Parameterized Complexity of Path Set Packing

TL;DR

The paper analyzes Path Set Packing (PSP) through the lens of parameterized complexity, establishing both hardness and algorithmic results. It shows $W[1]$-hardness for PSP with respect to vertex cover number and also for the combined parameter pathwidth plus maximum degree plus solution size, answering open questions from prior work. On the positive side, it provides an $ ext{FPT}$ algorithm when parameterized by feedback vertex number plus maximum degree, and another $ ext{FPT}$ algorithm when parameterized by treewidth plus maximum degree plus maximum path length; in addition, it offers a $4$-approximation algorithm running in $ ext{FPT}$ time parameterized by the feedback edge number. Collectively, these results delineate the parameterized boundary between hardness and tractability for PSP and related Set Packing variants, while leaving open whether PSP is fixed-parameter tractable by the feedback edge number. $k$-Colored reductions and careful graph decompositions underpin the constructions, with all relevant math expressed in $...$ format throughout the work.

Abstract

In Path Set Packing, the input is an undirected graph $G$, a collection $\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the parameterized complexity of Path Set Packing with respect to both natural and structural parameters. We show that the problem is $W[1]$-hard with respect to vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in COCOON 2018. On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in $\calp$. These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a $4$-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.

Figures (5)

• Figure 1: An example of vertex selection gadget $H_i$, the darkened edges forms a long path $P_{x_{i,2},V_{i,1}}=(x_{i,2},c_{i,1},v_{i,1,1},c_{i,2},v_{i,1,2},......., c_{i,k},v_{i,1,k})$.
• Figure 2: An example of inter gadget edges, and darkened edges forms a short path $P_{v_{i,i',j},v_{j,j',i}}$ corresponding to an edge $v_{i,i'}v_{j,j'}$ in $G$.
• Figure 3: An example of path $P_i$, edge verification paths $P^e_{i,1}$$P^e_{i,2}$, and $P^e_{i,k}$, also the edges between vertices of vertex selection paths and edge verification paths.
• Figure 4: Example induced graphs of $G$. The darkened edges in $G[X\cup S\cup T]$ represents the edge set ${\cal A}=E(G[X]) \cup E(X,S\cup T)$. The darkened components in $G[S\cup T]$ forms the set $\cal D$, and the lighter components forms $\cal T$.
• Figure 5: The darkened edges in $G[X\cup S\cup T]$ represents the edge set ${\cal B}=E(G[X]) \cup E(X,S)$. The darkened components in $G[S\cup T]$ forms the set $\cal D$, and the lighter components forms $\cal T$. The graph $G-\cal B$ is a forest.

Theorems & Definitions (61)

• corollary thmcountercorollary
• corollary thmcountercorollary
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• definition thmcounterdefinition: DBLP:books/sp/CyganFKLMPPS15
• definition thmcounterdefinition: DBLP:books/sp/CyganFKLMPPS15
• theorem thmcountertheorem: DBLP:books/sp/CyganFKLMPPS15