Approximability of Connected Factors

Abstract

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard -finding a minimal connected 2-factor is just the traveling salesman problem (TSP).Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected d-factor. We give a 3-approximation for all d and improve this to an (r + 1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP.Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.

On the Complexity Landscape of Connected f-Factor Problems

“…Since |Q | > g(n) + 1, we conclude that the number n of vertices of G is greater than (n/g(n) − 2g(n) 2 + 1)(g(n) + 1). Rearranging for n we obtain that n < 2g(n) 4 + 2g(n) 3 + g(n) 2 + g(n) < 6g(n) 4 which contradicts our assumption that n ≥ 6g(n) 4 . Since Q i is a proper refinement of Q i+1 for every i with 1 ≤ i < k and |Q 0 | = 1, we infer that the length of the sequence S is at most g(n) + 1.…”

“…, (H k , Q k ) satisfying properties (M1)-(M4). Recall our assumption that n ≥ 6g(n) 4 . Let (G, f ) be an instance of CONNECTED g-BOUNDED f -FACTOR.…”

“…Complexity Class n , ∀ > 0 NPC [4] n polylog(n) QP (Theorem 2) n log n RP (Theorem 3) n c , ∀c ≥ 3 P (Theorem 1) Table 1 The table depicting the known as well as new results on the complexity landscape of the CON-NECTED f -FACTOR problem.…”

A Characterization for the Existence of Connected $f$-Factors of $\textit{ Large}$ Minimum Degree

“…We refer this connected regular spanning subgraph as a connected d-factor. Further, recently Cornelissen et al [3] showed that for every 0 < ǫ < 1, for f (v) ≥ n 1−ǫ the connected f -factor problem is NP-Complete.…”

“…For the case when f (v) ≥ ⌈ n 2 ⌉ − 1, Cornelissen et al [3] have also shown a simple test for the existence of a connected f -factor. We observe that when f (v) ≥ ⌈n/2⌉ for every v in V , any f -factor of a given graph G is connected and has diameter at most 2.…”