arXiv:2605.12457 · v2
Layer-Based Width for PAFP
摘要
The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with two of its vertices labeled $s,t$ and a set $\mathcal F$ of forbidden vertex pairs, there is an $s$--$t$ path that contains at most one endpoint from each pair in $\mathcal F$. We initiate the study of PAFP through a layer-based width measure. We first look at the directed union digraph $G\cup\mathcal F$, which is obtained by adding one arc per forbidden pair to $G$, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be the set of vertices at directed shortest-path distance $d$ from $s$, and define the BFS-width from $s$ as $\max_d |L_d|$. We show that if $G\cup\mathcal F$ has BFS-width $b$ from $s$, and $p$ is the maximum, over BFS-layer cuts, of the smaller of the numbers of distinct heads and distinct tails of backward arcs crossing the cut, then PAFP is FPT parameterized by $b+p$. Here a backward arc goes from a later BFS layer to an earlier one. In contrast, we show that bounded BFS-width alone does not suffice: PAFP remains $\mathsf{NP}$-complete when the union digraph has BFS-width 2. We also study layer-based width parameters on the input graph. We show that if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs, and this result is tight. Moreover, we study exact-length layers in the input graph, where the $d$-th exact-length layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. We show that for DAGs of exact-length width at most $2$, PAFP is polynomial-time decidable by a $2$-SAT encoding of fixed-length paths. Unlike previously known polynomial-time regimes our 2 input -graph tractability results simultaneously allow unrestricted forbidden pairs and input graphs with exponentially many $s$--$t$ paths.
作者备注：This version strengthens the backward arc theorem of the previous version by achieving FPT while bounding the BFS-width and the maximum over all BFS-layer cuts of the minimum of the number of heads crossing the BFS-layer cut and the number of tails crossing the BFS-layer cut, whereas the older theorem bounded the BFS-width with the total number of backward arcs in the union digraph
期刊信息：37th International Workshop on Combinatorial Algorithms (IWOCA 2026), Lecture Notes in Computer Science, vol. 16587, pp. 311-326, Springer (2026)