Directed Ear Anonymity

TL;DR

This work introduces ear anonymity, a directed-graph parameter that generalizes funnels, and studies its computational complexity. It proves $NP$-hardness for ear anonymity in general while giving a linear-time ($O(m(n+m))$) algorithm for DAGs by reducing to an interval-hitting problem over blocking subpaths; it also develops minor-closure properties and connections to directed treewidth. A key results is that Ear Identifying Sequence is $\Sigma_2^p$-hard, achieved via a chain of reductions from $\Sigma_2^p$-complete problems and linkage-related problems, highlighting hierarchical complexity beyond NP. The paper ends with open questions about membership in the polynomial hierarchy, dtw-based parameterized tractability, and the search for compact witnesses to certify ear anonymity bounds.

Abstract

We define and study a new structural parameter for directed graphs, which we call \emph{ear anonymity}. Our parameter aims to generalize the useful properties of \emph{funnels} to larger digraph classes. In particular, funnels are exactly the acyclic digraphs with ear anonymity one. We prove that computing the ear anonymity of a digraph is \NP/-hard and that it can be solved in $O(m(n + m))$-time on acyclic digraphs (where \(n\) is the number of vertices and \(m\) is the number of arcs in the input digraph). It remains open where exactly in the polynomial hierarchy the problem of computing ear anonymity lies, however for a related problem we manage to show $Σ_2^p$-completeness.

Figures (9)

• Figure 1: For any subset of at most 5 arcs of the cycle $( v_1, v_2, v_3, v_4, v_5, v_6, v_1 )$ we can find some path visiting such arcs which is distinct from the cycle considered.
• Figure 2: A digraph with two maximal ears. While the ear $( v_1, v_2, v_3 )$ contains all vertices of the ear $( v_1, v_3 )$, both ears admit an identifying sequence of length 1.
• Figure 3: The path $(v_2, u)$ is a deviation for the path $P = (v_1, v_2, v_3)$. The unique identifying sequence of length one for $P$ is $((v_2, v_3))$.
• Figure 4: The path $( v_2, u, v_3 )$ is a bypass for $P = ( v_1, v_2, v_3, v_4 )$. Note that there is exactly one identifying sequence of length 1 for $P$, namely $( ( v_2, v_3 ) )$.
• Figure 5: The set $\{ (v_1, v_2), (v_5, v_6), (v_8, v_9) \}$ is a hitting set of size 3 for the blocking interval set for $P = (v_1, v_2, \ldots, v_9)$, yet $\text{\upshape{ea}}_{D}(P) = 4$, witnessed by the sequence $((v_1, v_2), (v_5, v_6), (v_7, v_8), (v_8, v_9))$.

Theorems & Definitions (45)

• Definition 2.1: milani2020efficient
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• proof
• Definition 3.6
• Definition 4.1
• Definition 4.2
• Definition 4.3