Hamiltonian path problem

Last updated

The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected 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. [1]

Contents

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-complete problems, as shown in Michael Garey and David S. Johnson's book Computers and Intractability: A Guide to the Theory of NP-Completeness and Richard Karp's list of 21 NP-complete problems. [2] [3]

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:

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. [3] A search procedure by Frank Rubin [5] 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 Bellman, Held, and Karp can be used to solve the problem in time O(n2 2n). 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. [6] [7]

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(1.657n); for bipartite graphs this algorithm can be further improved to time O(1.415n). [8]

Backtracking

For graphs of maximum degree three, a careful backtracking search can find a Hamiltonian cycle (if one exists) in time O(1.251n). [9]

Boolean satisfiability

Hamiltonian paths can be found using a SAT solver. The Hamiltonian path is NP-Complete meaning it can be mapping reduced to the 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 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. [10]

An optical solution to the Hamiltonian problem has been proposed as well. [11] 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; 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:

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

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. [20] However, finding this second cycle does not seem to be an easy computational task. Papadimitriou defined the complexity class PPA to encapsulate problems such as this one. [21]

Polynomial time verifier

The proposed solution {s,w,v,u,t} forms a valid Hamiltonian Path in the graph G. Hamiltonian Path Problem.jpg
The proposed solution {s,w,v,u,t} forms a valid Hamiltonian Path in the graph G.

The Hamiltonian path problem is NP-Complete meaning a proposed solution can be verified in polynomial time. [1]

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, c. For the Hamiltonian Path problem, c would consist of a string of vertices where the first vertex is the start of the proposed path and the last is the end. [22] The algorithm will determine if c is a valid 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. [22] [23]

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. [22]

Applications

Networks on chip

Networks on chip (NoC) are used in computer systems and processors serving as communication for on-chip components. [24] The performance of NoC is determined by the method it uses to move packets of data across a network. [25] 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 to each end node and send packets across the corresponding path. Utilizing this strategy guarantees deadlock and livelock free routing, increasing the efficiency of the NoC. [26]

Computer graphics

Rendering engines are a form of software used in computer graphics to generate images or models from input data. [27] In 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." [28] 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.

Related Research Articles

<span class="mw-page-title-main">Hamiltonian path</span> Path in a graph that visits each vertex exactly once

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details.

<span class="mw-page-title-main">Clique problem</span> Task of computing complete subgraphs

In computer science, the clique problem is the computational problem of finding cliques in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique, finding a maximum weight clique in a weighted graph, listing all maximal cliques, and solving the decision problem of testing whether a graph contains a clique larger than a given size.

<span class="mw-page-title-main">Eulerian path</span> Trail in a graph that visits each edge once

In graph theory, an Eulerian trail is a trail in a finite graph that visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:

<span class="mw-page-title-main">Spanning tree</span> Tree which includes all vertices of a graph

In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T.

<span class="mw-page-title-main">Independent set (graph theory)</span> Unrelated vertices in graphs

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.

<span class="mw-page-title-main">Vertex cover</span> Subset of a graphs vertices, including at least one endpoint of every edge

In graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.

<span class="mw-page-title-main">Cubic graph</span> Graph with all vertices of degree 3

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles. Equivalently, each FVS contains at least one vertex of any cycle in the graph. The feedback vertex set number of a graph is the size of a smallest feedback vertex set. The minimum feedback vertex set problem is an NP-complete problem; it was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.

<span class="mw-page-title-main">Feedback arc set</span> Edges that hit all cycles in a graph

In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph. Removing these edges from the graph breaks all of the cycles, producing an acyclic subgraph of the given graph, often called a directed acyclic graph. A feedback arc set with the fewest possible edges is a minimum feedback arc set and its removal leaves a maximum acyclic subgraph; weighted versions of these optimization problems are also used. If a feedback arc set is minimal, meaning that removing any edge from it produces a subset that is not a feedback arc set, then it has an additional property: reversing all of its edges, rather than removing them, produces a directed acyclic graph.

Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.

In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs.

<span class="mw-page-title-main">Crossing number (graph theory)</span> Fewest edge crossings in drawing of a graph

In graph theory, the crossing numbercr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing.

In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.

<span class="mw-page-title-main">Maximum cut</span> Problem of finding a maximum cut in a graph

In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible. Finding such a cut is known as the max-cut problem.

<span class="mw-page-title-main">Hamiltonian decomposition</span>

In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs. In the undirected case a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected.

Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.

<span class="mw-page-title-main">Graph power</span> Graph operation: linking all pairs of nodes of distance ≤ k

In graph theory, a branch of mathematics, the kth powerGk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.

