We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has ((m^2 n) (m n)) crossings. In d-dimensions, with d >= 4, this bound becomes (m^2 n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some d-dimensional grid of volume N is n^ (n). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2^ (n) lower bound and the n^O(n) upper bound.
The work most closely related to the current work, in that it presents an extremal result relating crossings, volume, and number of edges, is that of Bose al @cite_27 , who show that the maximum number of edges in a crossing-free @math -D geometric grid graph, @math , with vertex set @math , is exactly For a fixed volume, @math , maximizing gives @math , in which case becomes @math . We state this here as lemma since we make use of it several times. boser immediately yields the upper-bound @math (see the beginning of counting ). It also yields the lower-bound @math since, if a geometric grid graph @math has @math we can remove an edge an edge from @math that eliminates at least one crossing. Since this can be repeated until @math has @math edges, this implies that @math has at least @math crossings.
@cite_27
2161190897 1485502831 2098322943 2161036821
We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g. In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings.