10 research outputs found

    Graph Theory

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    Digraph Colouring and Arc-Connectivity

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    The dichromatic number χ(D)\vec\chi(D) of a digraph DD is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by λ(D)\lambda(D) the maximum local edge connectivity of a digraph DD. Neumann-Lara proved that for every digraph DD, χ(D)λ(D)+1\vec\chi(D) \leq \lambda(D) + 1. In this paper, we characterize the digraphs DD for which χ(D)=λ(D)+1\vec\chi(D) = \lambda(D) + 1. This generalizes an analogue result for undirected graph proved by Stiebitz and Toft as well as the directed version of Brooks' Theorem proved by Mohar. Along the way, we introduce a generalization of Haj\'os join that gives a new way to construct families of dicritical digraphs that is of independent interest.Comment: 34 pages, 11 figure

    Approximation Algorithms for Path TSP, ATSP, and TAP via Relaxations

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    Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations. We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for the s-t path graph-TSP that achieves the best known approximation factor of 1.5. Then, we turn our attention to the general metric s-t path TSP. [An, Kleinberg, and Shmoys, STOC 2012] improved on the long standing 5/3-approximation factor and presented an algorithm that achieves an approximation factor of (1+\sqrt{5})/2 \approx 1.61803. Later, [Sebo, IPCO 2013] further improved the approximation factor to 8/5. We present a simple, self-contained analysis that unifies both results. Additionally, we compare two different LP relaxations of the s-t path TSP, namely the path version of the Held-Karp LP relaxation for TSP and a weaker LP relaxation, and we show that both LPs have the same (fractional) optimal value. Also, we show that the minimum cost of integral solutions of the two LPs are within a factor of 3/2 of each other. Furthermore, we prove that a half-integral solution of the stronger LP relaxation of cost c can be rounded to an integral solution of cost at most 3c/2. Finally, we give an instance that presents obstructions to two natural methods that aim for an approximation factor of 3/2. The Sherali-Adams (SA) system and the Lasserre (Las) system are two popular Lift-and-Project systems that tighten a given LP relaxation in a systematic way. In the second part of the thesis, we study the Asymmetric Traveling Salesman Problem (ATSP) and unweighted Tree Augmentation Problem, respectively, in the framework of the SA system and the Las system. For ATSP, our focus is on negative results. For any fixed integer t>=0 and small \epsilon, 0<\epsilon<<1, we prove that the integrality ratio for level t of the SA system starting with the standard LP relaxation of ATSP is at least 1+(1-\epsilon)/(2t+3). For a further relaxation of ATSP called the balanced LP relaxation, we obtain an integrality ratio lower bound of 1+(1-\epsilon)/(t+1) for level t of the SA system. Also, our results for the standard LP relaxation extend to the path version of ATSP. For the unweighted Tree Augmentation Problem, our focus is on positive results. We study this problem via the Las system. We prove an upper bound of (1.5+\epsilon) on the integrality ratio of a semidefinite programming (SDP) relaxation, where \epsilon>0 can be any small constant, by analyzing a combinatorial algorithm. This SDP relaxation is derived by applying the Las system to an initial LP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Las system via the decomposition result of [Karlin, Mathieu, and Nguyen, IPCO 2011]

    Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture

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    In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any nn-vertex graph can be decomposed into O(n)O(n) cycles and edges. We improve upon the previous best bound of O(nloglogn)O(n\log\log n) cycles and edges due to Conlon, Fox and Sudakov, by showing an nn-vertex graph can always be decomposed into O(nlogn)O(n\log^{*}n) cycles and edges, where logn\log^{*}n is the iterated logarithm function.Comment: Final version, accepted for publicatio

    Towards the Erdős-Gallai cycle decomposition conjecture

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    In the 1960's, Erdős and Gallai conjectured that the edges of any n-vertex graph can be decomposed into O(n) cycles and edges. We improve upon the previous best bound of O(nloglogn) cycles and edges due to Conlon, Fox and Sudakov, by showing an n-vertex graph can always be decomposed into O(nlog∗n) cycles and edges, where log∗n is the iterated logarithm function

    Path and cycle decompositions of dense graphs

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    We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on nn vertices can be decomposed into at most n2\left\lceil \frac{n}{2}\right\rceil paths, while a conjecture of Haj\'{o}s states that any Eulerian graph on nn vertices can be decomposed into at most n12\left\lfloor \frac{n-1}{2}\right\rfloor cycles. The Erd\H{o}s-Gallai conjecture states that any graph on nn vertices can be decomposed into O(n)O(n) cycles and edges. We show that if GG is a sufficiently large graph on nn vertices with linear minimum degree, then the following hold. (i) GG can be decomposed into at most n2+o(n)\frac{n}{2}+o(n) paths. (ii) If GG is Eulerian, then it can be decomposed into at most n2+o(n)\frac{n}{2}+o(n) cycles. (iii) GG can be decomposed into at most 3n2+o(n)\frac{3 n}{2}+o(n) cycles and edges. If in addition GG satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such GG. (iv) GG can be decomposed into max{odd(G)2,Δ(G)2}+o(n)\max \left\{\frac{odd(G)}{2},\frac{\Delta(G)}{2}\right\}+o(n) paths, where odd(G)odd(G) is the number of odd-degree vertices of GG. (v) If GG is Eulerian, then it can be decomposed into Δ(G)2+o(n)\frac{\Delta(G)}{2}+o(n) cycles. All bounds in (i)-(v) are asymptotically best possible.Comment: 48 pages, 2 figures; final version, to appear in the Journal of the London Mathematical Societ

    Path and cycle decompositions of graphs and digraphs

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    In this thesis, we make progress on five long standing conjectures on path and cycle decompositions of graphs and digraphs. Firstly, we confirm a conjecture of Jackson from 1981 by showing that the edges of any sufficiently large regular bipartite tournament can be decomposed into Hamilton cycles. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs. Secondly, we determine the minimum number of paths required to decompose the edges of any sufficiently large tournament of even order, thus resolving a conjecture of Alspach, Mason, and Pullman from 1976. We also prove an asymptotically optimal result for tournaments of odd order. Finally, we give asymptotically best possible upper bounds on the minimum number of paths, cycles, and cycles and edges required to decompose the edges of any sufficiently large dense graph. This makes progress on three famous conjectures from the 1960s: Gallai's conjecture, Hajós' conjecture, and the Erdős-Gallai conjecture, respectively. This includes joint work with António Girão, Daniela Kühn, Allan Lo, and Deryk Osthus

    Proceedings of the 10th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications

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