Editing Hamiltonian path problem

{{short description|Problem of finding a cycle through all vertices of a graph}}
{{about|the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph|the general graph theory concepts|Hamiltonian path}}

The '''Hamiltonian path problem''' is a topic discussed in the fields of [[Computational complexity theory|complexity theory]] and [[graph theory]]. It decides if a [[Directed graph|directed]] or [[Undirected graph|undirected]] [[Graph (discrete mathematics)|graph]], ''G'', contains a [[Hamiltonian path]], a path that visits every vertex in the graph exactly once. The problem may specify the start and end of the path, in which case the starting vertex ''s'' and ending vertex ''t'' must be identified.<ref name=":12">{{Cite book |last=Sipser |first=Michael |title=Introduction to the Theory of Computation |publisher=Cengage Learning |year=2013 |edition=3rd |pages=292–314}}</ref>

The '''Hamiltonian cycle problem''' is similar to the Hamiltonian path problem, except it asks if a given graph contains a [[Hamiltonian cycle]]. This problem may also specify the start of the cycle. The Hamiltonian cycle problem is a special case of the [[travelling salesman problem]], obtained by setting the distance between two cities to one if they are adjacent and two otherwise, and verifying that the total distance travelled is equal to ''n.'' If so, the route is a Hamiltonian cycle.

The Hamiltonian path problem and the Hamiltonian cycle problem belong to the class of [[NP-completeness|NP-complete]] problems, as shown in [[Michael Garey]] and [[David S. Johnson|David S. Johnson's]] book [[Computers and Intractability|Computers and Intractability: A Guide to the Theory of NP-Completeness]] and [[Richard M. Karp|Richard Karp's]] list of [[Karp's 21 NP-complete problems|21 NP-complete problems]].<ref>{{Cite book |last1=Garey |first1=Michael R |title=Computers and Intractability: A Guide to the Theory of NP-Completeness |last2=Johnson |first2=David S. |publisher=W. H. Freeman and Company |year=1979 |pages=60}}</ref><ref name="Held 1965 136–147">{{Cite journal |last1=Held |first1=M. |last2=Karp |first2=R. M. |date=1965 |title=The construction of discrete dynamic programming algorithms |url=http://dx.doi.org/10.1147/sj.42.0136 |journal=IBM Systems Journal |volume=4 |issue=2 |pages=136–147 |doi=10.1147/sj.42.0136 |issn=0018-8670}}</ref>

== Reductions ==

=== Reduction from the path problem to the cycle problem ===
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows:

* In one direction, the Hamiltonian path problem for graph ''G'' can be related to the Hamiltonian cycle problem in a graph ''H'' obtained from ''G'' by adding a new [[universal vertex]] ''x'', connecting ''x'' to all vertices of ''G''. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a Hamiltonian cycle. 
* In the other direction, the Hamiltonian cycle problem for a graph ''G'' is equivalent to the Hamiltonian path problem in the graph ''H'' obtained by adding terminal ([[degree (graph theory)|degree]]-one) vertices ''s'' and ''t'' attached respectively to a vertex v of G and to ''v','' a [[Edge contraction#Vertex cleaving|cleaved copy]] of ''v'' which gives ''v' ''the same neighbourhood as ''v''. The Hamiltonian path in ''H'' running through vertices {{tmath|s-v-x-\cdots-y-v'-t}} corresponds to the Hamiltonian cycle in ''G'' running through {{tmath|v-x-\cdots-y(-v)}}.<ref>[https://math.stackexchange.com/q/1290804 Reduction from Hamiltonian cycle to Hamiltonian path]</ref>

==Algorithms==

=== Brute force ===
To decide if a graph has a Hamiltonian path, one would have to check each possible path in the input graph G. There are ''n''! different sequences of vertices that ''might'' be Hamiltonian paths in a given ''n''-vertex graph (and are, in a [[complete graph]]), so a [[brute force search]] algorithm that tests all possible sequences would be very slow.

=== Partial paths ===
An early exact algorithm for finding a Hamiltonian cycle on a directed graph was the enumerative algorithm of Martello.<ref name="Held 1965 136–147"/> A search procedure by Frank Rubin<ref>{{citation |last=Rubin |first=Frank |title=A Search Procedure for Hamilton Paths and Circuits |journal=[[Journal of the ACM]] |volume=21 |issue=4 |pages=576–80 |year=1974 |doi=10.1145/321850.321854 |s2cid=7132716|doi-access=free }}</ref> divides the edges of the graph into three classes: those that must be in the path, those that cannot be in the path, and undecided. As the search proceeds, a set of decision rules classifies the undecided edges, and determines whether to halt or continue the search. Edges that cannot be in the path can be eliminated, so the search gets continually smaller. The algorithm also divides the graph into components that can be solved separately, greatly reducing the search size. In practice, this algorithm is still the fastest.

=== Dynamic programming ===
Also, a [[dynamic programming]] algorithm of [[Held-Karp algorithm|Bellman, Held, and Karp]] can be used to solve the problem in time O(''n''<sup>2</sup> 2<sup>''n''</sup>). In this method, one determines, for each set ''S'' of vertices and each vertex ''v'' in ''S'', whether there is a path that covers exactly the vertices in ''S'' and ends at ''v''. For each choice of ''S'' and ''v'', a path exists for (''S'',''v'') if and only if ''v'' has a neighbor ''w'' such that a path exists for (''S'' − ''v'',''w''), which can be looked up from already-computed information in the dynamic program.<ref>{{Cite journal |last=Bellman |first=Richard |date=January 1962 |title=Dynamic Programming Treatment of the Travelling Salesman Problem |journal=Journal of the ACM |language=en |volume=9 |issue=1 |pages=61–63 |doi=10.1145/321105.321111 |s2cid=15649582 |issn=0004-5411|doi-access=free }}</ref><ref>{{Cite journal |last1=Held |first1=Michael |last2=Karp |first2=Richard M. |date=March 1962 |title=A Dynamic Programming Approach to Sequencing Problems |url=http://epubs.siam.org/doi/10.1137/0110015 |journal=Journal of the Society for Industrial and Applied Mathematics |language=en |volume=10 |issue=1 |pages=196–210 |doi=10.1137/0110015 |issn=0368-4245}}</ref>

=== Monte Carlo ===
Andreas Björklund provided an alternative approach using the [[inclusion–exclusion principle]] to reduce the problem of counting the number of Hamiltonian cycles to a simpler counting problem, of counting cycle covers, which can be solved by computing certain matrix determinants. Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary ''n''-vertex graphs by a [[Monte Carlo algorithm]] in time O(1.657<sup>''n''</sup>); for [[bipartite graph]]s this algorithm can be further improved to time [[Big O notation#Little-o notation|O]](1.415<sup>''n''</sup>).<ref>{{Cite book |last=Bjorklund |first=Andreas |title=2010 IEEE 51st Annual Symposium on Foundations of Computer Science |chapter=Determinant Sums for Undirected Hamiltonicity |date=October 2010 |chapter-url=http://dx.doi.org/10.1109/focs.2010.24 |pages=173–182 |publisher=IEEE |doi=10.1109/focs.2010.24|arxiv=1008.0541 |isbn=978-1-4244-8525-3 }}</ref>

=== Backtracking ===
For graphs of maximum degree three, a careful backtracking search can find a Hamiltonian cycle (if one exists) in time O(1.251<sup>''n''</sup>).<ref>{{Citation |last1=Iwama |first1=Kazuo |title=An Improved Exact Algorithm for Cubic Graph TSP |date=2007 |url=http://link.springer.com/10.1007/978-3-540-73545-8_13 |work=Computing and Combinatorics |volume=4598 |pages=108–117 |editor-last=Lin |editor-first=Guohui |access-date=2023-10-07 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-540-73545-8_13 |isbn=978-3-540-73544-1 |last2=Nakashima |first2=Takuya|series=Lecture Notes in Computer Science }}</ref>

=== Boolean satisfiability ===
Hamiltonian paths can be found using a [[SAT solver]]. The Hamiltonian path is NP-Complete meaning it can be [[Mapping reducibility|mapping reduced]] to the [[3-SAT|3-SAT problem]]. As a result, finding a solution to the Hamiltonian Path problem is equivalent to finding a solution for 3-SAT.

