Maximum Rectilinear Crossing Number

The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many variants have been studied, differing in both the methods of counting edge crossings and the types of drawings admitted in the problem domain [9, 8, 10]. Recently there has been interest in characterizing both the minimum and maximum number of edge crossings possible in particular graph classes for various edge crossing variants (e.g., [7, 12, 6])—in particular, the maximum rectilinear crossing number (MRCN) of a graph G, denoted by CR(G) (e.g., [2]). There exist multiple terminologies and notations for this problem (as in, e.g., [12, 6]) but we adopt the conventions of [2]. In this paper we investigate the corresponding algorithmic problem, where, given a graph G we seek a straight-line drawing maximizing the number of edge crossings. We prove that the problem is NP-hard. Then we present an efficient derandomization of a known randomized 1/3-approximation algorithm [12], extended to a more general edge-weighted setting. We note the distinction between CR(G) and CR ◦ (G), where the latter denotes the maximum crossing number taken only over convex straight-line drawings. But since all convex drawings which preserve vertex ordering also preserve edge crossings, we will only examine the simple convex drawing where the vertices of G are placed along the circumference of a circle of arbitrary size. In fact the statement that CR(G) = CR ◦ (G) is an open conjecture, first posed in [2]. A number of approximation results are known for MRCN in the special cases where G is k-colorable, for k ∈ {2, 3, 4}, or where G is a triangulation [6]. In [12] a related optimization problem is considered, motivated by the “Planarity Game”, in which, given a drawing of a planar graph with many edge crossings, the player tries to rearrange the vertices in order to eliminate all edge crossings. It is shown in that paper that the problem of finding a drawing that is maximally difficult for the game player—in the sense of requiring the largest number of single-vertex moves to remove all edge crossings—is NP-Hard.