Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with @math crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most @math crossings, where @math is the maximum degree in G. This result implies an @math -approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.
Computing the crossing number of a graph was shown to be NP-complete by Garey and Johnson @cite_2 . Hlin e n ' y @cite_26 showed that the problem remains NP-complete even on cubic graphs. Combining the reduction from @cite_2 with the inapproximability result for Minimum-Linear-Arrangement of @cite_40 , it follows that there is no PTAS for crossing number unless NP-hard problems can be solved in randomized subexponential time. Using a reduction from Min-Uncut, and assuming the Unique Games Conjecture, one can obtain that the crossing number is hard to approximate within any constant factor.
@cite_40 @cite_26 @cite_2
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It was proved by [M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983) 312-316] that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NP-hard to determine the crossing number of a simple 3-connected cubic graph. In particular, this implies that the minor-monotone version of the crossing number problem is also NP-hard, which has been open till now. Assuming that NP spl nsube spl cap sub spl epsi > 0 BPTIME(2 sup n spl epsi ), we show that graph min-bisection, densest subgraph and bipartite clique have no PTAS. We give a reduction from the minimum distance of code problem (MDC). Starting with an instance of MDC, we build a quasi-random PCP that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in some precise sense. Among the several new techniques introduced, we give a way of certifying that a given polynomial belongs to a given subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial and it can be checked by reading a constant number of its values. Assuming that NP @math @math BPTIME( @math ), we show that graph min-bisection, dense @math -subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values. We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.