=== Unconventional methods ===
Because of the difficulty of solving the Hamiltonian path and cycle problems on conventional computers, they have also been studied in unconventional models of computing. For instance, [[Leonard Adleman]] showed that the Hamiltonian path problem may be solved using a [[DNA computing|DNA computer]]. Exploiting the parallelism inherent in chemical reactions, the problem may be solved using a number of chemical reaction steps linear in the number of vertices of the graph; however, it requires a factorial number of DNA molecules to participate in the reaction.<ref>{{citation |last=Adleman |first=Leonard |title=Molecular computation of solutions to combinatorial problems |date=November 1994 |journal=[[Science (journal)|Science]] |volume=266 |issue=5187 |pages=1021–1024 |bibcode=1994Sci...266.1021A |citeseerx=10.1.1.54.2565 |doi=10.1126/science.7973651 |jstor=2885489 |pmid=7973651 |author-link=Leonard Adleman}}.</ref>

An optical solution to the Hamiltonian problem has been proposed as well.<ref name="oltean_hamiltonian2">{{cite conference |author=Mihai Oltean |date=2006 |title=A light-based device for solving the Hamiltonian path problem |conference=Unconventional Computing |publisher=Springer LNCS 4135 |pages=217–227 |arxiv=0708.1496 |doi=10.1007/11839132_18}}</ref> The idea is to create a graph-like structure made from optical cables and beam splitters which are traversed by light in order to construct a solution for the problem. The weak point of this approach is the required amount of energy which is exponential in the number of nodes.

==Complexity==
The problem of finding a Hamiltonian cycle or path is in [[FNP (complexity)|FNP]]; the analogous [[decision problem]] is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of [[Karp's 21 NP-complete problems]]. They remain NP-complete even for special kinds of graphs, such as:

* [[bipartite graph]]s,<ref>{{Cite web|url=https://cs.stackexchange.com/q/18335 |title=Proof that the existence of a Hamilton Path in a bipartite graph is NP-complete|website=Computer Science Stack Exchange|access-date=2019-03-18}}</ref>
* undirected [[planar graph]]s of maximum degree three,<ref>{{citation
 | last1 = Garey | first1 = M. R. | author1-link = Michael Garey
 | last2 = Johnson | first2 = D. S. | author2-link = David S. Johnson
 | last3 = Stockmeyer | first3 = L. | author3-link = Larry Stockmeyer
 | contribution = Some simplified NP-complete problems
 | doi = 10.1145/800119.803884
 | pages = 47–63
 | title = Proc. 6th ACM Symposium on Theory of Computing (STOC '74)
 | year = 1974| s2cid = 207693360 }}.</ref> 
* directed planar graphs with indegree and outdegree at most two,<ref>{{citation
 | last = Plesńik | first = J.
 | issue = 4
 | journal = [[Information Processing Letters]]
 | pages = 199–201
 | title = The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two
 | url = http://www.aya.or.jp/~babalabo/DownLoad/Plesnik%208.4.192-196.pdf
 | volume = 8
 | year = 1979
 | doi = 10.1016/0020-0190(79)90023-1}}.</ref> 
* [[Bridgeless graph|bridgeless]] undirected planar 3-[[regular graph|regular]] [[bipartite graph]]s, 
* 3-connected 3-regular bipartite graphs,<ref>{{citation
 | last1 = Akiyama | first1 = Takanori
 | last2 = Nishizeki | first2 = Takao | author2-link = Takao Nishizeki
 | last3 = Saito | first3 = Nobuji
 | issue = 2
 | journal = Journal of Information Processing
 | mr = 596313
 | pages = 73–76
 | title = NP-completeness of the Hamiltonian cycle problem for bipartite graphs
 | url = 
 | volume = 3
 | year = 1980–1981}}.</ref> 
* subgraphs of the [[Lattice graph#Square grid graph|square grid graph]],<ref>{{citation
 | last1 = Itai | first1 = Alon
 | last2 = Papadimitriou | first2 = Christos
 | last3 = Szwarcfiter | first3 = Jayme
 | issue = 11
 | journal = [[SIAM Journal on Computing]]
 | pages = 676–686
 | title = Hamilton Paths in Grid Graphs
 | doi = 10.1137/0211056
 | volume = 4
 | year = 1982| citeseerx = 10.1.1.383.1078}}.</ref> 
* cubic subgraphs of the square grid graph.<ref>{{citation
 | last = Buro | first = Michael
 | title = Computers and Games
 | contribution = Simple Amazons endgames and their connection to Hamilton circuits in cubic subgrid graphs
 | contribution-url = http://skatgame.net/mburo/ps/amaend.pdf
 | volume = 2063
 | pages = 250–261
 | doi = 10.1007/3-540-45579-5_17| series = Lecture Notes in Computer Science
 | date = 2001
 | isbn = 978-3-540-43080-3
 | citeseerx = 10.1.1.40.9731
 }}.</ref>

