Mending Partial Solutions with Few Changes

TL;DR

This paper explores the complexity of transforming partial solutions into complete solutions in graph problems, introducing the concept of mending volume and revealing a rich hierarchy of problem complexities in trees.

Contribution

It introduces the notion of mending volume for LCL problems in trees and demonstrates the existence of infinite hierarchies of problem complexities based on mending volume.

Findings

Existence of LCL problems with mending volume Θ(n^α) for infinitely many α in (0,1]

Existence of LCL problems with mending volume Θ(log^k n) for infinitely many k ≥ 1

Distinct complexities for existential, expected, and deterministic mending volumes

Abstract

In this paper, we study the notion of mending, i.e. given a partial solution to a graph problem, we investigate how much effort is needed to turn it into a proper solution. For example, if we have a partial coloring of a graph, how hard is it to turn it into a proper coloring? 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 $0 < \alpha \le 1$, there is an LCL problem with mending volume $\Theta(n^\alpha)$, and for infinitely many values $k \ge 1$, there is an LCL problem with mending volume $\Theta(\log^k n)$. 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. We define three variants of the theme: (1) existential mending volume, i.e., how many nodes need to be modified, (2) expected mending volume, i.e., how many nodes we need to explore to find a patch if we use randomness, and (3) deterministic mending volume, i.e., how many nodes we need to explore if we use a deterministic algorithm. We show that all three notions are distinct from each other, and we analyze the landscape of the complexities of LCL problems for the respective models.