Local algorithms in (weakly) coloured graphs

A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(\Delta+1)/2$, where $\Delta$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.

💡 Research Summary

This paper investigates the power and limits of constant‑time (local) distributed algorithms for two classic combinatorial optimization problems—minimum dominating set (MDS) and maximum matching (MM)—under structural assumptions on the input graph’s vertex colouring. The authors work within the standard LOCAL model, where each node can exchange arbitrary messages with its neighbours in synchronous rounds, and a “local” algorithm finishes after a constant number of rounds, i.e., each node’s decision depends only on its O(1)-hop neighbourhood.

The study focuses on two families of coloured graphs. A weakly 2‑coloured graph is a graph whose vertices are coloured red or blue such that no two vertices of the same colour share a common neighbour. This is weaker than a proper 2‑colouring (bipartite graph) because vertices of the same colour may be adjacent, but they cannot be “twins” that see the same neighbourhood. A proper 2‑coloured graph, on the other hand, is a standard bipartite graph where each colour class forms an independent set.

Main contributions

1. Weakly 2‑coloured graphs

   • For both MDS and MM the authors present deterministic O(1)-round algorithms that achieve an approximation factor of ((\Delta+1)/2), where (\Delta) denotes the maximum degree of the graph.
   • They prove a matching lower bound: no local algorithm can guarantee a factor better than ((\Delta+1)/2) for either problem. The lower bound is established by constructing “saturated‑star” instances and arguing that any node that does not see its full (\Delta)-hop neighbourhood cannot distinguish between configurations that force a sub‑optimal choice, thereby incurring at least the stated loss.

2. Proper 2‑coloured (bipartite) graphs

   • For MDS, the extra information provided by a proper 2‑colouring does not improve the approximation ratio; the same ((\Delta+1)/2) bound remains optimal. The authors show that any algorithm that respects the bipartite structure can be transformed into one for weakly 2‑coloured graphs without loss.
   • For MM, however, the bipartite structure can be exploited to obtain a local approximation scheme: for any (\varepsilon>0) there exists a deterministic algorithm that runs in (O(\log (1/\varepsilon))) rounds and returns a matching of size at least ((1-\varepsilon))·OPT. The scheme iteratively searches for short augmenting paths (bounded length depending on (\varepsilon)) within the constant‑radius neighbourhoods, flips them, and converges to a near‑optimal matching. This result demonstrates that proper 2‑colouring can dramatically improve the approximability of MM while offering no benefit for MDS.

Algorithmic ideas

The MDS algorithm for weakly 2‑coloured graphs proceeds by each colour class locally selecting vertices of maximum degree, marking their neighbours, and repeating a constant number of times. Because vertices of the same colour cannot share a neighbour, the selected set dominates the entire graph, and a simple counting argument yields the ((\Delta+1)/2) factor.

The MM algorithm for weakly 2‑coloured graphs builds two independent matchings, one within each colour class, using a deterministic priority rule (e.g., smallest identifier). Conflicts are resolved locally, and the union of the two matchings is shown to be a ((\Delta+1)/2)‑approximation. The algorithm needs only constant‑radius information, making it a true local algorithm.

The MM approximation scheme for bipartite graphs starts from any local matching and repeatedly augments it using short augmenting paths that can be discovered in constant‑radius neighbourhoods thanks to the bipartite partition. By limiting the path length to (O(\log (1/\varepsilon))), the number of augmentation phases is also bounded, yielding an overall (O(\log (1/\varepsilon))) round algorithm with approximation ratio arbitrarily close to 1.

Lower bound technique

The lower bound construction uses a “star” centred at a high‑degree vertex together with a carefully coloured periphery. Because a node’s view is limited to a constant radius, it cannot differentiate between instances where the centre must be in the dominating set (or matched) and instances where it can be omitted without affecting optimality. This indistinguishability forces any local algorithm to make a conservative choice that incurs at least a factor of ((\Delta+1)/2) loss. The argument is formalised for both MDS and MM, showing that the presented algorithms are optimal among all constant‑time deterministic (and, by standard derandomisation arguments, also randomized) LOCAL algorithms.

Implications and future directions

The paper introduces weakly 2‑coloured graphs as a natural intermediate model between arbitrary graphs and bipartite graphs, highlighting how modest structural information can both enable and limit local computation. The tightness results for MDS and MM illustrate that, in the LOCAL model, the amount of global information that can be inferred from a constant‑radius view is fundamentally bounded by the maximum degree. The contrast between the two problems in the bipartite setting—where colour information helps MM but not MDS—suggests that problem‑specific properties dictate whether additional symmetry breaking (via colouring) can be leveraged.

Future research avenues include extending the analysis to weakly (k)-coloured graphs for (k>2), exploring randomized local algorithms that might achieve better expected ratios, and investigating the impact of communication constraints (e.g., the CONGEST model) on these approximation thresholds. Moreover, the local approximation scheme for MM could inspire similar schemes for other matching‑related problems such as edge‑cover or vertex‑cover in bipartite graphs, potentially leading to a broader theory of “local PTAS” under structural graph assumptions.