This thesis broadly focuses on the design and analysis of algorithmic tools leading to approximation algorithms with provably good performance guarantees. We were especially interested in problems that have highly practical importance yet possess a fundamental structure, such as integer covering, network design, graph partitioning, and discrete location problems. In particular, our main contrib...

In this paper, we investigate erasure-resilient codes coming from Steiner 2-designs with block size k which can correct up to any k erasures. In view of applications it is desirable that such a code can also correct as many erasures of higher order as possible. Our main result is that the erasure-resilient code constructed from an affine space with block size q – a special Steiner 2-design – ca...

Let P be a k-colored set of n points in general position on the plane, where k ≥ 2. A k-colored quadrangulation of P is a maximal straight-edge plane graph with vertex set P satisfying the property that every interior face is a properly colored quadrilateral, i.e., no edge connects vertices of the same color. It is easy to check that in general not every set of points admits a k-colored quadran...

Let G = (V, E) be a plane graph with nonnegative edge weights, and letN be a family of k vertex sets N1, N2, . . . , Nk ⊆ V , called nets. Then a noncrossing Steiner forest forN in G is a set T of k trees T1, T2, . . . , Tk in G such that each tree Ti ∈ T connects all vertices, called terminals, in net Ni , any two trees in T do not cross each other, and the sum of edge weights of all trees is ...

A vertex-colored graph G is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a vertex-rainbow u−v geodesi...

This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d 5. Here we give a polynomial time approximation algorithm of ratio O(log) for constant d, which is asymptotically optimal unless P = N P , and ...

The acyclic directed Steiner tree problem (ADSP) requires a minimal outward tree within an acyclic digraph with edge costs G = (V; E; d) which connects a root r with a distinguished subset S V , #S = k. The best possible performance guarantee of any polynomial approximation algorithm for ADSP cannot be less than 1 4 log k unless ~ P NP. The presented series of heuristics A n has a performance g...

In this paper, we consider a variation of the Euclidean Steiner Tree Problem in which the space underlying the set of nodes has a specified non-uniform cost structure. This problem is significant in many practical situations, such as laying cable networks, where the cost for laying a cable can be highly dependent on the location and the nature of the area through which it is to be laid. We empi...

We show that the diameter of connected $k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}-1$, while for $k=3$, it $\frac{57n}{23\delta}+O\left(1\right)$.