On the Complexity Landscape of Connected f-Factor Problems

Abstract

Let G be an undirected simple graph having n vertices and let $f:V(G)\rightarrow \{0,\dots , n-1\}$ be a function. An f-factor of G is a spanning subgraph H such that $d_H(v)=f(v)$ for every vertex $v\in V(G)$. 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 connectedf-factor is easily seen to generalize Hamiltonian Cycle and hence is $\mathsf {NP}$-complete. In fact, the Connected f-Factor problem remains $\mathsf {NP}$-complete even when we restrict f(v) to be at least $n^{\epsilon }$ for each vertex v and constant $0\le \epsilon <1$; 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 $\frac{n}{3}$ 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 $\frac{n}{(\log n)^c}$, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if $c\le 1$. Furthermore, we show that when $c>1$, the problem is $\mathsf {NP}$-intermediate.