Hamiltonian Cycles and paths Bin Zhou

Definitions Hamiltonian cycle (HC): is a cycle which passes once and exactly once through every vertex of G (G can be digraph). Hamiltonian path: is a path which passes once and exactly once through every vertex of G (G can be digraph). A graph is Hamiltonian iff a Hamiltonian cycle (HC) exists.

History Invented by Sir William Rowan Hamilton in 1859 as a game Since 1936, some progress have been made Such as sufficient and necessary conditions be given

History G.A. Dirac, 1952, If G is a simple graph with n(>=3) vertices, and if the degree of each is at least 1/2n, then G is Hamiltonian O.Ore, 1960 If G is a simple graph with n(>=3) vertices, and if the sum of the degrees of each pair of non-adjacent vertices is at least n, then G is Hamiltonian

History Bondy and Chvatal, 1976 For G to be Hamiltonian, it is necessary and sufficient that [G] n be Hamiltonian. ([G] n is gotten from G by adding edges joining non- adjacent vertices whose sum of degrees is equal to, or greater than n)

History Fraudee, Dould, Jacobsen, Schelp (1989) If G is a 2-connected graph such that for every pair of nonadjacent nodes u and v, |d(u)+d(v)|>=(2n-1)/3 then G is Hamiltonian

Application Hamiltonian cycles in fault random geometric network In a network, if Hamiltonian cycles exist, the fault tolerance is better.

Hamiltonian problem is NPC This is a well known NP complete problem For general graph, we can not find an exactly linear time complexity algorithm to find a Hamiltonian cycle or path

HC algorithms For general graphs, no efficient algorithm NP-complete for perfect graphs, planar bipartite graphs, grid graphs, 3-connected planar graphs For some special graphs, exist efficient algorithms. N. Ghiba, T. Nishizeki (1989) Polynomial algorithm for 4-connected planar graphs. G.Gutin (1997) Polynomial algorithm for quasi-transitive digraphs

Some Algorithms for HC L. Pósa (1976) Rotational transformation B. Bollobás, T.I.Fenner, and A. M. Frieze Cycle extension (HAM) (1987) Silvano Martello Algorithm 595 (1983)

Two classes of algorithms Heuristic algorithm Pósa, UHC, DHC, HAM, etc Backtrack algorithm 595HAM, KTC, MultiPath

Backtrack Algorithm Recurse(Path p, endpoint e) While (e has unvisited neighbors) { GetNewNode x; (add x node to P) PruneGraph. (Prune graph. If result graph does not permit a HC forming, remove x from P and continue) FormCycle (If P includes all nodes then try to form cycle. Fail, remove x and continue; Succ, return success) BackTrack: Recurse(P,x) } Return fail.

Backtrack Algorithm Search all the potential solutions Employ pruning of some kind to restrict the amount of researching Advantage: Find all solution, can decide HC exists or not Disadvantage Worst case, needs exponential time. Normally, take a long time

Heuristic Algorithm Initialize path P While { Find new unvisited node. If found { Extend path P and pruning on the graph. If this choice does not permit HC, remove the extended node. } else Transform Path. Try all possible endpoints of this path Form cycle. Try to find HC }

Heuristic Algorithm Advantage: Fast. Linear or low-order polynomial time Disadvantage Maybe can not find the HC

Ham heuristic algorithm Try to extend existing path and never decrease the path length Do cycle extension Do rotational transformation

Ham algorithm Start from a random node and find a neighbor to get a path P. |P|=2 Do { Change partial path array A. oldlength=|P|. While |P|==oldlength { Find neighbors of P’s endpoints.Try to extend P. For (each neighbor) do { If Extendable Extend and continue; Else Do cycle extension or rotational transformation; } Check termination condition and change P } }}

Cycle Extention x1x1 xixi x i+1 u xkxk Path P: xixi x1x1 xkxk x i+1 u Path P’:

Rotational transformation x1x1 xixi x i+1 xkxk Path P: xixi x1x1 x i+1 xkxk Path P’:

Results

Problems The program can not check large graph due to the memory restriction May be need more conditions to decide the probability of HC exists We can solve large problem using parallel computing