A binary tanglegram is a pair of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics or software engineering, it is required that the individual trees are drawn crossing-free. A natural optimization problem, denoted tanglegram layout problem, is thus to minimize the number of crossings between inter-tree edges. The tanglegram layout problem is NP-hard and is currently considered both in application domains and theory. In this paper we present an experimental comparison of a recursive algorithm of , our variant of their algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer quadratic program that yields optimal solutions.
In graph drawing the so-called (2SCM) is an important NP-hard problem that occurs when computing layered graph layouts. Such layouts have been introduced by @cite_5 and are widely used for drawing hierarchical graphs. In 2SCM, vertices of a bipartite graph are to be placed on two parallel lines (called ) such that for each vertex on one line all its adjacent vertices lie on the other line. As in TL the objective is to minimize the number of edge crossings provided that edges are drawn as straight-line segments. In one-sided crossing minimization (1SCM) the order of the vertices on one of the layers is fixed. Even 1SCM is NP-hard @cite_10 . J "unger and Mutzel @cite_8 performed an experimental comparison of exact and heuristic algorithms for both 1SCM and 2SCM. The main findings were that for 1SCM the exact solution can be computed quickly for up to 60 vertices in the free layer, and for 2SCM an iterated barycenter heuristic is the method of choice for instances with more than 15 vertices in each layer.
@cite_5 @cite_10 @cite_8
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A 2-layer drawing represents a bipartite graph where each vertex is a point on one of two parallel lines, no two vertices on the same line are adjacent, and the edges are straight-line segments. In this paper we study 2-layer drawings where any two crossing edges meet at right angle. We characterize the graphs that admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is ( NP )-complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings. Consider a bipartite graph G = (O, D, E) where O and D are origin and destination node sets respectively (O = D = n), and E is a set of edges (i, j), i ∈ O and j ∈ D(E= m). Suppose we draw the origin nodes and the destination nodes arranged in two columns and the edges as straight line segments between origins and destinations. A noncrossing matching is a subset of edges M ⊑ E such that no two edges of M intersect (including intersections at nodes). The Maximum Non Crossing Matching (MNCM) is the problem of finding the noncrossing matching of maximum cardinality. Let p denote the cardinality of the MNCM. Problems arising in several fields can be modelled as MNCM: for example the 3-Side Switch Box in VLSI design has been presented in [2] where an O(n 2) time algorithm is proposed. The problem can be reduced to the one of finding the longest increasing subsequence in a permutation of size m [1]. An algorithm for this problem has been proposed by Fredman and slightly improved by Widmayer and Wong[7]; in our case the algorithm complexity is O(m + (m - p) log (p). In this paper some labeling algorithms for the MNCM, which work directly on the bipartite graph, will be proposed. The overall complexity is improved to O(m log log n) or to O(m + minnp, (m - p) log p). Finally the weighted case is considered. We present here 2-approximation algorithms for several node deletion and edge deletion biclique problems and for an edge deletion clique problem. The biclique problem is to find a node induced subgraph that is bipartite and complete. The objective is to minimize the total weight of nodes or edges deleted so that the remaining subgraph is bipartite complete. Several variants of the biclique problem are studied here, where the problem is defined on bipartite graph or on general graphs with or without the requirement that each side of the bipartition forms an independent set. The maximum clique problem is formulated as maximizing the number (or weight) of edges in the complete subgraph. A 2-approximation algorithm is given for the minimum edge deletion version of this problem. The approximation algorithms given here are derived as a special case of an approximation technique devised for a class of formulations introduced by Hochbaum. All approximation algorithms described (and the polynomial algorithms for two versions of the node biclique problem) involve calls to a minimum cut algorithm. One conclusion of our analysis of the NP-hard problems here is that all of these problems are MAX SNP-hard and at least as difficult to approximate as the vertex cover problem. Another conclusion is that the problem of finding the minimum node cut-set, the removal of which leaves two cliques in the graph, is NP-hard and 2-approximable. An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of a given (undirected or directed) graph G=(V,E). There are four versions of the problem, depending on whether G is undirected or directed, and whether the spanning subgraph is required to be k-node connected (k-NCSS) or k-edge connected (k-ECSS). The approximation guarantees are as follows: min-size k-NCSS of an undirected graph 1+[1 k], min-size k-NCSS of a directed graph 1+[1 k], min-size k-ECSS of an undirected graph 1+[7 k], & min-size k-ECSS of a directed graph 1+[4 spl radic k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(k|E| sup 2 ). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or a-edge connected) spanning subgraph whose size is within a factor of (1.5+ spl epsiv ) of minimum, where spl epsiv >0 is a constant.