2-Approximation algorithm for finding a spanning tree with maximum number of leaves

We study the problem of finding a spanning tree with maximum number of leaves. We present a simple 2-approximation algorithm for the problem, improving on the previous best performance ratio of 3 achieved by algorithms of Ravi and Lu. Our algorithm can be implemented to run in linear time using simple data structures. We also study the variant of the problem in which a given subset of vertices are required to be leaves in the tree. We provide a 5/2-approximation algorithm for this version of the proble

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

Approximation algorithms for bounded facility location

The bounded $k$-median problem is to select in an undirected graph $G=(V,E)$ a set $S$ of $k$ vertices such that the maximum distance from a vertex $v \in V$ to $S$ is at mos t a given bound $d$ and the average distance from vertices $V$ to $S$ is minimized. We present random ized algorithms for several versions of this problem. We also study the bounded version of the uncapacitated facility location problem. For this latter problem we present extensions of known deterministic algorithms for the unbounded version, a nd we prove some inapproximability results

Robustness analysis in combinatorial optimization

The robustness function of an optimization problem measures the maximum change in the value of its optimal solution that can be produced by changes of a given total magnitude on the values of the elements in its input. The problem of computing the robustness function of matroid optimization problems is studied under two cost models: the discrete model, which allows the removal of elements from the input, and the continuous model, which permits finite changes on the values of the elements in the input. For the discrete model, an $O(\log k)$-approximation algorithm is presented for computing the robustness function of minimum spanning trees, where $k$ is the number of edges to be removed. The algorithm uses as key subroutine a 2-approximation algorithm for the problem of dividing a graph into the maximum number of components by removing $k$ edges from it. For the continuous model, a number of results are presented. First, a general algorithm is given for computing the robustness function of any matroid. The algorithm runs in strongly polynomial time on matroids with a strongly polynomial time independence test. Faster algorithms are also presented for some particular classes of matroids: (1) an $O(n^3m^2 \log (n^2/m))$-time algorithm for graphic matroids, where $m$ is the number of elements in the matroid and $n$ is its rank, (2) an $O(mn(m+n^2)|E|\log(m^2/|E|+2))$-time algorithm for transversal matroids, where $|E|$ is a parameter of the matroid, (3) an $O(m^2n^2)$-time algorithm for scheduling matroids, and (4) an $O(m \log m)$-time algorithm for partition matroids. For this last class of matroids an optimal algorithm is also presented for evaluating the robustness function at a single point

A 3/2-Approximation Algorithm for the Graph Balancing Problem with Two Weights

In the pursuit of finding subclasses of the makespan minimization problem on unrelated parallel machines that have approximation algorithms with approximation ratio better than 2, the graph balancing problem has been of current interest. In the graph balancing problem each job can be non-preemptively scheduled on one of at most two machines with the same processing time on either machine. Recently, Ebenlendr, Krčál, and Sgall (Algorithmica 2014, 68, 62–80.) presented a 7 / 4 -approximation algorithm for the graph balancing problem. Let r , s ∈ Z + . In this paper we consider the graph balancing problem with two weights, where a job either takes r time units or s time units. We present a 3 / 2 -approximation algorithm for this problem. This is an improvement over the previously best-known approximation algorithm for the problem with approximation ratio 1.652 and it matches the best known inapproximability bound for it

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