Algorithms For Longest Chains In Pseudo- Transitive Graphs

A directed acyclic graph G = (V, E) is pseudo-transitive with respect to a given subset of edges E1, if for any edge ab in E1 and any edge bc in E, we have ac in E. We give algorithms for computing longest chains and demonstrate geometric applications that unify and improves some important past results. (For specific applications see the introduction.)

💡 Research Summary

The paper introduces a new graph-theoretic concept—pseudo‑transitivity—to capture “weak transitivity” properties that arise in many geometric and combinatorial problems but are not covered by the classical theory of partially ordered sets (posets). A directed acyclic graph G=(V,E) together with a distinguished subset of edges E₁⊆E is called pseudo‑transitive if for every edge a→b in E₁ and every edge b→c in E the edge a→c also belongs to E. When E₁=∅ the definition is vacuous (any DAG becomes pseudo‑transitive), and when E₁=E the graph is fully transitive, i.e., a poset. The authors focus on two settings: (i) general pseudo‑transitive graphs, and (ii) “strongly” pseudo‑transitive graphs where both E₁ and E−E₁ are themselves partial orders.

The main technical contributions are algorithms for computing the longest chain (i.e., a directed path whose every pair of vertices is connected by an edge) under these assumptions. For a general pseudo‑transitive graph the authors show that the length ω(G) of a longest chain can be obtained in O(n·m) time, where n=|V| and m=|E|. The method splits the graph into two sub‑graphs G₁=(V,E₁) and G₂=(V,E−E₁). A topological order of the DAG is computed, and a standard DP over this order yields the longest chain in G₂. The pseudo‑transitivity condition then allows one to extend these chains through edges of E₁ in a single pass, leading to the O(n·m) bound.

When both E₁ and E−E₁ are partial orders (the strongly pseudo‑transitive case), the authors go further and handle weighted vertices. They define a chain as either “splitable” (there exists a splitting vertex j such that all edges before j belong to E₁) or “degenerate” (no such split exists). Four n×n matrices are maintained: ωₓ,ᵧ (maximum‑weight chain from x to y), ω̄ₓ,ᵧ (maximum‑weight degenerate chain), and two auxiliary matrices for special splitable cases. By processing edges in increasing order of the difference of their topological numbers, each matrix entry is computed once, yielding an overall running time of O(n·m) (worst‑case O(n³) for dense graphs). The algorithm thus finds a maximum‑weight chain in polynomial time, a problem that is NP‑hard in arbitrary DAGs.

The paper demonstrates the utility of these results through several geometric applications. Given a finite family P of bounded closed subsets of ℝᵏ and a hyperplane h orthogonal to the x‑axis, the authors construct a pseudo‑transitive graph G(P,h) whose underlying undirected graph is isomorphic to the complement of the intersection graph of P. In this construction, edges of E₁ correspond to pairs where one object lies strictly to the left of the other, while edges of E represent disjointness. Using the algorithms above, they obtain:

• An O(n³) exact algorithm for the maximum independent set of axis‑parallel unit‑height rectangles, improving slightly on the earlier dynamic‑programming approach of Agarwal, Kreveld, and Suri.
• An O(n³) exact algorithm for the largest set of pairwise disjoint line segments that all share an endpoint on a common line ℓ and make an acute angle with ℓ. For arbitrary angles they achieve a ½‑approximation in the same time bound.
• An O(n³) exact algorithm for maximum independent sets in circle graphs.

These results subsume and modestly improve several earlier works, such as those by Pach and Töröcsik on disjoint line segments, Biro and Trotter on segment orders, and Agarwal–Mustafa on planar segment independent sets. The authors also note that while the theoretical bounds are polynomial, the constants hidden in the O‑notation can be large for dense graphs, and some proof details (e.g., the exact construction of E in the geometric reduction) are sketched rather than fully formalized.

In summary, the paper contributes a novel structural notion—pseudo‑transitivity—that bridges the gap between full transitivity (posets) and arbitrary DAGs. It provides concrete O(n·m) algorithms for longest (and weighted) chains in both the general and strongly pseudo‑transitive settings, and shows that many geometric independence problems can be modeled within this framework, yielding exact or improved polynomial‑time solutions. The work opens avenues for further research on tighter algorithms for dense instances, approximation schemes for broader classes of objects, and the exploration of pseudo‑transitivity in other combinatorial optimization contexts.