58th Annual IEEE Symposium on Foundations of Computer Science Polylogarithmic approximation for minimum planarization (almost) Ken-ichi Kawarabayashi Anastasios Sidiropoulos National Institute of Informatics Department of Computer Science 2-1-2, Hitotsubashi, Chiyoda-ku University of Illinois at Chicago Tokyo, Japan Chicago, USA Email: k [email protected] Email: [email protected] Abstract—In the minimum planarization problem, given some of bounded degree [1]. We present the first non-trivial ap- n-vertex graph, the goal is to find a set of vertices of minimum proximation algorithms for this problem on general graphs. cardinality whose removal leaves a planar graph. This is a Our main results can be summarized as follows: fundamental problem in topological graph theory. We present a 32 O log (1) n-approximation algorithm for this problem on general Theorem 1.1: There exists a O(log n)-approximation graphs with running time nO(log n/ log log n). We also obtain algorithm for the minimum vertex planarization problem ε O /ε O n/ n a O(n )-approximation with running time n (1 ) for any with running time n (log log log ). arbitrarily small constant ε>0. Prior to our work, no Theorem 1.2: For any arbitrarily small constant ε> non-trivial algorithm was known for this problem on general 0, there exists a O(nε)-approximation algorithm for the graphs, and the best known result even on graphs of bounded nΩ(1) minimum vertex planarization problem with running time degree was a -approximation [1]. O(1/ε) As an immediate corollary, we also obtain improved ap- n . proximation algorithms for the crossing number problem on Applications to crossing number: The crossing number / ε graphs of bounded degree. Specifically, we obtain O(n1 2+ )- of a graph G, denoted cr(G), is the minimum number of / O approximation and n1 2 log (1) n-approximation algorithms in crossings in any drawing of G into the plane (see [2]). Prior time nO(1/ε) and nO(log n/ log log n) respectively. The previ- / O to our work, the best-known approximation for the crossing ously best-known result was a polynomial-time n9 10 log (1) n- approximation algorithm [2]. number of bounded-degree graphs was due to Chuzhoy [2]. Our algorithm introduces several new tools including an Given a bounded-degree graph, her algorithm computes a 10 O(1) efficient grid-minor construction for apex graphs, and a new drawing with (cr(G)) log n crossings, which implies O method for computing irrelevant vertices. Analogues of these a n9/10 log (1) n-approximation. We now explain how our tools were previously available only for exact algorithms. Our result on minimum planarization implies an improved ap- work gives efficient implementations of these ideas in the setting proximation algorithm for crossing number on bounded- of approximation algorithms, which could be of independent interest. degree graphs. It is easy to show that for any graph G, mvp(G) ≤ cr(G), simply by removing one endpoint of one Keywords -minimum planarization; approximation algo- edge involved in each crossing in some optimal drawing. rithm; polylogarithmic approximation; quasi-polynomial time Thus, using our α-approximation algorithm for minimum planarization, we can compute a planarizing set of size at most α · cr(G). Thus, in graphs of maximum degree Δ, I. INTRODUCTION we can compute some F ⊂ E(G), with |F |≤αΔcr(G), \ In the minimum planarization problem, given a graph G, such that G F is planar. Chimani and Hlineny´ [6] (see the goal is to find a set of vertices of minimum cardinality also [7]) have given a polynomial-time algorithm which ⊂ whose removal leaves a planar graph. This is a fundamental given some graph G and some F E(G), such that \ problem in topological graph theory, which been extensively G F is planar, computes a drawing of G with at most 3 ·| |· 3 ·| |2 studied over the past 40 years. It generalizes planarity, and O(Δ F cr(G)+Δ F ) crossings. Combining has connections to several other problems, such as crossing this with our result we immediately obtain an algorithm O(log n/ log log n) number and Euler genus. The problem is known to be fixed- with running time n , which given a graph parameter tractable [3]–[5], but very little is known about its G of bounded degree, computes a drawing of G with at 2 O(1) approximability. most (cr(G)) log n crossings. Similarly, we obtain an algorithm with running time nO(1/ε), which given a graph G of bounded degree, computes a drawing of G with at A. Our contribution O most (cr(G))2nε log (1) n crossings, for any fixed ε>0. Prior to our work, no