Upward Planarity Testing via SAT

Abstract

Abstract.A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph's triconnectivity structure which was able to solve sparse graphs.In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

Social choice meets graph drawing: How to get subexponential time algorithms for ranking and drawing problems

“…All this might also give a reason why heuristics or also translations to other areas (mathematical optimization and SAT solving are prominent examples) work very well on this particular problem and also some of its generalizations and variants (see Refs. [27][28][29][30][31][32][33] and the references cited therein for some examples). This also proves the practical interest in this kind of graph drawing problems.…”

Upward Planarity Testing: A Computational Study

“…In [3], a SAT formulation based on ordered embeddings has been proposed. We consider a strict total (vertical) order of the vertices together with a strict partial (horizontal) order of the edges; edges are comparable w.r.t.…”

“…The collection of the above rules allows a satisfying truth assignment to τ and σ if and only if G is upward planar [3]. Given such an assignment, it is trivial to construct the embedding in linear time.…”

“…Given the SAT formulations, it is straight-forward to construct integer linear programs (using only binary variables) along the same lines, for which to test feasibility. This has been done for OE in [3], leading to a vastly weaker practical performance than the SAT approach. We note that FPSS (and the hybridizations) also allow such ILPs.…”

“…There, the embedding enumeration is coupled with a sophisticated method to obtain upper and lower bounds for partially embedded graphs, allowing for a branch-and-bound algorithm. Finally, the most recent approach [3] is to formulate the problem as an integer linear program (ILP) or boolean satisfiability problem (SAT), to be solved with a corresponding solver.…”