Flip Distances Between Graph Orientations

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10435475" target="_blank" >RIV/00216208:11320/21:10435475 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=p4Ljxf5xJ1" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=p4Ljxf5xJ1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00453-020-00751-1" target="_blank" >10.1007/s00453-020-00751-1</a>

Result language

angličtina

Original language name

Flip Distances Between Graph Orientations

Original language description

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called α-orientations of a graph G, in which every vertex v has a specified outdegree α(v) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two α-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569-576, 2019).

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

<a href="/en/project/GA19-08554S" target="_blank" >GA19-08554S: Structures and algorithms in highly symmetric graphs</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Algorithmica

ISSN

0178-4617

e-ISSN

—

Volume of the periodical

83

Issue of the periodical within the volume

1

Country of publishing house

US - UNITED STATES

Number of pages

28

Pages from-to

116-143

UT code for WoS article

000552959900001

EID of the result in the Scopus database

2-s2.0-85088645770