A linear time algorithm for a variant of the max cut problem in series parallel graphs

Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $\delta (U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $\Omega$ such that $w(\Omega)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.Comment: 6 page

A successive algorithm for the Chinese Postman Problem

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1991.Thesis (Master's) -- Bilkent University, 1991.Includes bibliographical refences.The Chinese Postman Problem being one of the well known problems in combinatorial optimization has many applications in real life problems such as mail delivery, road maintenance and bus scheduling. In this thesis work, we present a successive algorithm for the solution of Chinese Postman Problem. Additionally, we present efficient data structures for the existing algorithms in literature and for the implementation of our successive algorithm.Narin, NoyanM.S

Partitioning planar graphs: a fast combinatorial approach for max-cut

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. and that of Berman et al. The running time of the former can be bounded by O(|V|^(3/2)log|V|). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O(V{\ensuremath{|}}{\^{ }}(3/2)(log{\ensuremath{|}}V{\ensuremath{|}}){\^{ }}(3/2))alpha({\ensuremath{|}}V{\ensuremath{|}}), where alpha({\ensuremath{|}}V{\ensuremath{|}}) is the inverse Ackermann function, it can solve large instances in practice. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by O({\ensuremath{|}}V{\ensuremath{|}}{\^{ }}(3/2)log{\ensuremath{|}}V{\ensuremath{|}}), similar to the bound achieved by Shih et al. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 10{\^{ }}6 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of Shih et al. and Berman et al. It turns out that our algorithm is considerably faster in practice than the one by Shih et al. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of Berman et al. However, whereas the procedure of generating the expanded graph in Berman et al. is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task

A simple MAX-CUT algorithm for planar graphs

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time can be bounded by O(|V|^(1.5)log|V|), similar to the fastest known methods. However, our transformation yields a much smaller associated graph than that of the known methods. Furthermore, it can be computed fast. As the practical running time strongly depends on the size of the associated graph, it can be expected that our algorithm is considerably faster than the methods known in the literature. More specifically, our program can determine maximum cuts in huge realistic and random planar graphs with up to 10^6 nodes

A simple MAX-CUT algorithm for planar graphs

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time can be bounded by O(|V|^(1.5)log|V|), similar to the fastest known methods. However, our transformation yields a much smaller associated graph than that of the known methods. Furthermore, it can be computed fast. As the practical running time strongly depends on the size of the associated graph, it can be expected that our algorithm is considerably faster than the methods known in the literature. More specifically, our program can determine maximum cuts in huge realistic and random planar graphs with up to 10^6 nodes