[Paper Review] An $O(\log OPT)$-approximation for covering and packing minor models of ${ heta}_r$

Dimitris Chatzidimitriou, Jean‐Florent Raymond+2 more|arXiv (Cornell University)|Oct 14, 2015

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper presents an O(log OPT)-approximation algorithm for vertex and edge packing and covering problems involving minor models of the graph θr (two vertices connected by r parallel edges). Using algorithmic techniques rooted in protrusion decomposition and tree-partition width, the authors prove that the class of graphs containing θr as a minor has both the vertex and edge Erdős–Pósa property with a gap of O(k log k), and they provide the first approximation algorithms for edge versions of these problems for general r.

ABSTRACT

Given two graphs $G$ and $H$, we define $ extsf{v-cover}_{H}(G)$ (resp. $ extsf{e-cover}_{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also $ extsf{v-pack}_{H}(G)$ (resp. $ extsf{v-pack}_{H}(G)$) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of $G$ that contain a minor isomaorphic to $H$. We denote by $ heta_r$ the graph with two vertices and $r$ parallel edges between them. When $H= heta_r$, the parameters $ extsf{v-cover}_{H}$, $ extsf{e-cover}_{H}$, $ extsf{v-pack}_{H}$, and $ extsf{v-pack}_{H}$ are NP-hard to compute (for sufficiently big values of $r$). Drawing upon combinatorial results in [Minors in graphs of large $ heta_r$-girth, Chatzidimitriou et al., arXiv:1510.03041], we give an algorithmic proof that if $ extsf{v-pack}_{ heta_r}(G)\leq k$, then $ extsf{v-cover}_{ heta_r}(G) = O(k\log k)$, and similarly for $ extsf{v-pack}_{ heta_r}$ and $ extsf{e-cover}_{ heta_r}$. In other words, the class of graphs containing ${ heta_r}$ as a minor has the vertex/edge Erd\H{o}s-P\'osa property, for every positive integer $r$. Using the algorithmic machinery of our proofs, we introduce a unified approach for the design of an $O(\log { m OPT})$-approximation algorithm for $ extsf{v-pack}_{ heta_r}$, $ extsf{v-cover}_{ heta_r}$, $ extsf{v-pack}_{ heta_r}$, and $ extsf{e-cover}_{ heta_r}$ that runs in $O(n\cdot \log(n)\cdot m)$ steps. Also, we derive several new Erd\H{o}s-P\'osa-type results from the techniques that we introduce.

Motivation & Objective

• To design an efficient approximation algorithm for vertex and edge packing and covering of minor models of θr.
• To establish the vertex and edge Erdős–Pósa property for minor models of θr with an optimal O(k log k) gap.
• To extend algorithmic techniques based on tree-partition width and protrusion decomposition to graph classes with bounded tree-partition width.
• To generalize results on the Erdős–Pósa property to minor models of θr,r′ and provide new approximation algorithms for edge variants.
• To explore the broader applicability of the method to other planar graphs and their minor models.

Proposed method

• Reduction of the problem to graphs of bounded tree-partition width using protrusion decomposition.
• Application of a structural result that graphs of large tree-partition width contain minor models of k-walls, k-fans, k-paths, or k-stars.
• Use of Corollary 5 to bound the tree-partition width of graphs with small e-coverΘr,r′(G), linking it to the existence of edge-disjoint Θr,r′-subdivisions.
• Leveraging Theorem 3 to relate the edge covering number to the edge packing number via a gap function f2(h, tk).
• Design of a unified O(n · log n · m)-time algorithm for approximating v-packθr, v-coverθr, e-packθr, and e-coverθr.
• Proof of the edge-Erdős–Pósa property for minor models of θr,r′, extending prior results limited to θr.

Experimental results

Research questions

• RQ1 Can an O(log OPT)-approximation algorithm be designed for both vertex and edge packing and covering of minor models of θr for any r ≥ 1?
• RQ2 What is the tightest possible gap function for the vertex and edge Erdős–Pósa property in the case of minor models of θr?
• RQ3 Does the edge-Erdős–Pósa property hold for minor models of θr,r′, and can it be leveraged for approximation algorithms?
• RQ4 Can the algorithmic framework based on tree-partition width and protrusion decomposition be generalized to other planar graphs beyond θr?
• RQ5 Is there a polylogarithmic approximation for vertex and edge packing and covering of minor models of any fixed planar graph?

Key findings

• The paper establishes that the class of graphs containing θr as a minor has the vertex and edge Erdős–Pósa property with a gap of O(k log k), matching the known lower bound.
• An O(log OPT)-approximation algorithm is designed for v-packθr, v-coverθr, e-packθr, and e-coverθr, running in O(n · log n · m) time.
• The edge-Erdős–Pósa property is extended to minor models of θr,r′ for all r, r′ ∈ ℕ with r′ ≤ r, a result not previously known.
• For graphs of bounded tree-partition width, the vertex and edge Erdős–Pósa property holds with a linear gap, improving upon general bounds.
• The algorithmic machinery based on protrusion decomposition and tree-partition width is shown to be generalizable beyond θr, suggesting broader applicability to other planar graph classes.