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1 Introduction
The NP-hard Maximum Planar Subgraph (MPS) problem is an established question in graph theory, already discussed in the classical textbook by Garey and Johnson [LiuGeldmacher1979, GareyJohnson1979]. Given a graph , we ask for a largest subset of edges such that induces a planar graph. By contrast, the closely related maximal planar subgraph problem asks for a set of edges that we cannot extend without violating planarity and is trivially solvable in polynomial time. The inverse measure of MPS that counts the minimum number of edges that must be removed to obtain a planar subgraph, is called the skewness of and denoted by .
There are several reasons why this problem has received a good deal of attention: Graph theoretically, skewness is a very natural and common measure of non-planarity (like crossing number or genus). Algorithmically, finding a large planar subgraph is central to the planarization method [batiniTalamoTamassia1984, chimaniGutwenger2012] that is heavily used in graph drawing: one starts with a large (favorably maximum) planar subgraph and re-inserts the deleted edges, typically to obtain a low number of overall crossings. In fact, this gives an approximation of the crossing number with ratio roughly [chimaniHlineny17], where is the maximum node degree. Furthermore, several graph problems become easier when the input’s skewness is small or constant. E.g., we can compute a maximum flow in time [hochsteinWeihe2007]111[hochsteinWeihe2007] considers the crossing number; the algorithm trivially works also for the stronger parameter skewness.—the same runtime complexity as on planar graphs if the skewness is constant.
There are several practical heuristic approaches to tackle the problem
[ChimaniKleinWiedera2016]. However, MPS is MaxSNP-hard, i.e., there is an upper bound on the obtainable approximation ratio unless [CalinescuFernandesFinklerKarloff1998], and there are further limits known for specific algorithmic approaches [Schmid2017, ChimaniHedtkeWiedera2016]. Already a spanning tree gives an approximation ratio of , the best known ratio is [CalinescuFernandesFinklerKarloff1998], and only recently a practical -approximation algorithm emerged [Schmid2017].Considering exact algorithms, options are scarce. Over two decades ago, an integer linear program based on Kuratowski’s characterization of planarity was introduced in [Mutzel1994], which remained the only non-trivial exact algorithm. Only very recently, [ChimaniHedtkeWiedera2018] showed the existence of potentially feasible alternatives to the Kuratowski-based approach, but the former still constitutes the practically by far most efficient (and theoretically most thoroughly explored) model. All known ILP models (including those discussed in this paper) can also directly solve the weighted MPS, i.e., identify the heaviest planar subgraph w.r.t. given edge weights.
Contribution.
In this paper, we strengthen the Kuratowski model by introducing new constraints and supplementary variables, based on analyzing the cycles occurring in the solutions; see Section 3. In particular, we show in Section LABEL:sec:hierarchy that starting with the original Kuratowski model and considering cycles of growing lengths yields a natural hierarchy of ever stronger LP-relaxations. In Section LABEL:sec:furtherstrengthening, we establish additional constraint classes using our cycle variables to further strengthen the LP-relaxations, both theoretically and practically. We show the latter property in an experimental evaluation in Section LABEL:sec:experiments. We defer the proofs of some lemmata to the appendix, in which case we mark the lemma with ‘’.11todo: 1Refer reader to full version?
2 Preliminaries
Graph Notation.
Our non-planar input graph is called . Generally, we consider an undirected graph , with nodes and edges , which are cardinality-2 subsets of . We use to denote all edges incident to node in and define the node degree . If is a subgraph of , we write . A (sub)graph is a cycle if it is connected and all its nodes have degree 2. The girth of is the length of its smallest cycle. The union of two (non-disjoint) graphs is denoted by . For and we define node- and edge-induced subgraphs and , respectively. We further use .
Given a planar drawing of some planar graph , the cyclic adjacency order around each node in defines an embedding of . The disjoint regions bounded by edges in correspond to the faces of ; the infinite region, bounded only on the inside, is called outer face. The degree of any face is the number of half-edges (“sides” of edges) that occur on the boundary of ; a bridge occurs twice on the same face.
Linear Programming.
