On the Randomized Locality of Matching Problems in Regular Graphs

TL;DR

This work establishes tight locality boundaries for matching problems in regular graphs within distributed models. It shows that a $(1+\epsilon)$-approximate maximum matching can be computed locally with locality independent of global graph parameters, and, in dense regimens, with $O(\log(1/\epsilon))$ rounds in CONGEST. A key technical contribution is a novel martingale-based analysis of Luby’s algorithm on the line graph, including a Regular-Graph Preservation Lemma that maintains near-regularity across rounds and yields a constant fraction of progress per round. Additionally, the paper proves node-averaged complexity bounds for maximal matching in regular graphs and provides lower bounds showing near-optimality in various regimes, creating a comprehensive view of locality for matching in regular graphs with implications for both theory and practical distributed systems.

Abstract

The main goal in distributed symmetry-breaking is to understand the locality of problems; i.e., the radius of the neighborhood that a node needs to explore in order to arrive at its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. For approximate matching, we develop randomized algorithms to show that $(1 + ε)$-approximate matching in regular graphs is truly local; i.e., the locality depends only on $ε$ and is independent of all other graph parameters. Furthermore, as long as the degree $Δ$ is not very small (namely, as long as $Δ\geq \text{poly}(1/ε)$), this dependence is only logarithmic in $1/ε$. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes $n$ or the degree $Δ$. We show matching lower bounds for both results. For maximal matching, our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only $O(1)$. Central to our main technical contribution is a novel martingale-based analysis for the $\approx 40$-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby's algorithm on the line graph of a $Δ$-regular graph results in an almost $Δ/2$-regular graph.

Figures (6)

• Figure 1: We focus on a node $u$ that has $\Delta$ neighbors. The set of neighbors of $u$ is denoted by $N(u)$, and the set of nodes at distance 2 from $u$ is denoted by $N^2(u)$. We denote the size of $N^2(u)$ by $k$. The set of edges between $N(u)$ and $N^2(u)$ is denoted by $A$. Observe that each $v\in N(u)$ has exactly $\Delta-1$ edges incident to it in $A_u$. However, this is not necessarily the case for the nodes in $N^2(u)$, that can have anywhere between $1$ to $\Delta$ incident edges in $A_u$.
• Figure 2: (Degree Two Case) These are the two ways to our (path) gadget may be inserted between $u$ and $u'$. In $r$ rounds, the shaded innermost node $v^i_r$ can only learn about the internal contents of the gadget and nothing else in the surrounding graph, no matter whether the gadget was inserted forwards or backwards. The gadget is connected to two anchor nodes via the blue curved edges.
• Figure 3: (Degree Two Case) For even $k$, we generate a distribution of counterexamples by inserting $k$ gadgets between $k$ anchor nodes $u^{1, 2}, u^{2, 3}, \cdots, u^{k-1, k}, u^{k, 1}$. Each gadget is independently and uniformly at random chosen to be inserted forwards or backwards. We then pair up gadgets, starting with $\{g^{1}, g^{2}\}$.
• Figure 4: (Even Degree Case) This is our node subgadget $h^{v}$ for degree $\Delta = 4$, which takes on the role of a node $v$ from the original construction. The red dashed lines emphasize missing edges (if they were not missing, the gadget would internally look like biclique $K_{\Delta, \Delta + 1}$). The subgadget is connected to two other subgadgets $h^{v'}, h^{v"}$ via blue curved edges to correspond with the original node being connected to two other nodes $v'$ and $v"$. To maintain the bipartiteness of the resulting graph, the nodes being replaced by these node subgadgets must all be from even-length cycles.
• Figure 5: (General Degree Case) These are nodes from five adjacent layers of our base graph, which is bipartite, regular, and has degree five. The edges are colored blue and red so that every node has three incident blue edges and two incident red edges. Each of these colored edges $(u, v)$ will be replaced by an edge subgadget $h^{u, v}$.

Theorems & Definitions (96)

• Theorem 1.1
• Theorem 1.2: Main Result I
• Theorem 1.3: Informal version of \ref{['thm:lower-general']}
• Theorem 1.4: Main Result I Is Essentially Optimal
• Theorem 1.5: Main Result II
• Lemma 2.1: Local Regular-Graph Preservation Lemma - Informal
• Lemma 2.2: Regular-Graph Preservation Lemma - Informal
• Theorem 3.1: Main Result I
• Proposition 3.1: Proposition 3 in KawarabayashiKS20
• Theorem 3.2