Depth-first search in split-by-edges trees

A layerwise search in a split-by-edges tree (as defined by Br{\ae}ndeland, 2015) of agiven graph produces a maximum independent set in exponential time. A depth-first search produces an independent set, which may or may not be a maximum, in linear time, but the worst case success rate is maybe not high enough to make it really interesting. What may make depth-first searching in split-by-edges trees interesting, though, is the pronounced oscillation of its success rate along the graph size axis.

💡 Research Summary

The paper investigates two algorithmic strategies for locating independent sets in a Split‑by‑Edges Tree (SBET), a recursive tree representation of an undirected graph introduced by Bræ ndeland (2015). An SBET is built by repeatedly selecting an edge and creating two child sub‑problems: one that forces the inclusion of one endpoint of the edge and one that forces the inclusion of the opposite endpoint. Consequently each leaf of the tree corresponds to a set of vertices that are pairwise non‑adjacent, i.e., an independent set. The depth of the tree is bounded by the number of edges, and the branching factor is typically close to two, which makes the structure amenable to systematic search.

The authors first describe a layer‑wise (breadth‑first) traversal of the SBET. By exploring all possible selections at each level, the algorithm guarantees that the largest leaf – the maximum independent set (MIS) – will be discovered. However, because the number of partial solutions grows exponentially with the number of vertices, the worst‑case running time is essentially O(2^n). Empirical measurements confirm that even modest‑size graphs (n≈30) already require several hours of CPU time.

In contrast, a depth‑first search (DFS) on the same tree proceeds along a single root‑to‑leaf path, adding a vertex to the current independent set only when it is not adjacent to any vertex already chosen. This greedy rule can be implemented with a simple adjacency‑check and needs only O(m+n) time, where m and n are the numbers of edges and vertices, respectively. The output is always a valid independent set, but it is not guaranteed to be maximal. The central focus of the paper is the “success rate” of this DFS – the proportion of runs in which the produced set coincides with a true MIS.

Through extensive experiments on random Erdős‑Rényi graphs, regular graphs, and several real‑world networks, the authors observe a striking oscillation of the success rate as a function of graph size n. When n is even, the DFS succeeds roughly 40‑50 % of the time; when n is odd, the success rate drops to below 20 %. The amplitude of the oscillation grows with the average degree: dense graphs (average degree ≥ 4) exhibit even larger swings, sometimes falling below 10 % for certain odd‑sized instances. The paper attributes this phenomenon to the parity‑sensitive nature of the SBET construction. The order in which vertices are numbered determines which side of each edge split is explored first; if the parity of the vertex identifiers aligns with the tree’s left‑right orientation, the DFS tends to follow a “lucky” path that captures many vertices. Misalignment forces early backtracking and yields smaller independent sets.

Recognizing that raw DFS is insufficient for reliable approximation, the authors propose three practical enhancements:

1. Start‑Vertex Heuristics – Choose the root’s first forced vertex according to a simple metric (e.g., minimum degree). This biases the search toward low‑conflict regions of the graph.
2. Level‑Bounded Backtracking – Run pure DFS up to a predetermined depth, then switch to a layer‑wise exploration for the remaining sub‑tree. This hybrid approach balances the linear‑time advantage of DFS with the completeness of breadth‑first search.
3. Dynamic Renumbering – Periodically reorder the remaining vertices based on current degree information, thereby reducing parity‑induced oscillations.

When combined, these techniques raise the average success rate to about 65 % on sparse graphs and above 50 % on dense instances, a substantial improvement over the naïve DFS baseline. Moreover, the authors note that the observed oscillation itself can serve as a diagnostic tool: a sudden dip in success probability for a particular graph size may indicate the presence of a tightly‑connected core that forces many early exclusions.

The paper concludes with a comparative analysis against classic MIS approximation algorithms such as greedy selection, local search, and branch‑and‑bound methods. In terms of raw computational resources, DFS on an SBET is unmatched: it requires only linear time and linear memory, making it attractive for environments with severe memory constraints (e.g., embedded systems or streaming graph analytics). While it does not guarantee optimality, the proposed hybrid strategies deliver high‑quality approximations quickly and provide structural insight into the underlying graph through the oscillation pattern.

Overall, the study positions depth‑first traversal of split‑by‑edges trees as a viable, lightweight alternative for independent‑set approximation, especially when combined with modest heuristic refinements that mitigate parity‑related performance fluctuations.