Greedy Graph Construction Algorithm

• GGC is an algorithmic paradigm that incrementally builds graphs using local greedy decisions to optimize properties like girth, regularity, and connectivity.
• It applies to diverse areas such as randomized regular graph construction, spanner design, clustering, and Markov network learning with proven theoretical guarantees.
• Methodologies involve selecting candidate edges or merging clusters based on local criteria to efficiently approximate complex graph problems.

The Greedy Graph Construction Algorithm (GGC) encompasses a set of algorithmic paradigms that construct graphs or subgraphs via stepwise, local, greedy decisions. GGC variants play pivotal roles in graph theory, combinatorial optimization, learning graphical models, algorithmic geometry, and clustering, with rigorous analyses governing girth, regularity, stretch, and connectivity properties. The term "GGC" is now applied to broad classes: randomized regular graph construction, spanner design, cut minimization, Markov network learning, and Steiner forest approximation. This entry surveys the defining methodologies, theoretical guarantees, common algorithmic motifs, representative domains, and limits of GGC approaches.

1. Algorithmic Foundations and Archetypes

GGC is characterized by the incremental addition of edges, nodes, or merges, directed by problem-specific local greedy rules—typically optimizing some proxy or constraint at each step.

High-Girth Regular Graph Construction

The Linial–Simkin construction generates a $k$-regular $n$-vertex graph with prescribed minimum girth $g = c\log_{k-1}(n)$ ($k\geq3$, $c\in(0,1)$). GGC is initialized with a Hamilton cycle. At each step, two unsaturated vertices at distance at least $g-1$ are randomly selected and connected, provided no degree nor girth constraints are violated. The process continues until full $k$-regularity or stalling. The output, with high probability, achieves the desired local degree and global cycle-length properties (Linial et al., 2019).

Spanner Construction

In both the standard greedy multiplicative $k$-spanner algorithm and its geometric analogues, edges are considered in order (by weight or distance); an edge $(u,v)$ is added iff no path of length $\leq k$ connects $n$0 and $n$1 in the current subgraph. This guarantees shortest-path lengths inflate at most by stretch $n$2, enforces girth $n$3, and yields sparsity (Chen, 2024, Alewijnse et al., 2014).

Recent extensions add hybrid $n$4-spanner constructions, greedy $n$5-spanners, and fault-tolerant variants using a similar local-path-deficit criterion with path-length and clustering-based charging. In the Euclidean setting, geometry-aware GGC leverages local characterizations and well-separated pair decompositions to limit candidate edge pairs and reduce the expected complexity on random point sets (Popova et al., 17 Mar 2026, Alewijnse et al., 2014).

Clustering and Graph Cut

For graph partitioning, the GGC paradigm merges clusters that yield the largest decrease in a discrete cut-objective at each step, restricting candidates to neighbor clusters to ensure efficiency and determinism. In normalized-cut, this produces unique, monotonic improvement in the global criterion, contrasting with stochastic, multi-stage or heuristic alternatives (Nie et al., 2024).

Markov Network Structure Learning

In graphical model recovery, GGC sequentially adds nodes to the candidate neighborhood of each variable, always choosing the node that most reduces empirical conditional entropy. The process halts locally when further reductions fall below a threshold, giving provable sample complexity and correctness guarantees under degree and girth constraints (Netrapalli et al., 2012).

Steiner Forest

The gluttonous GGC for the Steiner Forest connects closest active terminal pairs by shortest paths under the punctured metric induced by current supernodes, greedily merging clusters to reduce active demand. This yields constant-factor approximation by amortized analysis and supports cost-sharing and stochastic generalizations (Gupta et al., 2014).

2. Theoretical Guarantees and Structural Lemmas

A central tenet of GGC methods is their rigorous analysis—the results rely on probabilistic