An $$O(\log \mathrm{OPT})$$ O ( log OPT ) -Approximation for Covering/Packing Minor Models of $$\theta _{r}$$ θ r

Abstract

Given two graphs G and H, we define v-coverH (G) (resp. e-coverH (G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also v-pack H (G) (resp. e-pack H (G)) is the maximum number of vertex-(resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H. We denote by θr the graph with two vertices and r parallel edges between them. When H = θr, the parameters v-coverH , e-coverH , v-pack H , and e-pack H are NP-hard to compute (for sufficiently big values of r). Drawing upon combinatorial results in [7], we give an algorithmic proof that if v-pack θr (G) ≤ k, then v-cover θr (G) = O(k log k), and similarly for e-pack θr and e-cover θr . In other words, the class of graphs containing θr as a minor has the vertex/edge Erdős-Pósa property, for every positive integer r. Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(log OPT)-approximation algorithm for v-pack θr , v-cover θr , e-pack θr , and e-cover θr that runs in O(n · log(n) · m) steps. Also, we derive several new Erdős-Pósa-type results from the techniques that we introduce. Pósa, proved in 1965 [12] that there is a function f : N → N such that for each positive integer k, every graph either contains k vertex-disjoint cycles or it contains f (k) vertices that intersect every cycle in G. Moreover, they proved that the "gap" of this min-max relation is f (k) = O(k · log k) and that this gap is optimal. This result initiated an interesting line of research on the duality between coverings and packings of combinatorial objects. To formulate this duality, given a class C of connected graphs, we define v-cover C (G) (resp. e-cover C (G)) as the minimum cardinality of a set S of vertices (resp. edges) such that each C-subgraph of G contains some element of S. Also, we define v-pack C (G) (resp. e-pack C (G)) as the maximum number of vertex-(resp. edge-) disjoint C-subgraphs of G. Packings and coverings. Paul Erdős and LajosWe say that C has the vertex Erdős-Pósa property (resp. the edge Erdős-). Using this terminology, the original result of Erdős and Pósa says that the set of all cycles has the vertex Erdős-Pósa property with gap O(k · log k). The general question in this branch of Graph Theory is to detect instantiations of C which have the vertex/edge Erdős-Pósa property (in short, v/e-EP-property) and when this is the case, minimize the gap function f . Several theorems of this type have been proved concerning different instantiations of C such as odd cycles [24,30], long cycles [3], and graphs containing cliques as minors [10] (see also [18,22,34] for results on more general combinatorial structures and [31, 33] for surveys).A general class that is known to have the v-EP-property is the class C H of the graphs that contain some fixed planar graph H as a minor 1 . This fact was proven by Robertson and Seymour in [35] and the best known general gap is f (k) = O(k · log O(1) k) due to the results of [8] -see also [14][15][16] for b...

Cutwidth: Obstructions and Algorithmic Aspects

“…The proof is omitted as it is very similar to the one in [5] where a similar encoding was defined in order to treat the topological minor relation. To see the main idea, recall that F q,h (G i ) registers all different "partial occurrences" of graphs of ≤ h vertices (and therefore also of graphs of H) in G i , for all possible ways to obtain G i from G i after removing at most q edges.…”

“…For minor-closed parameters pathwidth and treewidth, Lagergren [18] showed that any minimal minor obstruction to admitting a path decomposition of width k has size at most single-exponential in O(k 4 ), whereas for tree decompositions he showed an upper bound double-exponential in O(k 5 ) . Less is known about immersion-closed parameters, such as cutwidth.…”

Recent techniques and results on the Erdős–Pósa property

“…Tree partitions have been introduced in [See85] (see also [Hal91]) and tree partition width has been defined for simple graphs in [DO96]. The extension of this definition for multigraphs is due to [CRST15a].…”

“…for every pair x, y of positive reals. The following argument has been first used in [FST11] (see also [CC13a,RST16,CRST15a]).…”

“…We then apply induction on either the graph where a small cycle has been deleted (in the first case), or on the reduced graph (in the second case). This technique has been successfully applied in [FJW13,CRST15a], for instance.…”