[Paper Review] 1.85 Approximation for Min-Power Strong Connectivity

Gruiă Cälinescu|arXiv (Cornell University)|May 15, 2012

Complexity and Algorithms in Graphs17 references3 citations

TL;DR

This paper presents a 1.85-approximation algorithm for the Min-Power Strong Connectivity problem in bidirected graphs, improving upon the previous best ratio of 2. It combines techniques from Robins-Zelikovsky's Steiner Tree work and Caragiannis et al.'s broadcast power assignment, using fractional star covers and a refined linear programming relaxation to achieve tighter bounds on the integrality gap and algorithm performance.

ABSTRACT

Given a directed simple graph G=(V,E) and a nonnegative-valued cost function the power of a vertex u in a directed spanning subgraph H is given by the maximum cost of an arcs of H exiting u. The power of H is the sum of the power of its vertices. Power Assignment seeks to minimize the power of H while H satisfies some connectivity constraint. In this paper, we assume E is bidirected (for every directed edge e in E, the opposite edge exists and has the same cost), while H is required to be strongly connected. This is the original power assignment problem introduced by Chen and Huang in 1989, who proved that bidirected minimum spanning tree has approximation ratio at most 2 (this is tight). In Approx 2010, we introduced a Greedy approximation algorithm and claimed a ratio of 1.992. Here we improve the analysis to 1.85. The proof also shows that a natural linear programming relaxation, introduced by us in 2012, has the same 1.85 integrality gap.

Motivation & Objective

• To improve the approximation ratio for the Min-Power Strong Connectivity problem beyond the long-standing 2-approximation.
• To develop a tighter analysis of the Greedy algorithm using fractional covering techniques inspired by Steiner Tree and broadcast power assignment.
• To establish that the natural linear programming relaxation for the problem has an integrality gap of at most 1.85.
• To simplify and strengthen earlier approximation methods by replacing integer covers with fractional covers in the analysis.

Proposed method

• The algorithm uses a greedy selection of stars (directed trees of height 1) to cover edges of a spanning tree, minimizing total power consumption.
• A fractional covering of the spanning tree edges is constructed using an optimal solution to a linear programming relaxation (LP2), assigning weights to stars based on their contribution to edge coverage.
• The analysis leverages a key lemma showing that for any collection of stars and spanning tree, there exists a fractional assignment of stars such that the total power is at most (1/2)opt*, and the coverage of uncovered tree edges is at least c(T) - f(B).
• The proof uses cut constraints and disjoint star sets on tree partitions to bound the contribution of stars to covering individual edges, ensuring that each uncovered edge is covered by stars with total weight at least 1.
• The method integrates techniques from Robins-Zelikovsky (2000) for Steiner Tree and Caragiannis et al. (2007) for broadcast power assignment, adapted via fractional covers and LP-based analysis.
• The integrality gap of the LP2 relaxation is shown to be at most 1.85 by demonstrating that the Greedy output is at most 1.85 times the LP optimum.

Experimental results

Research questions

• RQ1 Can the approximation ratio for Min-Power Strong Connectivity be improved beyond the known 2-approximation?
• RQ2 Can fractional covering techniques from Steiner Tree and broadcast power assignment be adapted to strengthen the analysis of the Greedy algorithm for strong connectivity?
• RQ3 What is the true integrality gap of the natural LP relaxation for Min-Power Strong Connectivity?
• RQ4 Can the use of fractional star covers replace the complex integer covers used in prior work to simplify and improve the analysis?

Key findings

• The paper achieves a 1.85-approximation for Min-Power Strong Connectivity, improving upon the previous best ratio of 2.
• The integrality gap of the natural LP relaxation (LP2) is proven to be at most 1.85, matching the approximation ratio of the Greedy algorithm.
• The analysis replaces integer covers with fractional covers, significantly simplifying the proof and improving the bound compared to earlier work.
• The bound of 1.85 is tight in the sense that no constant smaller than 1/2 can be used in the key covering lemma, as shown by a limiting example.
• The result demonstrates that the Greedy algorithm, when analyzed with fractional star covers and LP-based bounds, achieves a better approximation ratio than previously known.