Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.Comment: 17 pages, Conference version to appear in ESA 201

Makespan minimization in job shops: a polynomial time approximation scheme

In this paper we present a polynomial time approximation scheme for the job shop scheduling problem with fixed number of machines and fixed number of operationsper job. The polynomial time approximation scheme can be extended to the case of job shop problems with release and delivery times, multiprocessor job shops, and dag job shops

A linear time approximation scheme for the job shop scheduling problem

We study the preemptive and non-preemptive versions of the job shop scheduling pr oblem when the number of machines and the number of operations per job are fixed. We present linear time approximation schemes for both problems. These algorithms are the best possible for such problems in two regards: they achieve the best po ssible performance ratio since both problems are known to be strongly NP-hard; an d they have optimum asymptotic time complexity

Approximating Spanning Trees with Few Branches

Given an undirected, connected graph, the aim of the minimum branch-node spanning tree problem is to find a spanning tree with the minimum number of nodes with degree larger than 2. The problem is motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. Unfortunately, it is NP-hard to recognize instances that admit an objective value of zero, rendering the search for guaranteed approximation ratios futile. We suggest to investigate a complementary formulation, called maximum path-node spanning tree, where the goal is to find a spanning tree that maximizes the number of nodes with degree at most two. While the optimal solutions (and the practical applications) of both formulations coincide, our formulation proves more suitable for approximation. In fact, it admits a trivial 1/2-approximation algorithm. Our main contribution is a local search algorithm that guarantees a ratio of 6/11

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w: V ↦ → R≥1 on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G, w, k 〉 of Weighted Max Leaf in linear time into an equivalent instance 〈G ′ , w ′ , k ′ 〉 such that |V (G ′) | ≤ 5.5k and k ′ ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G = (V, E) that excludes some simple substructures always contains a spanning tree with at least |V |/5.5 leaves