A Space-Efficient Parameterized Algorithm for the Hamiltonian Cycle Problem by Dynamic Algebraization

Abstract

An NP-hard graph problem may be intractable for general graphs but it could be efficiently solvable using dynamic programming for graphs with bounded treewidth. Employing dynamic programming on a tree decomposition usually uses exponential space. In 2010, Lokshtanov and Nederlof introduced an elegant framework to avoid exponential space by algebraization. Later, Fürer and Yu modified the framework in a way that even works when the underlying set is dynamic, thus applying it to tree decompositions.

In this work, we design space-efficient algorithms to count the number of Hamiltonian cycles and furthermore solve the Traveling Salesman problem, using polynomial space while the time complexity is only slightly increased. This might be inevitable since we are reducing the space usage from an exponential amount (in dynamic programming solutions) to polynomial. We give an algorithm to count the number of Hamiltonian cycles in time $\mathcal {O}((4k)^d\, nM(n\log {n}))$ using $\mathcal {O}(kdn\log {n})$ space, where M(r) is the time complexity to multiply two integers, each of which being represented by at most r bits. Then, we solve the more general Traveling Salesman problem in time $\mathcal {O}((4k)^d poly(n))$ using space $\mathcal {O}(\mathcal {W}kdn\log {n})$, where k and d are the width and the depth of the given tree decomposition and $\mathcal {W}$ is the sum of weights. Furthermore, this algorithm counts the number of Hamiltonian Cycles.