On the Complexity Landscape of Connected f-Factor Problems

Let G be an undirected simple graph having n vertices and let f : V ( G ) → { 0 , ⋯ , n - 1 } be a function. An f -factor of G is a spanning subgraph H such that d H ( v ) = f ( v ) for every vertex v ∈ V ( G ) . The subgraph H is called a connected f -factor if, in addition, H is connected. A classical result of Tutte (Can J Math 6(1954):347–352, 1954 ) is the polynomial time algorithm to check whether a given graph has a specified f -factor. However, checking for the presence of a connected f -factor is easily seen to generalize Hamiltonian Cycle and hence is NP -complete. In fact, the Connected f -Factor problem remains NP -complete even when we restrict f ( v ) to be at least n ϵ for each vertex v and constant 0 ≤ ϵ < 1 ; on the other side of the spectrum of nontrivial lower bounds on f , the problem is known to be polynomial time solvable when f ( v ) is at least n 3 for every vertex v . In this paper, we extend this line of work and obtain new complexity results based on restrictions on the function f . In particular, we show that when f ( v ) is restricted to be at least n ( log n ) c , the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c ≤ 1 . Furthermore, we show that when c > 1 , the problem is NP -intermediate.