Cover and Conquer: Augmenting Decompositions for Connectivity Problems.
Arash Haddadan, Alantha Newman, R. Ravi · arXiv (Cornell University) · 2017
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.