Edge-Orders

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time

Scalable and Efficient Multipath Routing: Complexity and Algorithms

A fundamental unsolved challenge in multipath routing is to provide disjoint end-to-end paths, each one satisfying certain operational goals (e.g., shortest possible), without overwhelming the data plane with prohibitive amount of forwarding state. In this paper, we study the problem of finding a pair of shortest disjoint paths that can be represented by only two forwarding table entries per destination. Building on prior work on minimum length redundant trees, we show that the underlying mathematical problem is NP-complete and we present heuristic algorithms that improve the known complexity bounds from cubic to the order of a single shortest path search. Finally, by extensive simulations we find that it is possible to very closely attain the absolute optimal path length with our algorithms (the gap is just 1–5%), eventually opening the door for wide-scale multipath routing deployments

Hardness and Approximation of Submodular Minimum Linear Ordering Problems

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $f(\cdot)$ due to an ordering $\sigma$ of the items (say $[n]$), i.e., $\min_{\sigma} \sum_{i\in [n]} f(E_{i,\sigma})$, where $E_{i,\sigma}$ is the set of items mapped by $\sigma$ to indices $[i]$. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012], using Lov\'asz extension of submodular functions. We show a $(2-\frac{1+\ell_{f}}{1+|E|})$-approximation for monotone submodular MLOP where $\ell_{f}=\frac{f(E)}{\max_{x\in E}f(\{x\})}$ satisfies $1 \leq \ell_f \leq |E|$. Our theory provides new approximation bounds for special cases of the problem, in particular a $(2-\frac{1+r(E)}{1+|E|})$-approximation for the matroid MLOP, where $f = r$ is the rank function of a matroid. We further show that minimum latency vertex cover (MLVC) is $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest

Diffusion systems and heat equations on networks

A theoretical framework for sesquilinear forms defined on the direct sum of Hilbert spaces is developed in the first part. Conditions for the boundedness, ellipticity and coercivity of the sesquilinear form are proved. A criterion of E.-M. Ouhabaz is used in order to prove qualitative properties of the abstract Cauchy problem having as generator the operator associated with the sesquilinear form. In the second part we analyze quantum graphs as a special case of forms on subspaces of the direct sum of Hilbert spaces. First, we set up a framework for handling quantum graphs in the case of infinite networks. Then, the operator associated with such systems is identified and investigated. Finally, we turn our attention to symmetry properties of the associated parabolic problem and we investigate the connection with the physical concept of a gauge symmetry.Comment: 120 pages, PhD Thesi

Efficient Linked List Ranking Algorithms and Parentheses Matching as a New Strategy for Parallel Algorithm Design

The goal of a parallel algorithm is to solve a single problem using multiple processors working together and to do so in an efficient manner. In this regard, there is a need to categorize strategies in order to solve broad classes of problems with similar structures and requirements. In this dissertation, two parallel algorithm design strategies are considered: linked list ranking and parentheses matching