Abstract
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all oneedgedeleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory's core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexitytheoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NPhard to compute an optimal coloring for a graph from optimal colorings for all its onevertexdeleted subgraphs, and that this remains true even when optimal solutions for all oneedgedeleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) oneedgeadded supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal3UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing betavertexcritical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.
BibTeX  Entry
@InProceedings{burjons_et_al:LIPIcs:2019:11022,
author = {Elisabet Burjons and Fabian Frei and Edith Hemaspaandra and Dennis Komm and David Wehner},
title = {{Finding Optimal Solutions With Neighborly Help}},
booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
pages = {78:178:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771177},
ISSN = {18688969},
year = {2019},
volume = {138},
editor = {Peter Rossmanith and Pinar Heggernes and JoostPieter Katoen},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11022},
URN = {urn:nbn:de:0030drops110221},
doi = {10.4230/LIPIcs.MFCS.2019.78},
annote = {Keywords: Critical Graphs, Computational Complexity, Structural SelfReducibility, Minimality Problems, Colorability, Vertex Cover, Satisfiability, Reoptimizat}
}
Keywords: 

Critical Graphs, Computational Complexity, Structural SelfReducibility, Minimality Problems, Colorability, Vertex Cover, Satisfiability, Reoptimizat 
Collection: 

44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) 
Issue Date: 

2019 
Date of publication: 

20.08.2019 