The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

The Complexity of Routing with Few Collisions

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is whether one can connect the terminals by at least $p$ routes (e.g. paths) such that at most $k$ edges are time-wise shared among them. We study three types of routes: traverse each vertex at most once (paths), each edge at most once (trails), or no such restrictions (walks). We prove that for paths and trails the problem is NP-complete on undirected and directed graphs even if $k$ is constant or the maximum vertex degree in the input graph is constant. For walks, however, it is solvable in polynomial time on undirected graphs for arbitrary $k$ and on directed graphs if $k$ is constant. We additionally study for all route types a variant of the problem where the maximum length of a route is restricted by some given upper bound. We prove that this length-restricted variant has the same complexity classification with respect to paths and trails, but for walks it becomes NP-complete on undirected graphs

The Parameterized Complexity of the Minimum Shared Edges Problem

We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is contained in coNP/poly). In the proof of the fixed-parameter tractability of MSE parameterized by p, we employ the treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG 2013].Comment: 35 pages, 16 figure

Algorithms For Clustering Problems:Theoretical Guarantees and Empirical Evaluations

Clustering is a classic topic in combinatorial optimization and plays a central role in many areas, including data science and machine learning. In this thesis, we first focus on the dynamic facility location problem (i.e., the facility location problem in evolving metrics). We present a new LP-rounding algorithm for facility location problems, which yields the first constant factor approximation algorithm for the dynamic facility location problem. Our algorithm installs competing exponential clocks on clients and facilities, and connects every client by the path that repeatedly follows the smallest clock in the neighborhood. The use of exponential clocks gives rise to several properties that distinguish our approach from previous LP-roundings for facility location problems. In particular, we use \emph{no clustering} and we enable clients to connect through paths of \emph{arbitrary lengths}. In fact, the clustering-free nature of our algorithm is crucial for applying our LP-rounding approach to the dynamic problem. Furthermore, we present both empirical and theoretical aspects of the $k$-means problem. The best known algorithm for $k$-means with a provable guarantee is a simple local-search heuristic that yields an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that enables us (1) to exploit the geometric structure of $k$-means and (2) to satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a $6.357$-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of $k$-means where the underlying metric is not required to be Euclidean and for $k$-median in Euclidean metrics. We also improve the running time of our algorithm to almost linear running time and still maintain a provable guarantee. We compare our algorithm with {\sc K-Means++} (a widely studied algorithm) and show that we obtain better accuracy with comparable and even better running time