Finding Linear Arrangements of Hypergraphs with Bounded Cutwidth in Linear Time

Cutwidth is a fundamental graph layout parameter. It generalises to hypergraphs in a natural way and has been studied in a wide range of contexts. For graphs it is known that for a fixed constant k there is a linear time algorithm that for any given G, decides whether G has cutwidth at most k and, in the case of a positive answer, outputs a corresponding linear arrangement. We show that such an algorithm also exists for hypergraphs

Related Orderings of AT-Free Graphs

An ordering of a graph G is a bijection of V(G) to {1, . . . , |V(G)|}. In this thesis, we consider the complexity of two types of ordering problems. The first type of problem we consider aims at minimizing objective functions related to an ordering of the graph. We consider the problems Cutwidth, Imbalance, and Optimal Linear Arrangement. We also consider a problem of another type: S-End-Vertex, where S is one of the following search algorithms: breadth-first search (BFS), lexicographic breadth-first search (LBFS), depth-first search (DFS), and maximal neighbourhood search (MNS). This problem asks if a specified vertex can be the last vertex in an ordering generated by S. We show that, for each type of problem, orderings for one problem may be related to orderings for another problem of that type. We show that there is always a cutwidth-minimal ordering where equivalence classes of true twins are grouped for any graph, where true twins are vertices with the same closed neighbourhood. This enables a fixed-parameter tractable (FPT) algorithm for Cutwidth on graphs parameterized by the edge clique cover number of the graph and a new parameter, the restricted twin cover number of the graph. The restricted twin cover number of the graph generalizes the vertex cover number of a graph, and is the smallest value k ≥ 0 such that there is a twin cover of the graph T and k−|T| non-trivial components of G−T. We show that there is also always an imbalance-minimal ordering where equivalence classes of true twins are grouped for any graph. We show a polynomial time algorithm for this problem on superfragile graphs and subsets of proper interval graphs, both subsets of AT-free graphs. An asteroidal triple (AT) is a triple of independent vertices x, y, z such that between every pair of vertices in the triple, there is a path that does not intersect the closed neighbourhood of the third. A graph without an asteroidal triple is said to be AT-free. We also provide closed formulas for Imbalance on some small graph classes. In the FPT setting, we improve algorithms for Imbalance parameterized by the vertex cover number of the input graph and show that the problem does not have a polynomially sized kernel for the same parameter number unless NP ⊆ coNP/poly. We show that Optimal Linear Arrangement also has a polynomial algorithm for superfragile graphs and an FPT algorithm with respect to the restricted twin cover number. Finally, we consider S-End-Vertex, for BFS, LBFS, DFS, and MNS. We perform the first systematic study of the problem on bipartite permutation graphs, a subset of AT-free graphs. We show that for BFS and MNS, the problem has a polynomial time solution. We improve previous results for LBFS, obtaining a linear time algorithm. For DFS, we establish a linear time algorithm. All the results follow from the linear structure of bipartite permutation graphs

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

NASA SERC 1990 Symposium on VLSI Design

This document contains papers presented at the first annual NASA Symposium on VLSI Design. NASA's involvement in this event demonstrates a need for research and development in high performance computing. High performance computing addresses problems faced by the scientific and industrial communities. High performance computing is needed in: (1) real-time manipulation of large data sets; (2) advanced systems control of spacecraft; (3) digital data transmission, error correction, and image compression; and (4) expert system control of spacecraft. Clearly, a valuable technology in meeting these needs is Very Large Scale Integration (VLSI). This conference addresses the following issues in VLSI design: (1) system architectures; (2) electronics; (3) algorithms; and (4) CAD tools

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum