Distributed Backup Placement in One Round and its Applications to Maximum Matching Approximation and Self-Stabilization

In the distributed backup-placement problem each node of a network has to select one neighbor, such that the maximum number of nodes that make the same selection is minimized. This is a natural relaxation of the perfect matching problem, in which each node is selected just by one neighbor. Previous (approximate) solutions for backup placement are non-trivial, even for simple graph topologies, such as dense graphs. In this paper we devise an algorithm for dense graph topologies, including unit disk graphs, unit ball graphs, line graphs, graphs with bounded diversity, and many more. Our algorithm requires just one round, and is as simple as the following operation. Consider a circular list of neighborhood IDs, sorted in an ascending order, and select the ID that is next to the selecting vertex ID. Surprisingly, such a simple one-round strategy turns out to be very efficient for backup placement computation in dense networks. Not only that it improves the number of rounds of the solution, but also the approximation ratio is improved by a multiplicative factor of at least $2$. Our new algorithm has several interesting implications. In particular, it gives rise to a $(2 + \epsilon)$-approximation to maximum matching within $O(\log^* n)$ rounds in dense networks. The resulting algorithm is very simple as well, in sharp contrast to previous algorithms that compute such a solution within this running time. Moreover, these algorithms are applicable to a narrower graph family than our algorithm. For the same graph family, the best previously-known result has $O(\log {\Delta} + \log^* n)$ running time. Another interesting implication is the possibility to execute our backup placement algorithm as-is in the self-stabilizing setting. This makes it possible to simplify and improve other algorithms for the self-stabilizing setting, by employing helpful properties of backup placement.

💡 Research Summary

The paper addresses the distributed “backup‑placement” problem, a natural relaxation of perfect matching in which every node must select exactly one neighbor, but a node may be selected by several neighbors. The objective is to minimize the maximum number of nodes that select the same neighbor (i.e., the maximum indegree in the directed subgraph induced by the selections). Prior work achieved only poly‑logarithmic round complexities and approximation ratios that grow with the logarithm of the network size, even on dense topologies.

The authors focus on graphs whose neighbourhood independence is bounded by a constant c — that is, in the neighbourhood of any vertex the size of a maximum independent set is at most c. This family includes unit‑disk graphs, unit‑ball graphs, line graphs, claw‑free graphs, and many other dense network models that appear in wireless, sensor, and IoT settings.

The core contribution is a one‑round deterministic algorithm called next‑modulo. Each vertex v gathers the IDs of its neighbors, adds its own ID, sorts them, and selects the neighbor whose ID is the immediate successor of v’s ID on the circular list (wrapping around to the smallest ID if no larger ID exists). All vertices execute this rule simultaneously, requiring only a single communication round because the decision depends solely on locally known IDs.

The authors prove (Theorem 2.1) that in any graph with neighbourhood independence c, no vertex can be selected by more than c neighbors. Consequently the directed subgraph G′ formed by the selected edges has maximum indegree c (and maximum total degree at most c + 1). This yields an O(1)‑approximation for the backup‑placement problem, improving the previous bound of 2c + 1.

Using this subgraph, the paper derives a maximum‑matching approximation. Because Δ(G′) ≤ c + 1, a known maximal‑matching algorithm (Panconesi‑Rizzi, 2001) runs in O(Δ + log* n) = O(log* n) rounds on G′. Lemma 3.1 shows that any maximal matching in G′ captures at least a 1/(c + 1) fraction of the optimum matching size in the original graph, giving a (c + 1)‑approximation. By iterating the process a constant number k of times—each iteration computes a maximal matching on the current G′, removes matched vertices and their incident edges, and recomputes a fresh backup placement—the algorithm boosts the guarantee to a (2 + ε)‑approximation while still finishing in O(log* n) rounds (Lemma 3.2). The constant k depends only on ε and c, not on the network size.

A notable side result is that the algorithm is self‑stabilizing: because the rule depends only on immutable IDs and requires a single round, any arbitrary initial state converges to a correct backup placement after one synchronous round, and the subsequent matching phases inherit this property. This simplifies the design of fault‑tolerant distributed protocols for the same graph families.

The paper also discusses the breadth of applicable graph classes. For unit‑disk graphs, the neighbourhood independence is at most 6 (any disc of radius ½ can contain at most six mutually non‑adjacent points). For line graphs it is 2. Hence the algorithm works on many dense topologies that were previously handled only by more complex multi‑stage procedures.

In summary, the authors present a remarkably simple yet powerful technique: a one‑round “next‑modulo” neighbor selection that guarantees a constant‑factor bound on the backup‑placement load, and they leverage this to obtain fast, constant‑approximation algorithms for maximum matching (including a (2 + ε)‑approximation) in O(log* n) rounds. The approach improves both round complexity and approximation ratio over prior work, extends to a wide range of practical network models, and naturally yields self‑stabilizing protocols.