References

Commons-logo.svg Media related to Hamiltonian path problem at Wikimedia Commons

  1. 1 2 Sipser, Michael (2013). Introduction to the Theory of Computation (3rd ed.). Cengage Learning. pp. 292–314.
  2. ↑ Garey, Michael R; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company. p. 60.
  3. 1 2 Held, M.; Karp, R. M. (1965). "The construction of discrete dynamic programming algorithms". IBM Systems Journal. 4 (2): 136–147. doi:10.1147/sj.42.0136. ISSN   0018-8670.
  4. ↑ Reduction from Hamiltonian cycle to Hamiltonian path
  5. ↑ Rubin, Frank (1974), "A Search Procedure for Hamilton Paths and Circuits", Journal of the ACM , 21 (4): 576–80, doi: 10.1145/321850.321854 , S2CID   7132716
  6. ↑ Bellman, Richard (January 1962). "Dynamic Programming Treatment of the Travelling Salesman Problem". Journal of the ACM. 9 (1): 61–63. doi: 10.1145/321105.321111 . ISSN   0004-5411. S2CID   15649582.
  7. ↑ Held, Michael; Karp, Richard M. (March 1962). "A Dynamic Programming Approach to Sequencing Problems". Journal of the Society for Industrial and Applied Mathematics. 10 (1): 196–210. doi:10.1137/0110015. ISSN   0368-4245.
  8. ↑ Bjorklund, Andreas (October 2010). "Determinant Sums for Undirected Hamiltonicity". 2010 IEEE 51st Annual Symposium on Foundations of Computer Science. IEEE. pp. 173–182. arXiv: 1008.0541 . doi:10.1109/focs.2010.24. ISBN   978-1-4244-8525-3.
  9. ↑ Iwama, Kazuo; Nakashima, Takuya (2007), Lin, Guohui (ed.), "An Improved Exact Algorithm for Cubic Graph TSP", Computing and Combinatorics, Lecture Notes in Computer Science, vol. 4598, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 108–117, doi:10.1007/978-3-540-73545-8_13, ISBN   978-3-540-73544-1 , retrieved 2023-10-07
  10. ↑ Adleman, Leonard (November 1994), "Molecular computation of solutions to combinatorial problems", Science , 266 (5187): 1021–1024, Bibcode:1994Sci...266.1021A, CiteSeerX   10.1.1.54.2565 , doi:10.1126/science.7973651, JSTOR   2885489, PMID   7973651 .
  11. ↑ Mihai Oltean (2006). A light-based device for solving the Hamiltonian path problem. Unconventional Computing. Springer LNCS 4135. pp. 217–227. arXiv: 0708.1496 . doi:10.1007/11839132_18.
  12. ↑ "Proof that the existence of a Hamilton Path in a bipartite graph is NP-complete". Computer Science Stack Exchange. Retrieved 2019-03-18.
  13. ↑ Garey, M. R.; Johnson, D. S.; Stockmeyer, L. (1974), "Some simplified NP-complete problems", Proc. 6th ACM Symposium on Theory of Computing (STOC '74), pp. 47–63, doi:10.1145/800119.803884, S2CID   207693360 .
  14. ↑ Plesńik, J. (1979), "The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two" (PDF), Information Processing Letters , 8 (4): 199–201, doi:10.1016/0020-0190(79)90023-1 .
  15. ↑ Akiyama, Takanori; Nishizeki, Takao; Saito, Nobuji (1980–1981), "NP-completeness of the Hamiltonian cycle problem for bipartite graphs", Journal of Information Processing, 3 (2): 73–76, MR   0596313 .
  16. ↑ Itai, Alon; Papadimitriou, Christos; Szwarcfiter, Jayme (1982), "Hamilton Paths in Grid Graphs", SIAM Journal on Computing , 4 (11): 676–686, CiteSeerX   10.1.1.383.1078 , doi:10.1137/0211056 .
  17. ↑ Buro, Michael (2001), "Simple Amazons endgames and their connection to Hamilton circuits in cubic subgrid graphs" (PDF), Computers and Games, Lecture Notes in Computer Science, vol. 2063, pp. 250–261, CiteSeerX   10.1.1.40.9731 , doi:10.1007/3-540-45579-5_17, ISBN   978-3-540-43080-3 .
  18. ↑ Chiba, Norishige; Nishizeki, Takao (1989), "The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs", Journal of Algorithms, 10 (2): 187–211, doi:10.1016/0196-6774(89)90012-6
  19. ↑ Schmid, Andreas; Schmidt, Jens M. (2018), "Computing Tutte Paths", Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP'18), to appear.
  20. ↑ Thomason, A. G. (1978), "Hamiltonian cycles and uniquely edge colourable graphs", Advances in Graph Theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977), Annals of Discrete Mathematics, vol. 3, pp.  259–268, doi:10.1016/S0167-5060(08)70511-9, ISBN   9780720408430, MR   0499124 .
  21. ↑ Papadimitriou, Christos H. (1994), "On the complexity of the parity argument and other inefficient proofs of existence", Journal of Computer and System Sciences , 48 (3): 498–532, CiteSeerX   10.1.1.321.7008 , doi:10.1016/S0022-0000(05)80063-7, MR   1279412 .
  22. 1 2 3 Bun, Mark (November 2022). "Boston University Theory of Computation" (PDF).
  23. ↑ Bretscher, A (February 5, 2021). "University of Toronto CSCC63 Week 7 Lecture Notes" (PDF).
  24. ↑ Bahn, Jun Ho. "Overview of Network-on-Chip". University Of California Irvine.
  25. ↑ Satish, E. G. (2022). "Comparative Performance Analysis of Routing Topology for NoC Architecture". Emerging Research in Computing, Information, Communication and Applications. Lecture Notes in Electrical Engineering. Vol. 790. pp. 431–440. doi:10.1007/978-981-16-1342-5_34. ISBN   978-981-16-1341-8 – via Springer.
  26. ↑ Bahrebar, P.; Stroobandt, D. (2014). "Improving hamiltonian-based routing methods for on-chip networks: A turn model approach". 2014 Design, Automation & Test in Europe Conference & Exhibition: 1–4 – via IEEE.
  27. ↑ Garsiel, Tali; Irish, Paul (August 5, 2011). "How Browsers Work".
  28. ↑ Arkin, Esther M.; Mitchell, Joseph S. B.; Held, Martin; Skiena, Steven S. "Hamiltonian Triangulations for Fast Rendering" (PDF). Department of Computer Science Stony Brook University.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.