Parameterizing Path Partitions

TL;DR

Paper addresses the problem of partitioning a graph into a small number of paths ($PP$, $IPP$, $SPP$) and variants. The authors prove NP-hardness for $IPP$ and $SPP$ on DAGs and for $SPP$ on bipartite undirected graphs; they show $W[1]$-hardness for $(DAGSPP, DAGIPP)$ when parameterized by the number of paths, and XP membership for several cases. They provide $FPT$ algorithms parameterized by neighborhood diversity and by vertex cover, and extend results to directed graphs via directed neighborhood diversity. They also study dual parameterizations and edge-disjoint variants, outlining open questions. The work advances understanding of path partition problems and provides practical $FPT$ techniques for structural graph parameters.

Abstract

We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W[1]-hard on DAGs. We also show that SPP is in XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results transfer to the case of covering by edge-disjoint paths, and purely covering.

Figures (8)

• Figure 1: The vertex gadget, replacing $v_i$ in $G$ with nine vertices in $G'$,
• Figure 2: Two different vertex partitions of a $H(v_i)$ gadget into 3-vertex paths, corresponding to different triple selections in the construction of \ref{['thm:DAGSPP-NPhard']}.
• Figure 3: A sketch of $G' = (V',E')$; thick lines mean that all the edges across two sets are present. The thick dashed line represents all edges that have been present between $A$ and $B$ already in $G$.
• Figure 4: Gadget $G^{i,u}$, $0< u \leq n$, $i \in [k]$
• Figure 5: $G^{i,u}$ connected to $G^{j,v}$, $i<j$

Theorems & Definitions (90)

• Remark 1
• Remark 2
• proof
• Theorem 3.1
• proof
• Claim 1
• proof
• Claim 2
• proof
• Lemma 1: monnot2007path