claims: An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve ...
joro's user avatar
  • 1,955
1 vote
1 answer
1k views

Finding a Hamiltonian cycle from perfect matching of a bipartite graph

A disjoint vertex cycle cover of G can be found by a perfect matching on the bipartite graph, H, constructed from the original graph, G, by forming two parts G (L) and its copy G(R) with original ...
Ganapati Natarajan's user avatar
1 vote
1 answer
139 views

K-fold Traveling salesman problem - A variant of TSP

Consider a weighted graph $K_n$ and where the weights between vertices $i,j$ is $w_{ij}$. Consider a path, $\sigma$, passing through each vertex only once. Here $\sigma_i$ is the vertex in the $(i\%n)^...
Vivek Bagaria's user avatar
1 vote
0 answers
68 views

Finding a Hamiltonian cycle in a graph if we are guaranteed that there are not many of them in the graph

Problem: Given an undirected simple graph $G=(V,E)$ on $n$ vertices, such that there are not more than $c^n$ ($c<2$) Hamiltonian cycles in $G$, find a Hamiltonian Cycle in $G$ if there exists one. ...
jamal_asif's user avatar
1 vote
0 answers
223 views

Cheapest Insertion is $2$-approximation for TSP

Consider the Cheapest Insertion Algorithm on a complete graph with $n$ vertices, where each edge $uv$ has a weight $w(uv)$, and the weights satisfy the triangle inequality $w(xz)\leq w(xy)+w(yz)$ for ...
Ioana Roman's user avatar
1 vote
0 answers
59 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
Travis Black's user avatar
0 votes
0 answers
35 views

Analysis of a pruning algorithm for the Hamiltonian path problem

$Definition:$ A disconnecting path in a connected undirected graph $G$ is a path, deletion of which disconnects $G$ as shown in the illustration below. Now consider the following algorithm to decide ...
Yolov4's user avatar
  • 1
-1 votes
1 answer
402 views

A sufficient condition for non existance of hamiltonian cycle

I think i have a sufficient condition for non existance of hamiltonian cycle in a graph, I want to check if it has already been found, I tried googling for it and didnt find anything so far, how can i ...
Ofek Ron's user avatar
  • 125
-1 votes
1 answer
139 views

What are the known classes of undirected graphs such that every graph belonging to that class is guaranteed to have a Hamiltonian Path?

One trivial class of graphs is the class consisting of complete graphs or complete bipartite graphs with equal sized partitions. I would love to know if more such classes exist.
Vk1's user avatar
  • 137
-2 votes
1 answer
242 views

Connecting partial paths to form a hamiltonian cycle [closed]

For an undirected graph that consists of partial paths such that each vertex is a part of one of those paths and that there are edges between all the paths, is there an efficient algorithm to connect ...
Izzah Leari's user avatar
-5 votes
1 answer
825 views

Self-avoiding walk in Graph [closed]

Short question: How many self-avoiding-filling-polygons are there in a grid-graph of $n×n$? Long question: Edit: This question is not about p = np. I am searching for a way to calculate the numbers ...
Ilya Gazman's user avatar