[Paper Review] Cover and Conquer: Augmenting Decompositions for Connectivity Problems.

Arash Haddadan, Alantha Newman+1 more|arXiv (Cornell University)|Jul 17, 2017

Complexity and Algorithms in Graphs25 references3 citations

TL;DR

This paper presents improved approximation algorithms for connectivity problems like TSP and 2EC on node-weighted, subcubic, and 3-edge-connected graphs by leveraging cycle covers that cover small cuts evenly. It achieves a 7/5-approximation for TSP and a 13/10-approximation for 2EC on 3-edge-connected, cubic, node-weighted graphs, while also showing that the everywhere 18/19 vector is a convex combination of tour incidence vectors.

ABSTRACT

We explore and introduce tools to design algorithms with improved approximation guarantees for edge connectivity problems such as the traveling salesman problem (TSP) and the 2-edge-connected spanning multigraph problem (2EC) on (restricted classes of) weighted graphs. In particular, we investigate decompositions of graphs that cover small-cardinality cuts an even number of times. Such decompositions allow us to design approximation algorithms that consist of augmenting an initial subgraph (e.g. cycle cover or spanning tree) with a cheap addition to form a feasible solution. Boyd, Iwata and Takazawa recently presented an algorithm to find a cycle cover that covers all 3- and 4-edge cuts in a bridgeless, cubic graph with applications to 2EC on 3-edge-connected, cubic graphs [BIT 2013]. We present several applications of their algorithm to connectivity problems on node-weighted graphs, which has been suggested as an intermediate step between graph and general metrics. Specifically, on 3-edge-connected, cubic, node-weighted graphs, we present a $\frac{7}{5}$-approximation algorithm for TSP and a $\frac{13}{10}$-approximation algorithm for 2EC. Moreover, we use a cycle cover that covers all 3- and 4-edge cuts to show that the everywhere $\frac{18}{19}$ vector is a convex combination of incidence vectors of tours, answering a question of Seb{\H{o}}. Extending this approach to input graphs that are 2-edge connected necessitates finding methods for covering 2-edge cuts. We therefore present a procedure to decompose an optimal solution for the subtour linear program into spanning, connected subgraphs that cover each 2-edge cut an even number of times. We use this decomposition to design a simple $\frac{4}{3}$-approximation algorithm for the 2-edge connected problem on subcubic, node-weighted graphs.

Motivation & Objective

• To design better approximation algorithms for edge connectivity problems such as TSP and 2EC on restricted classes of weighted graphs.
• To extend the cycle cover approach of Boyd, Iwata, and Takazawa to node-weighted graphs, which serve as an intermediate model between graphs and general metrics.
• To address the challenge of covering 2-edge cuts in 2-edge-connected graphs by decomposing optimal subtour LP solutions into subgraphs covering such cuts evenly.
• To establish that the everywhere 18/19 vector is a convex combination of tour incidence vectors, resolving a question posed by Sebő.

Proposed method

• Use cycle covers that cover all 3- and 4-edge cuts an even number of times to construct feasible solutions for TSP and 2EC.
• Apply the cycle cover technique to 3-edge-connected, cubic, node-weighted graphs to achieve improved approximation ratios.
• Introduce a decomposition procedure that splits an optimal subtour LP solution into spanning, connected subgraphs covering each 2-edge cut an even number of times.
• Leverage this decomposition to design a 4/3-approximation algorithm for 2EC on subcubic, node-weighted graphs.
• Use the cycle cover framework to prove that the everywhere 18/19 vector is a convex combination of incidence vectors of tours.
• Combine initial subgraphs (e.g. cycle covers or spanning trees) with low-cost augmentations to form feasible, near-optimal solutions.

Experimental results

Research questions

• RQ1 Can cycle covers that cover small cuts evenly be used to improve approximation guarantees for TSP and 2EC on node-weighted graphs?
• RQ2 What is the best achievable approximation ratio for TSP and 2EC on 3-edge-connected, cubic, node-weighted graphs using such cycle covers?
• RQ3 How can 2-edge cuts be effectively handled in 2-edge-connected graph problems when standard cycle covers fail?
• RQ4 Can the everywhere 18/19 vector be expressed as a convex combination of tour incidence vectors, and what does this imply for TSP approximation?

Key findings

• A 7/5-approximation algorithm is achieved for the TSP on 3-edge-connected, cubic, node-weighted graphs using cycle covers that cover all 3- and 4-edge cuts evenly.
• A 13/10-approximation algorithm is developed for the 2EC problem on the same class of graphs.
• The paper proves that the everywhere 18/19 vector is a convex combination of incidence vectors of tours, answering a question posed by Sebő.
• A 4/3-approximation algorithm is designed for 2EC on subcubic, node-weighted graphs by decomposing the optimal subtour LP solution into subgraphs covering 2-edge cuts evenly.
• The decomposition method enables the handling of 2-edge cuts in 2-edge-connected graphs, extending prior approaches limited to 3-edge-connected graphs.
• The framework demonstrates the utility of even coverage of small cuts as a key design principle for connectivity problems.