License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.OPODIS.2022.21
URN: urn:nbn:de:0030-drops-176413
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Melnyk, Darya ; Suomela, Jukka ; Villani, Neven

Mending Partial Solutions with Few Changes

LIPIcs-OPODIS-2022-21.pdf (0.9 MB)


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.

BibTeX - Entry

  author =	{Melnyk, Darya and Suomela, Jukka and Villani, Neven},
  title =	{{Mending Partial Solutions with Few Changes}},
  booktitle =	{26th International Conference on Principles of Distributed Systems (OPODIS 2022)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-265-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{253},
  editor =	{Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-176413},
  doi =		{10.4230/LIPIcs.OPODIS.2022.21},
  annote =	{Keywords: mending, LCL problems, volume model}

Keywords: mending, LCL problems, volume model
Collection: 26th International Conference on Principles of Distributed Systems (OPODIS 2022)
Issue Date: 2023
Date of publication: 15.02.2023

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