However, for some special classes of graphs, the problem can be solved in polynomial time:

* [[k-vertex-connected graph|4-connected]] planar graphs are always Hamiltonian by a result due to [[W. T. Tutte|Tutte]], and the computational task of finding a Hamiltonian cycle in these graphs can be carried out in linear time<ref>{{citation
 | last1 = Chiba| first1 = Norishige
 | last2 = Nishizeki| first2 = Takao
 | title = The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs
 | doi = 10.1016/0196-6774(89)90012-6
 | pages = 187–211
 | journal = Journal of Algorithms
 | volume = 10
 | issue = 2
 | year = 1989}}</ref> by computing a so-called [[Tutte path]]. 
* Tutte proved this result by showing that every 2-connected planar graph contains a Tutte path. Tutte paths in turn can be computed in quadratic time even for 2-connected planar graphs,<ref>{{citation
 | last1 = Schmid| first1 = Andreas
 | last2 = Schmidt| first2 = Jens M.
 | contribution = Computing Tutte Paths
 | title = Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP'18), to appear.
 | year = 2018
}}</ref> which may be used to find Hamiltonian cycles and long cycles in generalizations of planar graphs.

Putting all of these conditions together, it remains open whether 3-connected 3-regular bipartite planar graphs must always contain a Hamiltonian cycle, in which case the problem restricted to those graphs could not be NP-complete; see [[Barnette's conjecture]].

In graphs in which all vertices have odd degree, an argument related to the [[handshaking lemma]] shows that the number of Hamiltonian cycles through any fixed edge is always even, so if one Hamiltonian cycle is given, then a second one must also exist.<ref>{{citation
 | last = Thomason
 | first = A. G.
 | contribution = Hamiltonian cycles and uniquely edge colourable graphs
 | doi = 10.1016/S0167-5060(08)70511-9
 | mr = 499124
 | pages = [https://archive.org/details/advancesingrapht0000camb/page/259 259–268]
 | series = Annals of Discrete Mathematics
 | title = Advances in Graph Theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977)
 | volume = 3
 | year = 1978
 | isbn = 9780720408430
 | url = https://archive.org/details/advancesingrapht0000camb/page/259
 }}.</ref> However, finding this second cycle does not seem to be an easy computational task. [[Christos Papadimitriou|Papadimitriou]] defined the [[complexity class]] [[PPA (complexity)|PPA]] to encapsulate problems such as this one.<ref>{{citation | last = Papadimitriou| first = Christos H.| author-link = Christos Papadimitriou | doi = 10.1016/S0022-0000(05)80063-7
| issue = 3
| journal = [[Journal of Computer and System Sciences]]
| pages = 498–532
| title = On the complexity of the parity argument and other inefficient proofs of existence
| volume = 48
| year = 1994 | mr = 1279412| citeseerx = 10.1.1.321.7008
}}.</ref>

== Polynomial time verifier ==
[[File:Hamiltonian Path Problem.jpg|thumb|221x221px|The proposed solution {s,w,v,u,t} forms a valid Hamiltonian Path in the graph G.]]
The Hamiltonian path problem is [[NP-completeness|NP-Complete]] meaning a proposed solution can be verified in [[Polynomial-time|polynomial time]].<ref name=":12"/>

A verifier [[algorithm]] for Hamiltonian path will take as input a graph G, starting vertex s, and ending vertex t. Additionally, verifiers require a potential solution known as a [[Certificate (complexity)|certificate]], c. For the Hamiltonian Path problem, c would consist of a [[String (computer science)|string]] of vertices where the first vertex is the start of the proposed path and the last is the end.<ref name=":0">{{Cite web |last=Bun |first=Mark |date=November 2022 |title=Boston University Theory of Computation |url=https://cs-people.bu.edu/mbun/courses/332_F22/slides/CS332-Lec22.pdf}}</ref> The algorithm will determine if c is a valid [[Hamiltonian path|Hamiltonian Path]] in G and if so, accept.