A Linear Program
(LP) is a vector
and a set of linear inequalities (constraints) that define a polyhedron in ; we ask for an element that maximizes . An Integer Linear Program (ILP) additionally requires the components of to be integral. For a given problem, one can establish different ILPs, so-called models. To solve an ILP model, one uses branch-and-bound, where dual bounds are obtained from (fractional) solutions to the LP-relaxation, i.e., the ILP without the integrality requirements. Clearly, strong such LP-bounds are desired. We say a model is at least as strong as a model , if ’s LP-relaxation gives no worse bounds than ’s. We say is stronger than if, additionally, there is an instance where gives a strictly better bound. If, in this case, arises from by adding some constraints , we say strengthen .It is often beneficial to consider only a relevant subset of constraints in the solving process, in particular when the class of constraints is (exponentially) large. The procedure is referred to as separation. We employ it on (fractional) LP-solutions for selected constraint classes.
Kuratowski Model (-Model).
The following ILP is due to Mutzel [Mutzel1994]. Jünger and Mutzel showed that both constraint classes below form facets of the planar subgraph polytope [JuengerMutzel1996]. We use solution variables (for all ) that are if and only if edge is deleted, i.e., not in the planar subgraph. (In [Mutzel1994], equivalent variables are used.) The objective minimizes the skewness—thus maximizes the planar subgraph—and is given by
Thereby, we may consider edge weights ; they are in case of the traditional unweighted MPS problem. For a given subset of edges, we define as a shorthand. We can always use Euler’s bound on the number of edges in planar graphs:
| (1) |
By Kuratowski’s theorem [Kuratowski1930], a graph is planar if and only if it neither contains a subdivision of a nor of a . Hence, it suffices to ask for any member of the (exponentially large) set of all Kuratowski subdivisions that at least one of its edges is deleted:
| (2) |
Clearly, (2) are too many constraints to use all explicitly. Instead, we identify a sufficient subset of constraints via a (heuristic) separation procedure: we round the fractional solution and obtain a graph that can be tested for planarity. If it is non-planar, we extract a Kuratowski subdivision. This method does neither guarantee to always find a violated constraint if there is any, nor that the identified subdivision in fact corresponds to a violated Kuratowski constraint. Still, since it has these guarantees on integral solutions, it suffices to obtain an exact algorithm. Over the years, the performance of this approach was improved by strong preprocessing [ChimaniGutwenger2009], finding multiple Kuratowski subdivision in linear time [ChimaniMutzelSchmidt2007], and strong primal heuristics [ChimaniKleinWiedera2016]. We use all these identically in all considered algorithms.
The Kuratowski-model forms the basis of our extensions. As such, we denote it, without any of the below extensions, by ‘-model’.
3 Stronger Constraints Based on Cycles
We now present new constraints for the planar subgraph polytope (or a lifted version thereof). All but the first class require the introduction of new variables based on cycles, leading to the cycle model. For each constraint class we first give some motivation and intuition for its feasibility, before discussing its technical details. We then describe—provided the class is large—separation routines that quickly identify violated constraints, and usually show that it strengthens our ILP model.
The appendix consists mainly of proofs for the strength of certain classes of constraints. Such proofs always proceed in the following manner: First, we describe an integrally edge-weighted graph that is used as input. Note that (integral) edge-weights are naturally obtained by contracting parallel -paths in the input and other preprocessing techniques. We restrict ourselves to instances that cannot be reduced by standard techniques [ChimaniGutwenger2009]. Next, we give an LP-feasible solution with objective value for the model that does not use the new constraints. In particular, we also show that the solution satisfies all traditional Kuratowski constraints (2). Finally, we show that there is no LP-feasible solution with objective value when using (a subset of) the new constraints.
Generalized Euler Constraints.
We know from [JuengerMutzel1996] that inequality is facet-defining for complete biconnected graphs. We are interested in a class of similar constraints for dense subgraphs with large girth. The following lemma is folklore: A planar graph has at most edges.
Proof.
Let , , and denote an embedding of . For any face of we require at least half-edges. Thus, the number of faces in is bounded by . Using Euler’s formula, we obtain , the claimed results follows when solving for . ∎
We can thus derive a feasible generalized Euler constraint for any subgraph :
| (3) |
We note that this bound can sometimes be improved: for constraints (3) to be satisfied with equality it is necessary that if
is odd and
otherwise [FernandezSiegerTait2017]. However, we did not implement this in our algorithms. 22todo: 2Do implement and test?
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