Partitioning planar graphs: a fast combinatorial approach for max-cut (original) (raw)

Journal of Graph Algorithms and Applications

Algorithmic graph theory is a classical area of research by now and has been rapidly expanding during the last three decades. In many different contexts of computer science and applications, modelling problems by graphs is a natural and canonical process. Graph-theoretic concepts and algorithms play an important role in many fields of application, e.g. in communication network design, VLSI-design, CAD, traffic optimization or network visualization.

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

Algorithmica, 2010

A divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(2 6.903 √ n ) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(2 9.8594 √ n ). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2 O( √ k) n O(1) time algorithm for parameterized Planar k−cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(2 13.6 √ k n + n 3 ).

An algorithm for enumerating the near-minimum weight s-t cuts of a graph Ahmet Balcioglu

2000

: We provide an algorithm for enumerating near-minimum weight s-t cuts in directed and undirected graphs, with applications to network interdiction and network reliability. "Near-minimum" means within a factor of l+epilson of the minimum for some epilson > 0. The algorithm is based on recursive inclusion and exclusion of edges in locally minimum-weight cuts identified with a maximum flow algorithm. We prove a polynomial-time complexity result when epilson = 0, and for epilson > 0 we demonstrate good empirical efficiency. The algorithm is programmed in Java, run on a 733 MHz Pentium III computer with 128 megabytes of memory, and tested on a number of graphs. For example, all 274,550 near-minimum cuts within 10% of the minimum weight can be obtained in 74 seconds for a 627 vertex 2,450 edge unweighted graph. All 20,806 near-minimum cuts within 20% of minimum can be enumerated in 61 seconds on the same graph with weights being uniformly distributed integers in the range...

The complexity of the matching-cut problem for planar graphs and other graph classes

Journal of Graph Theory, 2009

The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be N P-complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains N P-complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is N P-complete for planar graphs with girth five. The reduction is from planar graph 3-colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching-cuts are described. These classes include claw-free graphs, co-graphs, and graphs A preliminary version of this

An exact algorithm for MAX-CUT in sparse graphs

Operations Research Letters, 2007

The MAX-CUT problem consists in partitioning the vertex set of a weighted graph into two subsets. The objective is to maximize the sum of weights of those edges that have their endpoints in two different parts of the partition. MAX-CUT is a well known NP-hard problem and it remains NP-hard even if restricted to the class of graphs with bounded maximum degree ∆ (for ∆ ≥ 3). In this paper we study exact algorithms for the MAX-CUT problem. Introducing a new technique, we present an algorithmic scheme that computes maximum cut in weighted graphs with bounded maximum degree. Our algorithm runs in time O * (2 (1−(2/∆))n ). We also describe a MAX-CUT algorithm for general weighted graphs. Its time complexity is O * (2 mn/(m+n) ). Both algorithms use polynomial space.

An efficient matlab algorithm for graph partitioning

2004

This report describes a graph partitioning algorithm based on spectral factorization that can be implemented very efficiently with just a hand full of MATLAB commands. The algorithm is closely related to the one proposed by Phillips and Kokotović [4] for state-aggregation in Markov chains. The appendix contains a MATLAB script that implements the algorithm. This algorithm is available online at [3].

O(n/sup 2/) algorithms for graph planarization

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1989

Abstmct-In this paper we present two O ( n * ) planarization algorithms-PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on Lempel, Even, and Cederbaum's planarity testing algorithm [9] and its implementation using PQ-trees [8]. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. This algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G,, of a nonplanar graph G, algorithm MAXIMAL-PLANARIZE constructs a maximal planar subgraph of G which contains G,,. This latter algorithm can also be used to maximally planarize a biconnected planar graph.

A linear-time algorithm for edge-disjoint paths in planar graphs

Combinatorica, 1995

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the "classical" case where an instance additionally fulfills the so-called evenness-condition. The fastest algorithm for this problem known from the literature requires O (nb/3(loglogn)l/3) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in an O(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.

Computing Maximum C-Planar Subgraphs

Lecture Notes in Computer Science, 2009

Deciding c-planarity for a given clustered graph C = (G, T ) is one of the most challenging problems in current graph drawing research. Though it is yet unknown if this problem is solvable in polynomial time, latest research focused on algorithmic approaches for special classes of clustered graphs. In this paper, we introduce an approach to solve the general problem using integer linear programming (ILP) techniques. We give an ILP formulation that also includes the natural generalization of cplanarity testing-the maximum c-planar subgraph problem-and solve this ILP with a branch-and-cut algorithm. Our computational results show that this approach is already successful for many clustered graphs of small to medium sizes and thus can be the foundation of a practically efficient algorithm that integrates further sophisticated ILP techniques.