The Shortest Path Problem with Edge Information Reuse is NP-Complete

We show that the following variation of the single-source shortest path problem is NP-complete. Let a weighted, directed, acyclic graph $G=(V,E,w)$ with source and sink vertices $s$ and $t$ be given. Let in addition a mapping $f$ on $E$ be given that associates information with the edges (e.g., a pointer), such that $f(e)=f(e’)$ means that edges $e$ and $e’$ carry the same information; for such edges it is required that $w(e)=w(e’)$. The length of a simple $st$ path $U$ is the sum of the weights of the edges on $U$ but edges with $f(e)=f(e’)$ are counted only once. The problem is to determine a shortest such $st$ path. We call this problem the \emph{edge information reuse shortest path problem}. It is NP-complete by reduction from 3SAT.

💡 Research Summary

The paper introduces a novel variant of the single‑source shortest‑path problem, called the Edge Information Reuse Shortest Path (EIRSP) problem. The input consists of a weighted, directed, acyclic graph G = (V, E, w), two distinguished vertices s (source) and t (sink), a mapping f : E → ℕ that assigns “information” to each edge, and an integer bound K. The cost of a simple s‑t path U is defined as the sum of the weights of the edges on U, but each distinct piece of information is charged only once: if two edges e and e′ have f(e) = f(e′), the weight of the later edge on the path is ignored. The decision version asks whether there exists an s‑t path whose cost r(U) does not exceed K.

The authors first observe that the problem belongs to NP. Given a candidate path U, one can traverse it twice, maintaining a set of already‑seen information identifiers; the first traversal marks all f(e) as unused, the second adds the weight of an edge only if its information has not been seen before. This verification runs in polynomial time.

The core contribution is a proof that EIRSP is NP‑complete. The reduction is from 3‑SAT, a classic NP‑complete problem. For each Boolean variable x_i the construction creates a gadget consisting of four vertices u_i, u_i′, \bar{u}i′, and u{i+1}. Two parallel edges (u_i, u_i′) and (u_i, \bar{u}i′) each have weight 1; they represent the choice of setting x_i to true or false, respectively. The subsequent edges (u_i′, u{i+1}) and (\bar{u}i′, u{i+1}) have weight 0, allowing the path to continue regardless of the choice. The edges (u_i, u_i′) and (u_i, \bar{u}_i′) are assigned the same information identifiers as certain edges in clause gadgets, establishing a link between variable assignments and clause satisfaction.

For each clause C_j = (ℓ_{j,0} ∨ ℓ_{j,1} ∨ ℓ_{j,2}) a clause gadget with five vertices v_j, v_{0j}, v_{1j}, v_{2j}, v_{j+1} is built. Edges (v_j, v_{kj}) for k = 0,1,2 have weight 1 and share their information identifiers with the variable‑gadget edges that correspond to the literal ℓ_{j,k}. Additional zero‑weight edges (v_{kj}, v_{j+1}) allow the path to exit the clause gadget after selecting one of the three literals.

All variable gadgets are concatenated in order, followed by all clause gadgets; the source s is the first vertex u_0 and the sink t is the final vertex v_m. Any s‑t path must traverse each variable gadget, thereby incurring at least n units of cost (one per variable). If the original 3‑SAT instance is satisfiable, one can choose, for each clause, a literal that is true under the satisfying assignment. The corresponding edge in the clause gadget shares its information with the edge already used in the variable gadget, so its weight is not added a second time. Consequently the total cost of the constructed path equals exactly n. Conversely, if there exists an s‑t path of cost n, the path must have used exactly one weight‑1 edge per variable and, for each clause, a weight‑1 edge whose information had already been used, implying that the associated literal is true. Hence a satisfying assignment can be extracted from the path.

The reduction uses O(n + m) vertices and edges and can be performed in polynomial time, establishing that EIRSP is NP‑hard. Combined with the NP membership, the problem is NP‑complete even on directed acyclic graphs.

The paper also discusses an earlier, flawed reduction from the PARTITION problem (included in an appendix) and acknowledges its incorrectness. As an alternative, the authors note a simple reduction from the known NP‑complete problem of finding an s‑t path that uses the minimum number of distinct colors in a colored graph. By interpreting f as the edge color and setting all weights to 1, EIRSP becomes exactly this “minimum‑color‑path” problem, providing another route to NP‑completeness.

In summary, the authors present a rigorous NP‑completeness proof for the Edge Information Reuse Shortest Path problem, highlight the subtlety introduced by allowing edge costs to be charged only once per information class, and connect the problem to existing NP‑complete variants in colored‑graph theory. The work clarifies the computational difficulty of optimizing paths under information‑reuse constraints, even in the restricted setting of DAGs.