non-trivial approximation algorithm Combining this with existing approximation algorithms for for minimum planarization was known for general graphs. crossing number of graphs of bounded degree that are based The only prior result was a nΩ(1)-approximation for graphs on balanced separators, we obtain the following (see [2] for 0272-5428/17 $31.00 © 2017 IEEE 779 DOI 10.1109/FOCS.2017.77 details). A subgraph J of G is called flat if it admits some planar O Theorem 1.3: There exists a n1/2 log (1) n- drawing with outer face F , such that all edges between J approximation algorithm for the crossing number of graphs and G \ J have one endpoint in F . If some vertex v ∈ H of bounded degree, with running time nO(log n/ log log n). is surrounded by a flat subgrid of H of size Ω(k), then Furthermore there exists a n1/2+ε-approximation algorithm it is irrelevant; this means that by removing v,wedonot for the crossing number of graphs of bounded degree, with change any optimal solution. Thus, if such an irrelevant running time nO(1/ε), for any fixed ε>0. vertex v exists, we recurse on G \{v}, and return the optimum solution found. We define the face cover of some B. Related work set of vertices U to be the minimum number of faces of In the F-deletion problem, the goal is to compute a H that are needed to cover U. If there exists some vertex minimum vertex set S in an input graph G such that G−S is u such that the neighborhood of u has face cover of size F-minor-free. Characterizing graph properties for which the Ω(k), then u is universal; that is, removing u decreases corresponding vertex deletion problem can be approximated the size of some optimum planarizing set by 1. Thus, if within a constant factor or a polylogarithmic factor is a such a universal vertex u exists, we recurse on G \{u}, long standing open problem in approximation algorithms and return the optimum solution found, together with u. [8], [9]. In spite of a long history of research, we are still If the grid H is large enough, then we can always find far from resolving the status of this problem. Constant-factor either an irrelevant or a universal vertex. Thus, by repeatedly approximation algorithms for the vertex Cover problem (i.e., removing such vertices, we arrive at a graph of bounded F = C3) are known since 1970s [10], [11]. treewidth, where the problem can be solved using standard Yannakakis [12] showed that approximating the minimum dynamic programming techniques. vertex set that needs to be deleted in order to obtain a connected graph with some property P within factor B. Obtaining an approximation algorithm. n1−ε is NP-hard, for a very broad class of properties (see We now discuss the main challenges towards extend- [12]). There was not much progress on approximability/non- ing the above approach to the approximate setting. In O approximability of vertex deletion problems until quite order to simplify the exposition, we discuss the log (1)- ε recently. Fomin et al. [13] showed that for every graph approximation algorithm. The n -approximation is essen- property P expressible by a finite set of forbidden minors tially identical, after changing some parameters. F containing at least one planar graph, the vertex deletion 1. The small treewidth case. In the above fixed- problem for property P admits a constant factor approxi- parameter algorithms, the problem is eventually reduced to mation algorithm. They explicitly mentioned that the most the bounded-treewidth case. That is, one has to solve the interesting case is when F contains a non-planar graph problem on a graph of treewidth f(k), for some function f. (they said that perhaps the most interesting case is when Since the optimum k is assumed to be constant, this can be F = {K3,3,K5}), because there is no poly-logarithmic done in polynomial time (in fact, linear time). However, in factor approximation algorithm so far. Indeed, the planar our setting, k can be as large as Ω(n), and thus this approach graph case and the non-planar case for the family F may be is not applicable. Instead, we try to find some small balanced O quite different, as the graph minor theory suggests. The main vertex separator S. If the treewidth is at most k log (1) n, O result of this paper almost settles the most interesting case. then we can find some separator of size k log (1) n. In this We believe that our techniques can lead to further results on case, we recurse on all non-planar connected components of approximation algorithms for minor-free properties. G \ S, and we add S to the final solution.