New perspectives towards Woodall's Conjecture and the Generalised Berge-Fulkerson Conjecture

组合优化是组合学、运筹学和理论计算机科学的一个丰富的交叉领域。贪婪算法的出现，图中最大匹配的高效算法和多面体刻画，完全矩阵和完全图的理论，以及关于旅行推销员问题的令人难以置信的计算基准，只是该领域的一些亮点。组合优化长期以来一直作为整数和线性规划的补充工具包，只有从这个角度出发，才能实现该领域的真正力量。正如标题所建议的那样，组合优化从与组合学和最优化的联系中受益匪浅。没有什么比最小-最大定理中的这种联系更明显的了。最大-最小定理广义地说，最优化问题的最小值等于对偶优化问题的最大值。一个恰当的例子是福特和富尔克森的最大流量-最小切割定理，这一结果植根于冷战期间俄罗斯和东欧之间的铁路物流。该定理表明，网络中可以从信源发送到汇聚节点的最大流量等于我们需要切断的链路的最小容量，以隔离信宿和信源。图是包含顶点(或节点)和连接它们的边(或链接)的真实世界网络的抽象模型。如果链接是有向的(即单向的)，则我们处理有向图(有向图的缩写)。所提出的研究集中在(Di)图上的两个猜测的最小-最大关系。第一个被称为Woodall猜想，提出于20世纪70年代末。人们可以将有向图想象为城市中的单向道路网络；如果一个人可以从任何地点开车到任何其他地点，那么它是紧密相连的。为了保证这一要求，市议会可以在某些道路上允许双向交通，但希望在尽可能少的道路上这样做。1978年，卢切西和杨格在一个颇具影响力的极大极小定理中提出了这个最优化问题，之后伍德尔提出了自然的“对偶”变种。它推测，在任何有向图中，连接(双向道路)的最小大小等于不相交的最大数目(城市的两个部分，单向分开)。尽管这个猜想引起了极大的兴趣，但这个猜想仍然没有得到解决，解决这个猜想的努力已经导致了更广泛领域的一些关键发展，更具体地说，是整数和线性规划中的完全对偶积分系统和子模流的框架，以及组合优化。第二个悬而未决的问题是推广的Berge-Fulkerson猜想(GBFC)，也是在20世纪70年代末提出的。这个猜想的起源来自著名的四色问题：给定一张区域地图，正式地称为平面图，四种颜色是否足以给区域着色，从而使任何两个共享边界的区域被分配不同的颜色？在阿佩尔和哈肯肯定地回答了这个问题之后，GBFC作为所有图的自然扩张出现了。该猜想指出，在所谓的r-图中，最小度的两倍等于完美匹配的最大数目，使得每条边恰好被使用两次。(r-图是r-正则图上的图，具有一些温和的奇偶性和连通性条件。)对这一猜想的研究形成了匹配理论的主题，它在图论的学科领域中占有重要地位。这个猜想还与中国邮递员问题和著名的旅行商问题密切相关。项目提案利用了这两个问题之间以前未被探索的协同作用，最终归因于一个基本的共同线索：在这两个问题中，我们都得到了所谓的理想集覆盖线性规划公式，目标是找到具有有限浮点表示的对偶线性规划的最优解。

Combinatorial Optimisation is a rich area at the intersection of Combinatorics, Operational Research, and Theoretical Computer Science. The advent of the greedy algorithm, efficient algorithms and polyhedral characterisations of maximum matchings in graphs, the theory of perfect matrices and perfect graphs, and the incredible computational benchmarks on the travelling salesman problem are just some of the highlights of the area. Combinatorial Optimisation has long served as a complementary toolkit to Integer and Linear Programming, and only by taking this perspective would one achieve the true power of the area. Combinatorial Optimisation, as suggested by the title, benefits heavily from connections to Combinatorics and Optimisation.Nowhere is this connection more manifest than in a min-max theorem which, broadly speaking, states that the minimum of an optimisation problem is equal to the maximum of a dual optimisation problem. A case in point is the Max Flow-Min Cut theorem of Ford and Fulkerson, a result that that takes its roots in railroad logistics between Russia and Eastern Europe during the Cold War. The theorem shows that the maximum volume flow in a network that can be sent from a source to a sink node equals the minimum capacity of the links we need to cut to isolate the sink from the source. Graphs are abstract models of real-world networks that involve vertices (or nodes) and edges (or links) connecting them. If the links are directed (i.e. one-way), then we deal with a digraph (short for directed graph). The proposed research focuses on two conjectured min-max relations on (di)graphs.The first of these is known as Woodall's Conjecture, posed in the late 1970s. One can think of a digraph as a network of one-way roads in a city; it is strongly connected if one can drive from any location to any other one. To guarantee this requirement, the council may enable two-way traffic in certain roads, but would like to do so on the fewest possible roads. After this optimisation problem was addressed in an influential min-max theorem by Lucchesi and Younger in 1978, Woodall proposed the natural "dual" variant. It conjectures that in any digraph, the minimum size of a dijoin (roads to be turned two-way) equals the maximum number of disjoint dicuts (two parts of the city, one way separated). The conjecture remains unresolved despite significant interest, and efforts to tackle it have led to some crucial developments in the broader area, more specifically to the frameworks of Totally Dual Integral systems and Submodular Flows in Integer and Linear Programming, and Combinatorial Optimisation.The second unsolved problem is the Generalised Berge-Fulkerson Conjecture (GBFC), also posed in the late 1970s. The origins of the conjecture come from the famous Four-Colour Problem: Given a map of regions, known formally as a planar graph, are four colours sufficient to colour the regions such that any two regions sharing a border are assigned different colours? After this question was answered affirmatively by Appel and Haken, GBFC arose as a natural extension to all graphs. The conjecture states that in a so-called r-graph, twice the minimum degree is equal to the maximum number of perfect matchings such that every edge is used exactly twice. (An r-graph is a graph on an r-regular graph with some mild parity and connectivity conditions.) The study of this conjecture has shaped the topic of Matching Theory, important in the subject area of Graph Theory. The conjecture is also intimately linked to the Chinese Postman Problem and the famous Travelling Salesman Problem.The project proposal takes advantage of a previously unexplored synergy between the two problems, ultimately due to a basic common thread: in both problems we are given a so-called ideal set-covering linear programming formulation, and the goal is to find an optimal solution to the dual linear program with a finite floating point representation.