Nili Guttmann-Beck

Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem

Abstract. Let G=(V,E) be a complete undirected graph with vertex set V , edge set E , and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clustersV1, . . ., Vk . The clustered traveling salesman problem is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper we consider several variants of this basic problem and provide polynomial time approximation algorithms for them.

Approximation algorithms for min-sum p -clustering

We consider the following problem: Given a graph with edge lengths satisfying the triangle inequality, partition its node set into p subsets, minimizing the total length of edges whose two ends are in the same subset. For this problem we present an approximation algorithm which comes to at most twice the optimal value. For clustering into two equal-sized sets, the exact bound on the maximum possible error ratio of our algorithm is between 1.686 and 1.7.

Approximation algorithms for minimum K-cut

Abstract. Let G=(V,E) be a complete undirected graph, with node set V={v1, . . ., vn } and edge set E . The edges (vi,vj) ∈ E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = { ki }i=1p

Approximation Algorithms for Min-Max Tree Partition

(\sum_{i=1}^p k_i \leq |V|

Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

) , the minimum K-cut problem is to compute disjoint subsets with sizes { ki }i=1p , minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts.

The (K,k)-capacitated spanning tree problem

We consider the problem of partitioning the node set of a graph intopequal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded error ratio can be given for the problem unless P=NP. We present anO(n2) time algorithm for the problem, wherenis the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2p?1. We also present an improved algorithm that obtains as an input a positive integerx. It runs inO(2(p+x)pn2) time, and its error ratio is at most (2?x/(x+p?1))p.

On two restricted ancestors tree problems

Let G=(V,E) be a complete undirected graph with vertex set V, edge set E, and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clustersV 1, ..., V k . The clustered traveling salesman problem (CTSP) is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper, we consider several variants of this basic problem and provide polynomial time approximation algorithms for them.

Series-parallel orientations preserving the cycle-radius

Abstract This paper considers a generalization of the capacitated spanning tree problem, in which some of the vertices have capacity K , and the others have capacity k K . We prove that the problem can be approximated within a constant factor, and present better approximations when k is 1 or 2.

The (K, k)-capacitated spanning tree problem

We consider rooted minimum spanning tree problems subject to allowed ancestor relations which follow from network security constraints. In one problem nodes are associated with security labels that impose father-child relations. We prove that the feasible solutions define a matroid. In the other problem, there are permission constraints which impose descendant-ancestor relations. We show that even simple special cases of this problem are not approximable within a sub-logarithmic factor, and describe a square root approximation when the edge weights satisfy the triangle inequality.

Approximation algorithms for minimum tree partition

Let G be an undirected 2-edge connected graph with nonnegative edge weights and a distinguished vertex z. For every node consider the shortest cycle containing this node and z in G. The cycle-radius of G is the maximum length of a cycle in this set. Let H be a directed graph obtained by directing the edges of G. The cycle-radius of H is similarly defined except that cycles are replaced by directed closed walks. We prove that there exists for every nonnegative edge weight function an orientation H of G whose cycle-radius equals that of G if and only if G is series-parallel.