Characterization of removable elements with respect to having k disjoint bases in a matroid

AbstractThe well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with k edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with k disjoint bases. For any graph G that may not have k-edge-disjoint spanning trees, the problem of determining what edges should be added to G so that the resulting graph has k edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid M that has k disjoint bases, the set Ek(M) of elements in M such that for any e∈Ek(M), M−e also has k disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of Ek(M) in terms of the strength of M. Consequently, this yields a characterization of edge sets Ek(G) in a graph G with at least k edge-disjoint spanning trees such that ∀e∈Ek(G), G−e also has k edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set Ek(M) in a matroid M, or the edge subset Ek(G) for a connected graph G

Cycles and Bases of Graphs and Matroids

The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In [62], Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte\u27s result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In [3], Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In [55], Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in [8] introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n \u3e 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k \u3e 0, any matroid M can be embedded into a matroid M\u27 with the same rank (that is, r(M) = r( M\u27)) such that M\u27 has k disjoint bases. Further we determine the minimum value of |E( M\u27)| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases

Aspects of matroid connectivity and uniformity.

In approaching a combinatorial problem, it is often desirable to be armed with a notion asserting that some objects are more highly structured than others. In particular, focusing on highly structured objects may avoid certain degeneracies and allow for the core of the problem to be addressed. In matroid theory, the principle notion fulfilling this role of “structure” is that of connectivity. This thesis proves a number of results furthering the knowledge of matroid connectivity and also introduces a new structural notion, that of generalised uniformity. The first part of this thesis considers 3-connected matroids and the presence of elements which may be deleted or contracted without the introduction of any non-minimal 2-separations. Principally, a Wheels-and-Whirls Theorem and then a Splitter Theorem is established, guaranteeing the existence of such elements, provided certain well-behaved structures are not present. The second part of this thesis generalises the notion of a uniform matroid by way of a 2-parameter property capturing “how uniform” a given matroid is. Initially, attention is focused on matroids representable over some field. In particular, a finiteness result is established and a specific class of binary matroids is completely determined. The concept of generalised uniformity is then considered more broadly by an analysis of its relevance to a number of established matroid notions and settings. Within that analysis, a number of equivalent characterisations of generalised uniformity are obtained. Lastly, the third part of the thesis considers a highly structured class of matroids whose members are defined by the nature of their circuits. A characterisation is achieved for the regular members of this class and, in general, the infinitely many excluded series minors are determined

Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids

Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron

Maintaining Matroid 3-Connectivity With Respect to a Fixed Basis

We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M, there exists a set K [is a subset of] E(M) of four distinct elements such that for all k [is an element of the set] K, si(M=k) is 3-connected whenever k [is an element of the set] B, and co(M\k) is 3-connected whenever k [is an element of the set] E(M) - B. Moreover, we show that if no other elements of E(M) - K satisfy this property, then M necessarily has path-width 3

Supersymmetric Field Theories, Scattering Amplitudes and the Grassmannian

In this thesis we carry out a detailed investigation of a class of four-dimensional N=1 gauge theories, known as Bipartite Field Theories (BFTs), and their utility in integrable systems and scattering amplitudes in 4-dimensional N=4 Super-Yang-Mills (SYM). We present powerful combinatorial tools for analyzing the moduli spaces of BFTs, and find an interesting connection with the matching and matroid polytopes, which play a central role in the understanding of the Grassmannian. We use the tools from BFTs to construct (0+1)-dimensional cluster integrable systems, and propose a way of obtaining (1+1)- and (2+1)-dimensional integrable field theories. Using the matching and matroid polytopes of BFTs, we analyze the singularity structure of planar and non-planar on-shell diagrams, which are central to modern developments of scattering amplitudes in N=4 SYM. In so doing, we uncover a new way of obtaining the positroid stratication of the Grassmannian. We use tools from BFTs to understand the boundary structure of the amplituhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N=4 SYM theory. We provide the most comprehensive study of the geometry of the amplituhedron to date. We also present a detailed study of non-planar on-shell diagrams, constructing the on-shell form using two new, independent methods: a non-planar boundary measurement valid for arbitrary non-planar graphs, and a proposal for a combinatorial method to determine the on-shell form directly from the graph

Approximation Algorithms for Traveling Salesman Problems

The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and has minimum cost. We consider both the symmetric traveling salesman problem (TSP) where G is an undirected graph and the asymmetric traveling salesman problem (ATSP) where G is a directed graph. We also investigate the unit-weight special cases and the more general path versions, where we do not require the walk to be closed, but to start and end in prescribed vertices s and t. In this thesis we give improved approximation algorithms and better upper bounds on the integrality ratio of the classical linear programming relaxations for several of these traveling salesman problems. For this we use techniques arising from various parts of combinatorial optimization such as linear programming, network flows, ear-decompositions, matroids, and T-joins. Our results include a (22 + &epsilon)-approximation algorithm for ATSP (for any &epsilon > 0), the first constant upper bound on the integrality ratio for s-t-path ATSP, a new upper bound on the integrality ratio for s-t-path TSP, and a black-box reduction from s-t-path TSP to TSP

Online Budgeted Maximum Coverage

We study the Online Budgeted Maximum Coverage (OBMC) problem. Subsets of a weighted ground set U arrive one by one, where each set has a cost. The online algorithm has to select a collection of sets, under the constraint that their cost is at most a given budget. Upon arrival of a set the algorithm must decide whether to accept or to reject the arriving set, and it may also drop previously accepted sets (preemption). Rejecting or dropping a set is irrevocable. The goal is to maximize the total weight of the elements covered by the sets in the chosen collection. We present a deterministic 4/(1-r)-competitive algorithm for OBMC, where r is the maximum ratio between the cost of a set and the total budget. Building on that algorithm, we then present a randomized O(1)-competitive algorithm for OBMC. On the other hand, we show that the competitive ratio of any deterministic online algorithm is Omega(1/(sqrt{1-r})). We also give a deterministic O(Delta)-competitive algorithm, where Delta is the maximum weight of a set (given that the minimum element weight is 1), and if the total weight of all elements, w(U), is known in advance, we show that a slight modification of that algorithm is O(min{Delta,sqrt{w(U)}})-competitive. A matching lower bound of Omega(min{Delta,sqrt{w(U)}}) is also given. Previous to the present work, only the unit cost version of OBMC was studied under the online setting, giving a 4-competitive algorithm [Saha, Getoor, 2009]. Finally, our results, including the lower bounds, apply to Removable Online Knapsack which is the preemptive version of the Online Knapsack problem