A Space-efficient Parameterized Algorithm for the Hamiltonian Cycle Problem by Dynamic Algebraziation

An NP-hard graph problem may be intractable for general graphs but it could be efficiently solvable using dynamic programming for graphs with bounded width (or depth or some other structural parameter). Dynamic programming is a well-known approach used for finding exact solutions for NP-hard graph problems based on tree decompositions. It has been shown that there exist algorithms using linear time in the number of vertices and single exponential time in the width (depth or other parameters) of a given tree decomposition for many connectivity problems. 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, F\"urer 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 solve the Hamiltonian Cycle and the Traveling Salesman problems, 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 solution) to polynomial. We give an algorithm to solve Hamiltonian cycle in time $\mathcal{O}((4w)^d\, nM(n\log{n}))$ using $\mathcal{O}(dn\log{n})$ 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 $\mathcal{O}((4w)^d poly(n))$ using space $\mathcal{O}(\mathcal{W}dn\log{n})$, where $w$ 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.