Abstract
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 alpha-orientations of a graph G, in which every vertex v has a specified outdegree alpha (v), and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two alpha-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).
Highlights
The term flip is commonly used in combinatorics to refer to an elementary, local, reversible operation that transforms one combinatorial object into another
Such flip operations naturally yield a flip graph, whose vertices are the considered combinatorial objects, and two of them are adjacent if they differ by a single flip
A classical example is the flip graph of triangulations of a convex polygon [46, 57]; see Fig. 1
Summary
The term flip is commonly used in combinatorics to refer to an elementary, local, reversible operation that transforms one combinatorial object into another. It is known that computing the flip distance between two triangulations of a simple polygon [4] or of a point set [37] is NP-hard. The latter is known to be fixed-parameter tractable [33]. We consider vertex flips in c-orientations, inducing flip graphs that are distributive lattices and in particular subgraphs of skeletons of certain distributive polytopes These polytopes specialize to flip polytopes of planar -orientations, are generalized by the polytope of tensions of a digraph, and form part of the family of alcoved polytopes (see [21]). 4, we give the proof of our first main result, showing that computing the flip distance between -orientations and between perfect matchings is NP-hard even for planar graphs.
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