Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the red-blue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements. We note that Red-Blue Set Cover is closely related to several combinatorial optimization problems studied earlier. These include the group Steiner problem, directed Steiner problem, minimum label path, minimum monotone satisfying assignment and symmetric label cover. From the equivalence of Red-Blue Set Cover and MMSA3 it follows that, unless P=NP, even the restriction of Red-Blue Set Cover where every set contains only one blue and two red elements cannot be approximated to within O(2 log 1\Gammaffi n ) , where ffi = 1= log log c n, for any constant c ! 1=2 (where n = S). We give integer programming formulations of the problem and use them to obtain a 2 p n approximation algorithm for the restricted case of Red-Blue Set Cove...

In the Minimum Label Spanning Tree problem

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V,E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L: E → {c1,...,cq}, the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors. Keywords: labeled matching; bipartite graphs; NP-complete; approximate algorithms. 1

Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a non-negative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ N such that the edge set {e ∈ E: L(e) ∈ I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s-t path problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels, where s and t are provided as part of the input. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of well-known techniques, originally studied in the context of integer covering. 1

The ultimate goal of data mining is to extract knowledge from massive data. Knowledge is ideally represented as human-comprehensible patterns from which end-users can gain intuitions and insights. Axis-parallel hyper-rectangles provide interpretable generalizations for multi-dimensional data points with numerical attributes. In this dissertation, we study the fundamental problem of rectangle-based discriminative data generalization in the context of several useful data mining applications: cluster description, rule learning, and Nearest Rectangle classification. Clustering is one of the most important data mining tasks. However, most clustering methods output sets of points as clusters and do not generalize them into interpretable patterns. We perform a systematic study of cluster description, where we propose novel description formats leading to enhanced expressive power and introduce novel description problems specifying different trade-offs between interpretability and accuracy. We also present efficient heuristic algorithms for the introduced problems in the proposed formats. If-then rules are

We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first non-trivial approximation and hardness results for the Label Cut problem. Firstly, we present an O ( √ m)-approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NP-hard to approximate Label Cut within 2 log1−1 / log logc n n for any constant c < 1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions). 1

Abstract. We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both versions of the problem. For the maximization version we give a 1-approximation algorithm and show that it is APX-2 hard. For the minimization version, we show that it is not approximable within n 1−ǫ for every ǫ> 0. When every color appears in the graph at most r times and r is an increasing function of n the problem is not O(r 1−ǫ)-approximable. For fixed constant r we analyze a polynomialtime (r+Hr)/2-approximation algorithm (Hr is the r-th harmonic number), and prove APX-hardness. Analysis of the studied algorithms is shown to be tight. 1

Abstract not found

Given a connected, undirected graph G with labeled edges, the minimum-label spanning tree problem seeks a spanning tree on G to whose edges are attached the smallest possible number of labels. A greedy heuristic for this NP-hard problem greedily chooses labels so as to reduce the number of components in the subgraphs they induce as quickly as possible. A genetic algorithm for the problem encodes candidate solutions as permutations of the labels; an initial segment of such a chromosome lists the labels that appear on the edges in the chromosome’s tree. Three versions of the GA apply generic or heuristic crossover and mutation operators and a local search step. In tests on 27 randomly-generated instances of the minimum-label spanning tree problem, versions of the GA that apply generic operators, with and without the local search step, perform less well than the greedy heuristic, but a version that applies the local search step and operators tailored to the problem returns solutions that require on average 10 % fewer labels than the heuristic’s.

Wavelength rerouting has been suggested as a viable and cost-effective method to improve the blocking performance of wavelength-routed Wavelength-Division Multiplexing (WDM) networks. This method leads to the following combinatorial optimization problem, dubbed Venetian Routing. Given a directed multigraph G along with two vertices s and t and a collection of pairwise arc-disjoint paths, we wish to find an st-path which arc-intersects the smallest possible number of the given paths. In this paper we prove the computational hardness of this problem even in various special cases, and present several approximation algorithms for its solution. In particular we show a non-trivial connection between Venetian Routing and Label Cover.