Longest paths in Planar DAGs in Unambiguous Logspace

We show via two different algorithms that finding the length of the longest path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and also in the complement class co-UL. The result extends to toroidal DAGs as well.

💡 Research Summary

The paper “Longest paths in Planar DAGs in Unambiguous Logspace” investigates the computational complexity of the Long‑Path problem when the input graph is a planar directed acyclic graph (DAG). While Reachability and Distance are known to lie in UL ∩ co‑UL for planar directed graphs, Long‑Path is NP‑complete in general and its status for planar DAGs had not been settled. The authors prove that Long‑Path for planar DAGs (denoted PDLP) is also in UL ∩ co‑UL, and they extend the result to DAGs that can be embedded on a torus.

The paper presents two independent algorithms that achieve this result.

Algorithm 1 – Reduction to Distance.
The authors adapt a construction originally used for series‑parallel graphs. For each vertex u they define the predecessor set P_u (all vertices that can reach u) and a set of “boundary” edges E_u that cross from P_u to its complement. Because the graph is acyclic, every s‑to‑t path crosses each E_u exactly once. Each original edge (u,v) is replaced by a path of length
 l_uv = 2·(∑_{x∈V} outdeg(x)) − 1.
Consequently, any s‑to‑t path ρ of length |ρ| in the original graph is transformed into a path of length 2·|E| − |ρ| in the new graph. This transformation reverses the ordering of path lengths: the longest path in the original graph becomes the shortest path in the transformed graph, and vice‑versa. The transformation can be performed in logarithmic space with oracle access to Reachability on the original graph, because computing the sets P_u and E_u only requires Reach queries. Therefore, a Long‑Path instance (G,s,t,k) can be reduced to a Distance instance (G′,s,t,2·|E| − k). Since Distance on planar directed graphs is already known to be in UL ∩ co‑UL (Lemma 3), the reduction yields PDLP ∈ UL ∩ co‑UL.

Algorithm 2 – Double Inductive Counting.
The second approach follows the “double counting” technique used for planar shortest‑path computation (TW07) but adapts it to longest paths. The input graph is first transformed into a single‑source, single‑sink planar grid graph G′ of size O(n²) × O(n²) while preserving a one‑to‑one correspondence between original edges and marked edges in the grid. A carefully designed weighting scheme assigns each horizontal edge weight n⁴ + (mark·n⁸) and each vertical edge weight n⁴ + (mark·n⁸) + (up·col), where “mark” indicates whether the edge corresponds to an original edge, “up” is +1 for upward edges and –1 otherwise, and “col” is the column index. This weighting guarantees that for any length ℓ, all s‑to‑t paths of length ℓ in the original graph map to paths whose total weight lies in the interval (ℓ·n⁸, (ℓ+1)·n⁸). Moreover, within each interval the minimum‑weight and maximum‑weight paths are unique. Hence the weighted graph G″ is both min‑unique and max‑unique.

Using this max‑unique property, the authors implement a nondeterministic counting procedure inspired by RA97. For each integer k they maintain:

• S_k = { v | D(v) ≥ k }, where D(v) is the length of the longest v‑to‑t path,
• c_k = |S_k| (the number of vertices whose longest‑to‑t path is at least k),
• Σ_k = Σ_{v∉S_k} D(v) (the sum of longest‑to‑t lengths for vertices not in S_k).

The algorithm guesses the total sum M = Σ_{v∈V} D(v) (which equals the sum of all longest‑to‑t distances) and iteratively updates c_k and Σ_k using a subroutine “Update”. “Update” scans all vertices, checks whether D(v) ≥ k−1 and whether all out‑neighbors have D < k−1; if so, it decrements c_{k−1} and adds (k−1) to Σ_k. The crucial subroutine “Test” nondeterministically guesses, for each vertex, either a witness path of length ≥ k (if D(v) ≥ k) or a witness of length < k (if D(v) < k). Because the graph is max‑unique, exactly one global choice of witnesses corresponds to the true longest paths, making the whole computation unambiguous. If the guessed M equals the true total T, the algorithm accepts; otherwise it rejects. The authors prove that the procedure runs in UL and its complement also runs in UL, establishing PDLP ∈ UL ∩ co‑UL.

Extension to Toroidal DAGs.
Lemma 5 (ADR05) shows that Reachability on torus‑embeddable graphs log‑space many‑one reduces to planar Reachability. Using the same reduction, the authors obtain Corollary 6, which states that Distance and Long‑Path on toroidal DAGs are no harder than their planar counterparts. Consequently, the UL ∩ co‑UL upper bound also holds for toroidal DAGs.

Context and Related Work.
The paper situates its contributions among known results: Reachability is NL‑complete in general, but L‑complete for undirected graphs (Rei05) and lies in UL ∩ co‑UL for planar directed graphs (BTV07). Distance shares the same planar upper bound (TW07). For series‑parallel graphs, JT07 proved that Reach, Distance, and Long‑Path are all L‑complete. The present work can be viewed as a generalization of the series‑parallel reduction (Lemma 4) to arbitrary planar DAGs, albeit with a more involved construction.

Conclusion.
By providing two distinct log‑space unambiguous algorithms, the authors demonstrate that the Long‑Path problem on planar DAGs (and toroidal DAGs) is no harder than Reachability or Distance on the same class of graphs. This places PDLP squarely in UL ∩ co‑UL, matching the best known bounds for related path problems and opening avenues for further exploration of longest‑path computation in restricted graph families.