eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
16:1
16:18
10.4230/LIPIcs.MFCS.2024.16
article
Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints
Bandopadhyay, Susobhan
1
https://orcid.org/0000-0003-1073-2718
Banik, Aritra
1
Majumdar, Diptapriyo
2
https://orcid.org/0000-0003-2677-4648
Sahu, Abhishek
1
National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Indraprastha Institute of Information Technology Delhi, New Delhi, India
Given an undirected graph G and a set A β V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)β§΅A. An A-path is called an (A,π)-path if the length of the path is exactly π. In the (A, π)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, π)-paths in G or not.
The problem is already known to be fixed-parmeter tractable when parameterized by k+π via color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or π (Pβ-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of G. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance between G and a path graph (distance to path). We use a randomized reduction to achieve the mentioned result. Toward this, we prove a lemma similar to the influential "isolation lemma": Given a set system (X,β±) if the elements of X are assigned a weight uniformly at random from a set of values fairly large, then each subset in β± will have a unique weight with high probability. We believe that this result will be useful beyond the scope of this paper.
ALPP being hard even for structural parameters like distance to path+|A| rules out the possibility of any FPT algorithms for many well-known other structural parameters, including FVS+|A| and treewidth+|A|. There is a straightforward FPT algorithm for ALPP parameterized by vc, the vertex cover number of the input graph. Following this, we consider the parameters CVD(cluster vertex deletion)+|A| and CVD+|π| and show the problem to be FPT with respect to these parameters. Note that CVD is incomparable to the treewidth of a graph and has been in vogue recently.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.16/LIPIcs.MFCS.2024.16.pdf
Parameterized complexity
(A,π)-Path Packing
Kernelization
Randomized-Exponential Time Hypothesis
Graph Classes