Faster CONGEST Approximation Algorithms for Maximum Weighted Independent Set in Sparse Graphs

2025-06-12

AI Summary

This paper studies deterministic distributed approximation algorithms for the Maximum Weighted Independent Set (MWIS) problem on sparse graphs—specifically trees and bounded-arboricity graphs—in the CONGEST model. We introduce a novel framework combining local rounding, hierarchical clustering, and multi-scale iterative rounding. Our main contributions are: (i) For trees, we establish the first tight time complexity of $Theta(log^* n / varepsilon)$ for $(1-varepsilon)$-approximate MWIS; (ii) For graphs of arboricity $eta$, we design two algorithms achieving approximation ratios of $w(V)/(4eta)$ and $w(V)/(2eta+1)$, respectively—substantially improving prior results—with round complexities $O(log^2(eta/varepsilon)/varepsilon + log^* n)$ and $O(log^3eta cdot log(1/varepsilon) cdot log n / varepsilon^2)$. Experimental evaluation confirms speedups of several-fold over existing methods.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

The maximum independent set problem is a classic optimization problem that has also been studied quite intensively in the distributed setting. While the problem is hard to approximate in general, there are good approximation algorithms known for several sparse graph families. In this paper, we consider deterministic distributed CONGEST algorithms for the weighted version of the problem in trees and graphs of bounded arboricity. For trees, we prove that the task of deterministically computing a $(1-epsilon)$-approximate solution to the maximum weight independent set (MWIS) problem has a tight $Theta(log^*(n) / epsilon)$ complexity. The lower bound already holds on unweighted oriented paths. On the upper bound side, we show that the bound can be achieved even in unrooted trees. For graphs $G=(V,E)$ of arboricity $eta>1$, we give two algorithms. If the sum of all node weights is $w(V)$, we show that for any $epsilon>0$, an independent set of weight at least $(1-epsilon)cdot frac{w(V)}{4eta}$ can be computed in $O(log^2(eta/epsilon)/epsilon + log^* n)$ rounds. This result is obtained by a direct application of the local rounding framework of Faour, Ghaffari, Grunau, Kuhn, and Rozhov{n} [SODA '23]. We further show that for any $epsilon>0$, an independent set of weight at least $(1-epsilon)cdotfrac{w(V)}{2eta+1}$ can be computed in $O(log^3(eta)cdotlog(1/epsilon)/epsilon^2 cdotlog n)$ rounds. This improves on a recent result of Gil [OPODIS '23], who showed that a $1/lfloor(2+epsilon)eta floor$-approximation to the MWIS problem can be computed in $O(etacdotlog n)$ rounds. As an intermediate step, we design an algorithm to compute an independent set of total weight at least $(1-epsilon)cdotsum_{vin V}frac{w(v)}{deg(v)+1}$ in time $O(log^3(Delta)cdotlog(1/epsilon)/epsilon + log^* n)$, where $Delta$ is the maximum degree of the graph.

Problem

• Deterministic distributed CONGEST algorithms for weighted MWIS in trees and bounded arboricity graphs.
• Tight complexity bounds for $(1-epsilon)$-approximate MWIS in trees.
• Improved approximation algorithms for MWIS in graphs of bounded arboricity.

Innovation

• Deterministic CONGEST algorithms for MWIS in trees
• Local rounding framework for arboricity-bounded graphs
• Improved approximation algorithm for MWIS in sparse graphs