To decide this, the algorithm first verifies that all of the vertices in G appear exactly once in c. If this check passes, next, the algorithm will ensure that the first vertex in c is equal to s and the last vertex is equal to t. Lastly, to verify that c is a valid path, the algorithm must check that every edge between vertices in c is indeed an edge in G. If any of these checks fail, the algorithm will reject. Otherwise, it will accept.<ref name=":0" /><ref>{{Cite web |last=Bretscher |first=A |date=February 5, 2021 |title=University of Toronto CSCC63 Week 7 Lecture Notes |url=http://www.utsc.utoronto.ca/~bretscher/c63/lectures/w7.pdf}}</ref>

The algorithm can check in polynomial time if the vertices in G appear once in c. Additionally, it takes polynomial time to check the start and end vertices, as well as the edges between vertices. Therefore, the algorithm is a polynomial time verifier for the Hamiltonian path problem.<ref name=":0" />

== Applications ==

=== Networks on chip ===
[[Network on a chip|Networks on chip]] (NoC) are used in computer systems and [[Processor (computing)|processors]] serving as communication for on-chip components.<ref>{{Cite web |last=Bahn |first=Jun Ho |title=Overview of Network-on-Chip |url=https://newport.eecs.uci.edu/~nader/lab/NoCOverview.htm |website=University Of California Irvine}}</ref> The performance of NoC is determined by the method it uses to move [[Network packet|packets]] of data across a [[Computer network|network]].<ref>{{Cite book |last=Satish |first=E. G. |title=Emerging Research in Computing, Information, Communication and Applications |chapter=Comparative Performance Analysis of Routing Topology for NoC Architecture |series=Lecture Notes in Electrical Engineering |date=2022 |chapter-url=https://link.springer.com/chapter/10.1007/978-981-16-1342-5_34 |volume=790 |pages=431–440 |doi=10.1007/978-981-16-1342-5_34 |isbn=978-981-16-1341-8 |via=Springer}}</ref> The Hamiltonian Path problem can be implemented as a path-based method in [[multicast routing]]. Path-based multicast algorithms will determine if there is a Hamiltonian path from the start [[Node (networking)|node]] to each end node and send packets across the corresponding path. Utilizing this strategy guarantees [[deadlock (computer science)|deadlock]] and [[livelock]] free routing, increasing the efficiency of the NoC.<ref>{{Cite journal |last1=Bahrebar |first1=P. |last2=Stroobandt |first2=D. |date=2014 |title=Improving hamiltonian-based routing methods for on-chip networks: A turn model approach |journal=2014 Design, Automation & Test in Europe Conference & Exhibition |pages=1–4 |via=IEEE}}</ref>

=== Computer graphics ===
[[Rendering (computer graphics)|Rendering]] engines are a form of [[software]] used in [[computer graphics]] to generate images or models from input data.<ref>{{Cite web |last1=Garsiel |first1=Tali |last2=Irish |first2=Paul |date=August 5, 2011 |title=How Browsers Work |url=https://web.dev/howbrowserswork/}}</ref>  In [[Three-dimensional space|three dimensional]] graphics rendering, a common input to the engine is a [[polygon mesh]]. The time it takes to render the object is dependent on the rate at which the input is received, meaning the larger the input the longer the rendering time. For triangle meshes, however, the rendering time can be decreased by up to a factor of three. This is done through "ordering the triangles so that consecutive triangles share a face."<ref>{{Cite journal |last1=Arkin |first1=Esther M. |last2=Mitchell |first2=Joseph S. B. |last3=Held |first3=Martin |last4=Skiena |first4=Steven S. |title=Hamiltonian Triangulations for Fast Rendering |url=https://www3.cs.stonybrook.edu/~stripe/compgeompaper.pdf |journal=Department of Computer Science Stony Brook University}}</ref> That way, only one vertex changes between each consecutive triangle. This ordering exists if the [[dual graph]] of the triangular mesh contains a Hamiltonian path.

== References ==
{{commons category-inline}}
{{Reflist|30em}}

{{DEFAULTSORT:Hamiltonian Path Problem}}
[[Category:NP-complete problems]]
[[Category:Computational problems in graph theory]]
[[Category:Hamiltonian paths and cycles]]