eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-02-15
21:1
21:17
10.4230/LIPIcs.OPODIS.2022.21
article
Mending Partial Solutions with Few Changes
Melnyk, Darya
1
Suomela, Jukka
1
Villani, Neven
1
2
Aalto University, Finland
École Normale Supérieure Paris-Saclay, Université Paris-Saclay, France
In this paper, we study the notion of mending: given a partial solution to a graph problem, how much effort is needed to take one step towards a proper solution? For example, if we have a partial coloring of a graph, how hard is it to properly color one more node?
In prior work (SIROCCO 2022), this question was formalized and studied from the perspective of mending radius: if there is a hole that we need to patch, how far do we need to modify the solution? In this work, we investigate a complementary notion of mending volume: how many nodes need to be modified to patch a hole?
We focus on the case of locally checkable labeling problems (LCLs) in trees, and show that already in this setting there are two infinite hierarchies of problems: for infinitely many values 0 < α ≤ 1, there is an LCL problem with mending volume Θ(n^α), and for infinitely many values k ≥ 1, there is an LCL problem with mending volume Θ(log^k n). Hence the mendability of LCL problems on trees is a much more fine-grained question than what one would expect based on the mending radius alone.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol253-opodis2022/LIPIcs.OPODIS.2022.21/LIPIcs.OPODIS.2022.21.pdf
mending
LCL problems
volume model