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Questions tagged [hamiltonian-circuit]

36 questions
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Upper bound on the number of hamiltonian cycles on a $n \times n $ grid graph

What is the best upper bound that is known for the number of hamiltonian cycles on a $n \times n $ grid graph? I did some searching and found that the number of hamiltonian cycles on a planar graph with $n$ vertices is $O(\sqrt[4]{30}^n)$ where $n$…
Gaganpreet
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Find hamilton cycle in a directed graph reduced to sat problem

I need to find a Hamiltonian cycle in a directed graph using propositional logic, and to solve it by sat solver. So after I couldn't find a working solution, I found a paper that describes how to construct a CNF formula to find an Hamiltonian…
Dor Cohen
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Hamiltonian circuit for a family of graphs

For $\Sigma = \{a,b\}$, let $S_n = \{w\mid w \in \Sigma^{*} \land |w| = n\}$. Let $C_n \subset \Sigma^{*}$ be the language of circular strings that contain as substrings all elements of $S_n$. For instance, $bbaaaababaabbbba \in C_4$ I have…
André Souza Lemos
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An FPT algorithm for Hamiltonian cycle running parameterized by treewidth

I'm looking for an algorithm that solves the Hamiltonian cycle problem parameterized by treewidth. In particular, I'm curious about such an algorithm running in $\text{tw}(G)^{O(\text{tw}(G))} \cdot n$ time. In other words, once you have the…
Vandalo
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$k$-Opt TSP Local Search is NOT exact when $k = \lceil |V|/2 \rceil$

I've been self-studying the book Algorithms by Papadimitriou, Dasgupta and Vazirani. I am having a hard time with a question about local search involving the traveling salesman problem (TSP). We'll say a local search algorithm is exact if it always…
Fimpellizzeri
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Is Hamiltonian cycle problem on graphs with out-degree at most 3 NP hard?

I am trying to show a different form of Hamiltonian cycle problem is NP Hard. The problem is as follows. We have a directed graph and each node can have at most 3 outgoing edges. Determine if this graph has a Hamiltonian cycle. Is this problem NP…
Davis
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What is the best way to merge cycles to minimise total weight?

Suppose I have a vertex-disjoint set $S$ of simple cycles in a weighted undirected graph. So no vertex $v$ is contained in more than one cycle. A cycle $c$ is a closed path with no repeated vertices: $c=(v_1, v_2, \ldots, v_n)$, where $v_n=v_1$.…
Dmitry Kamenetsky
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Complexity of ANOTHER HAMILTONIAN CIRCUIT problem

All references I find about the ANOTHER HAMILTONIAN CIRCUIT problem: Given a graph and a hamiltonian circuit on it, is there another hamiltonian circuit on it? I was trying to reduce it to the hamiltonian circuit problem but I always need to add…
user1868607
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How many inputs does the Hadamard gate have?

Look at the diagram in the middle of page 6-3 here, http://stellar.mit.edu/S/course/6/fa14/6.845/courseMaterial/topics/topic3/lectureNotes/qctlec6/qctlec6.pdf I am confused as to how should one think of the Hadamard gate. As per the mathematics…
user6818
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Is determining the existence of a Hamiltonian cycle in a chordal graph NP-hard?

The Hamiltonian cycle problem asks if a given graph contains a Hamiltonian cycle. The Hamiltonian cycle problem belongs to the class of NP-complete problems. However, for some special classes of graphs, the problem can be solved in polynomial time,…
licheng
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$k$-Opt TSP Local Search is exact when $k = |V| - 1$

I've been self-studying the book Algorithms by Papadimitriou, Dasgupta and Vazirani. I am having a hard time with a question about local search involving the traveling salesman problem (TSP). We'll say a local search algorithm is exact if it always…
Fimpellizzeri
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Hamiltonian cycle in $C_n^k$ in polynomial time for constant $k$?

Let $C_n$ denote the cycle graph over $n$ vertices. Let $C_n^k$ denote the $k$-th power of the cycle graph, or namely that for two vertices $i,j$, $(i,j)\in Edges(C_n^k) \iff |i-j|\leq k$ for a constant $k$. Now given a subgraph $G$ of $C_n^k$, how…
AspiringMat
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Find the $k$-th lexicographically smallest hamiltonian circuit

Let's say we have given unweighted directed graph with $N$ nodes and $M$ edges, and we want to find the $K$-th hamiltonian circuit, ordered in lexicographical order. For example, if we have complete graph with $4$ nodes, and $12$ edges. The smallest…
someone12321
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Limited constant degree HamCycle

Let $G=(V,E)$ be a directed graph. I am interested in a "relaxed" version of the HamCycle problem. In my first case, the degree of each vertex is exactly 6, such that: 3 are outgoing edges and 3 are ingoing edges, for every vertex in $V$. I would…
Eric_
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Proving NP-hardness of Hamiltonian Cycle problem variant

I need to prove that determining whether a graph has a relaxed-Hamiltonian cycle (definition given ahead) is NP-hard. A relaxed-Hamiltonian cycle in $G$ is a closed walk $C$ that visits every vertex of $G$ exactly once, except for at most one…
Dhruv Deshmukh
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