Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights and distances. Finally, we provide bounds and constructions of two dimensional MCWCs

A Geometric Theory for Hypergraph Matching

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, our main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, we apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical we defer it to a subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical Society. 101 pages. v2: minor changes including some additional diagrams and passages of expository tex

Randomized and Deterministic Parameterized Algorithms and Their Applications in Bioinformatics

Parameterized NP-hard problems are NP-hard problems that are associated with special variables called parameters. One example of the problem is to find simple paths of length k in a graph, where the integer k is the parameter. We call this problem the p-path problem. The p-path problem is the parameterized version of the well-known NP-complete problem - the longest simple path problem. There are two main reasons why we study parameterized NP-hard problems. First, many application problems are naturally associated with certain parameters. Hence we need to solve these parameterized NP-hard problems. Second, if parameters take only small values, we can take advantage of these parameters to design very effective algorithms. If a parameterized NP-hard problem can be solved by an algorithm of running time in form of f(k)nO(1), where k is the parameter, f(k) is independent of n, and n is the input size of the problem instance, we say that this parameterized NP-hard problem is fixed parameter tractable (FPT). If a problem is FPT and the parameter takes only small values, the problem can be solved efficiently (it can be solved almost in polynomial time). In this dissertation, first, we introduce several techniques that can be used to design efficient algorithms for parameterized NP-hard problems. These techniques include branch and bound, divide and conquer, color coding and dynamic programming, iterative compression, iterative expansion and kernelization. Then we present our results about how to use these techniques to solve parameterized NP-hard problems, such as the p-path problem and the pd-feedback vertex set problem. Especially, we designed the first algorithm of running time in form of f(k)nO(1) for the pd-feedback vertex set problem. Thus solved an outstanding open problem, i.e. if the pd-feedback vertex set problem is FPT. Finally, we will introduce how to use parameterized algorithm techniques to solve the signaling pathway problem and the motif finding problem from bioinformatics

Sufficient conditions for the existence of specified subgraphs in graphs

A classical problem in combinatorics is, given graphs G and H, to determine if H is a subgraph of G. It is usually computationally complex to determine if H is a subgraph of G. Therefore, we often prove conditions that are sufficient to guarantee that a graph G contains H as a subgraph. In Chapter 2, we consider a theorem of Dirac and Erdős from 1963 that considers when a graph contains many disjoint cycles. Generalizing the seminal result of Corrádi and Hajnal, they prove that if a graph G contains many more vertices of degree at least 2k than vertices of degree at most 2k-2, then G contains k vertex-disjoint cycles. We strengthen their result, proving that if G contains 3k more vertices of high degree than vertices of low degree, then G contains k disjoint cycles and that this bound is sharp. Moreover, when G has many vertices, G is planar, or G contains few triangles, this value can be improved to 2k. The value 2k is the best possible, as shown by examples of Dirac and Erdős. In Chapter 3, we rephrase the problem of subgraphs in the language of graph packing. Two graphs G and G' pack if G is a subgraph of the complement of G' or, equivalently, if G' is a subgraph of the complement of G. Graph packing is a restatement of the subgraph problem that does not require one graph to be specified as the underlying graph and the other as the subgraph. Theorems of Sauer and Spencer and, independently, Bollobás and Eldridge prove that if G and G' together have few edges or if the maximum degree of G and the maximum degree of G' are small, then G and G' pack. We explore two results that combine bounds on the maximum degrees and number of edges in G and G'. Recently, Alon and Yuster proved that if G and G' are graphs on n vertices such that G has a bounded number of edges and G' has bounded degree, then G and G' pack. We characterize the pairs of graphs for which their theorem is sharp. In particular, we show that for sufficiently large n, if the vertex of maximum degree in G can be appropriately placed, then G can contain more edges and still pack with G'. We also consider a conjecture of Żak that states if the sum of the number of edges in G, the number of edges in G', and the degree of the largest vertex in G or G' is bounded above by 3n - 7, then G and G' pack. We prove that, up to an additive constant, this conjecture is correct. Using the notion of list packing, we prove that there is a constant C such that if the same sum is bounded above by 3n - C, then G and G' pack. This improves a theorem of Żak from 2014. Finally, we consider a generalization of finding a matching in a graph. The stable marriage problem was introduced by Gale and Shapley in 1962 and the generalization to multiple dimensions was first mentioned by Knuth in 1976. We consider a generalization of the Stable Marriage problem with s-dimensions and purely cyclic preferences (cyclic s-DSM). In 2004, Boros et al. showed that if there are at most s agents of each gender, then every instance of cyclic s-DSM admits a stable matching. In 2006, Eriksson et al. showed this is also true when s = 3 and there are 4 agents of each gender. We extend their result, proving that when there are s+1 agents of each gender, each instance of s-DSM admits a stable matching. We also provide a minimal example of an instance of s-DSM which admits no strongly stable matching

Exact Algorithms for NP-hard Problems on Networks: Design, Analysis, and Implementation

Die Arbeit befasst sich mit dem Entwurf von exakten Algorithmen für verschiedene NP-vollständige Optimierungsprobleme auf Graphen, wie etwa Vertex Cover, Independent Set, oder Dominating Set. Viele praxisbezogene Aufgaben, beispielsweise sogenannte 'facility location'-Probleme aus dem Bereich der Entscheidungsanalyse (decision analysis), sind durch entsprechende Netzwerk-Modellierung auf derartige Fragestellungen zurückzuführen. Im Vordergrund der Arbeit stehen Lösungsverfahren mit beweisbaren Laufzeitschranken. Wegen der solchen Problemen inhärenten, hohen kombinatorischen Komplexität müssen wir exponentielles Laufzeitverhalten unserer Algorithmen in Kauf nehmen, wollen dieses jedoch kleinstmöglich halten. Wir verfolgen dabei den jüngst vorgeschlagenen Ansatz sogenannter 'parametrisierter Algorithmen'. Vereinfacht gesagt handelt es sich hierbei um eine zweidimensionale Herangehensweise, bei welcher die Laufzeit nicht ausschlie3lich in der Grö3e der Eingabeinstanz, sondern überdies auch in der Grö3e eines sogenannten 'Problemparameters' gemessen wird. Dabei untersuchen wir sowohl von theoretischer, als auch von praktischer Seite unterschiedliche Methoden des Algorithmen-Designs: Datenreduktion, beschränkte Suchbäume, Separation von Graphen und das Konzept von Baumzerlegungen. Schlie3lich stellen wir ein Software-Paket vor, welches im Rahmen dieses Projektes entwickelt wurde und eine Vielzahl der entwickelten Algorithmen implementiert. Wir berichten über eine Reihe von empirischen Studien zur Auswertung der Praxistauglichkeit dieser Algorithmen.This thesis deals with the design of exact algorithms for various NP-complete optimization problems on graphs like Vertex Cover, Independent Set, or Dominating Set. We encounter such problems in a broad variety of application ranges, e.g., when modelling so-called facility location tasks in the area of decision analysis and network design. The main focus of this work is on solving algorithms with provable bounds on the running time. Due to the seemingly unavoidable inherent high combinatorial complexity of the problems under consideration, we are forced to deal with exponential running times; our goal, however, is to keep this exponential part as low as possible. To this end, we follow a recent approach of so-called 'fixed-parameter algorithms,' where the running time of an algorithm solving this problem shall be measured not only in the size of the input instance, but also in the size of a so-called 'problem parameter.' In this sense, whereas classical complexity theory offers a one-dimensional approach, parameterized complexity theory is a two-dimensional study of combinatorial problems. We investigate from both, a theoretical, as well as a practical point of view, various methods in the design of fixed-parameter algorithms: data reduction, bounded search trees, graph separation, and the concept of tree-decompositions. In addition, a software-package is presented, which was developed in our project and which implements most of our algorithms. Finally, we report on a first series of empirical studies underpinning the practical strength and usefulness of our algorithms

Analysis And Construction Of Extremal Circulant And Other Abelian Cayley Graphs

This thesis concerns the analysis and construction of extremal circulant and other Abelian Cayley graphs. For the purpose of this thesis, extremal graphs are understood as graphs with largest possible order for given degree and diameter, and the search for them is called the degree-diameter problem. The emphasis is on circulant graphs and on families of graphs defined for infinite diameter classes for given fixed degrees. Most studies in the degree-diameter problem have employed candidate algebraic structures to generate graphs that successively improve on previous best results. In contrast, this study has made extensive use of computer searches to find extremal graphs and graph families directly, and has then sought the algebra that describes them. In this way, the maximum degree for which largest-known circulant graph families have been discovered, with order greater than the legacy lower bound, has been increased from 7 to 20 and beyond. Topics covered include graphs in the following categories, undirected unless stated otherwise: circulant, other Abelian Cayley, bipartite circulant, arc-transitive circulant, directed circulant and mixed circulant; and their main properties such as distance partition, odd girth and automorphism group size. A major aspect of this thesis is the analysis of a matrix associated with each graph family, the lattice generator matrix, with newly discovered properties such as quasimaximality, radius maximality and eccentricity. Important new relationships between graph families of common dimension have also been discovered: translation, conjugation and transposition. An Extremal Order Conjecture is established for extremal undirected circulant and other Abelian Cayley graphs of any degree and diameter. An equivalent conjecture for directed circulant graphs and certain classes of mixed circulants is also established. Most of the extremal and largest-known graphs and graph families presented here have been discovered by the author and are documented comprehensively in the appendices

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum