Semi-Streaming Algorithms for Weighted $k$-Disjoint Matchings
S M Ferdous, Bhargav Samineni, Alex Pothen, Mahantesh Halappanavar, Bala Krishnamoorthy
TL;DR
The paper tackles weighted $k$-Disjoint Matching ($k$-DM) in the semi-streaming setting, where NP-hardness for $k\ge 2$ motivates scalable, memory-efficient approaches. It introduces two semi-streaming strategies: a primal-dual LP-based method achieving a $\frac{1}{3+\varepsilon}$-approximation and an $O(nk\log^2 n)$ space bound, and a reduction to MW$\!$b$-M that yields a $(\frac{1}{2+\varepsilon})(1-\frac{1}{k+1})$-approximation by combining a semi-streaming MW$b$M with an edge-coloring step. The authors prove dual feasibility, derive approximation guarantees, and analyze time/space complexities, with the practical benefit of drastically reduced memory and faster runtime compared to offline algorithms on large graphs including data-center network traces. Experimental results on 95 graphs show memory reductions from 6x to 512x and near-best solution quality (within about 5% of offline baselines), demonstrating the practical viability of semi-streaming $k$-DM for massive graphs. The work thus advances scalable optimization for reconfigurable networks and other large-scale systems where exact offline methods are impractical.
Abstract
We design and implement two single-pass semi-streaming algorithms for the maximum weight $k$-disjoint matching ($k$-DM) problem. Given an integer $k$, the $k$-DM problem is to find $k$ pairwise edge-disjoint matchings such that the sum of the weights of the matchings is maximized. For $k \geq 2$, this problem is NP-hard. Our first algorithm is based on the primal-dual framework of a linear programming relaxation of the problem and is $\frac{1}{3+\varepsilon}$-approximate. We also develop an approximation preserving reduction from $k$-DM to the maximum weight $b$-matching problem. Leveraging this reduction and an existing semi-streaming $b$-matching algorithm, we design a $(\frac{1}{2+\varepsilon})(1 - \frac{1}{k+1})$-approximate semi-streaming algorithm for $k$-DM. For any constant $\varepsilon > 0$, both of these algorithms require $O(nk \log_{1+\varepsilon}^2 n)$ bits of space. To the best of our knowledge, this is the first study of semi-streaming algorithms for the $k$-DM problem. We compare our two algorithms to state-of-the-art offline algorithms on 95 real-world and synthetic test problems, including thirteen graphs generated from data center network traces. On these instances, our streaming algorithms used significantly less memory (ranging from 6$\times$ to 512$\times$ less) and were faster in runtime than the offline algorithms. Our solutions were often within 5% of the best weights from the offline algorithms. We highlight that the existing offline algorithms run out of 1 TB memory for most of the large instances ($>1$ billion edges), whereas our streaming algorithms can solve these problems using only 100 GB memory for $k=8$.