On packing shortest cycles in graphs

Abstract

Abstract. We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g = 3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g ∈ {4, 5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g ≥ 6, both problems are APX-hard already for graphs with maximum degree 3.

Inapproximability of $H$-Transversal/Packing

“…Variants of k-Cycle Packing have also been considered in the literature. Rautenbach and Regen [50] studied k-Cycle Edge Packing on graphs with girth k and small degree. Chalermsook et al [11] studied a variant of k-Cycle Packing on directed graphs for k ≥ n 1/2 where we want to pack as many disjoint cycles of length at most k as possible, and proved that it is NP-hard to approximate within a factor of n 1/2− .…”

Maximum cycle packing using SPR-trees

“…Packing edge-disjoint cycles in graphs is a classical graph-theoretical problem. There is a large amount of literature concerning cycle packing problems for example [12], [11], [10], [1], [20], [7], [6], [19], [18]. In [14], [2] and [8] simple approximation algorithms are described since cycle packing problems are typically hard [14].…”

On maximum cycle packings in polyhedral graphs

“…A large amount of literature can be found concerning conditions that are sufficient for the existence of some number of disjoint cycles which may satisfy further restrictive conditions. For examples, we refer to publications [6], [9], [10], [12], [15], [16], [18], [20], [21], [23], [24]. The algorithmic problems concerning cycle packings are typically hard ( [5], [11], [20]) and approximation algorithms are described ( [11], [17]).…”

“…For examples, we refer to publications [6], [9], [10], [12], [15], [16], [18], [20], [21], [23], [24]. The algorithmic problems concerning cycle packings are typically hard ( [5], [11], [20]) and approximation algorithms are described ( [11], [17]). Several authors mention practical applications in computational biology ( [3], [8], [13]) or the design of optical networks ( [1]).…”