We prove that the maximum edge biclique problem in bipartite graphs is NP-complete. A biclique in a bipartite graph is a vertex induced subgraph which is complete. The problem of finding a biclique with a maximum number of vertices is known to be solvable in polynomial time but the complexity of finding a biclique with a maximum number of edges was still undecided. 1 Introduction Let G =(V,E) be a graph with vertex set V and edge set E.Apairof two disjoint subsets A and B of V is called a biclique if {a, b}#E for all a # A and b # B.Thustheedges{a, b} form a complete bipartite subgraph of G (which is not necessarily an induced subgraph if G is not bipartite). A biclique {A, B} clearly has |A| + |B| vertices and |A|#|B| edges. In this note we restrict ourselves to case that G is bipartite. The two colour classes of G will be denoted by V 1 and V 2 ,soV = V 1 # V 2 . Already in the book of Garey and Johnson [2] (problem GT24) the complexity of deciding whether or not a bipartit...

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

ABSTRACT. This paper is a short survey of feedback set problems. It will be published in

Many problems that are intractable for general graphs allow polynomial-time solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.

We solve an open problem posed by Eppstein in 1995 [14, 15] and re-enforced by Grohe [16, 17] concerning locally bounded treewidth in minor-closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minor-closed families of graphs with bounded local treewidth as precisely minor-closed families that minor-exclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apex-minor-free graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minor-closed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomial-time approximation schemes for a broad class of problems in apex-minor-free graphs, improving the running time from .

Given a subset of cycles of a graph, we consider the problem of finding a minimum-weight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimum-weight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem, in which one must remove a minimum-weight set of vertices so that the remaining graph is bipartite. We give a 9/4-approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties. This results in 9/4-approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primal-dual method for approximation algorithms as given in Goemans and Williamson [16]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.

in Sections 4.2 and 4.3. An asset of physical modeling that is often overlooked is its inherent exibility. For this reason, we conclude this chapter by listing examples of model speci cations tailored to speci c layout objectives. 4.1 The Springs Given a connected undirected graph with no particular background information, the following two criteria of readable layout seem to be generally agreed upon for the conventional two-dimensional straight-line representation. 1. Vertices should spread well on the page. 2. Adjacent vertices should be close. Only intuitive explanations can be oered. While uniform vertex distribution reduces clutter, the implied uniform edge lengths leave an undistorted impression of the graph. Since \clutter" and \distortion" already have physical connotations, it seems fairly natural to start thinking of a more speci c physical analogy. We are used to observing even spacing between repelling objects. This makes it natural to imagine vertices behaving l

Live range splitting techniques improve global register allocation by splitting the live ranges of variables into segments that are individually allocated registers. Load/store range analysis is a new technique for live range splitting that is based on reaching definition and live variable analyses. Our analysis localizes the profits and the register requirements of every access to every variable to provide a fine granularity of candidates for register allocation. Experiments on a suite of C and FORTRAN benchmark programs show that a graph coloring register allocator operating on load/store ranges often provides better allocations than the same allocator operating on live ranges. Experimental results also show that the computational cost of using load/store ranges for register allocation is moderately more than the cost of using live ranges. 1 Introduction Register allocation maps variables in an intermediate language program to either registers or memory locations in order to minimiz...

: Given a directed graph G = (V; A), the maximum acyclic subgraph problem is to compute a subset, A 0 , of arcs of maximum size or total weight so that G 0 = (V; A 0 ) is acyclic. We discuss several approximation algorithms for this problem. Our main result is an O(jAj + d 3 max ) algorithm that produces a solution with at least a fraction 1=2+\Omega\Gamma3 = p dmax ) of the number of arcs in an optimal solution. Here, d max is the maximum vertex degree in G. 1 Introduction Given a directed graph G = (V; A), V = f1; :::; ng, with arc weights w ij ? 0 (i; j) 2 A, the maximum acyclic subgraph problem is to find a subset A 0 ae A such that G 0 = (V; A 0 ) is acyclic and w(A 0 ) = P (i;j)2A 0 w ij is maximized. An alternative statement of this problem (the minimum feedback arc set problem) requires to find a minimum weight subset A 00 ae A such that every (directed) cycle of G contains at least one arc in A 00 . The problem is NP hard [Kp]. It belongs to the clas...

We describe new graph bipartization algorithms for layout modification and phase assignment of bright-field alternating phaseshifting masks (AltPSM) [25]. The problem of layout modification for phase-assignability reduces to the problem of making a certain layout-derived graph bipartite (i.e., 2-colorable). Previous work [3] solves bipartization optimally for the dark field alternating PSM regime. Only one degree of freedom is allowed (and relevant) for such a bipartization: edge deletion, which corresponds to increasing the spacing between features in order to remove phase conflict. Unfortunately, dark-field PSM is used only for contact layers, due to limitations of negative photoresists. Poly and metal layers are actually created using positive photoresists and bright-field masks. In this paper, we define a new graph bipartization formulation that pertains to the more technologically relevant bright-field regime. Previous work [3] does not apply to this regime. This formulation allows two degrees of freedom for layout perturbation: (i) increasing the spacing between features, and (ii) increasing the width of critical features. Each of these corresponds to node deletion in a new layout-derived graph that we define, called the feature graph. Graph bipartization by node deletion asks for a minimum weight node set A such that deletion of A makes the graph bipartite. Unlike bipartization by edge deletion, this problem is NP-hard. We investigate several practical heuristics for the node deletion bipartization of planar graphs, including one that has 9/4 approximation ratio. Computational experience with industrial VLSI layout benchmarks shows promising results.