### 数学代写|组合优化代写Combinatorial optimization代考|On k-edge-connected Polyhedra

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## 数学代写|组合优化代写Combinatorial optimization代考|Box-TDIness in Series-Parallel Graphs

Totally dual integral systems – introduced in the late 70 ‘s – are strongly connected to min-max relations in combinatorial optimization (Schrijver, 1998). A rational system of linear inequalities $A x \geq b$ is totally dual integral (TDI) if the maximization problem in the linear programming duality:
$$\min \left{c^{\top} x: A x \geq b\right}=\max \left{b^{\top} y: A^{\top} y=c, y \geq \mathbf{0}\right}$$
admits an integer optimal solution for each integer vector $c$ such that the optimum is finite. Every rational polyhedron can be described by a TDI system (Giles and Pulleyblank, 1979). For instance, $\frac{1}{q} A x \geq \frac{1}{q} b$ is TDI for some positive $q$.

However,

only integer polyhedra can be described by TDI systems with integer right-hand side (Edmonds and Giles, 1984). TDI systems with only integer coefficients yield min-max results that have combinatorial interpretation.

A stronger property is the box-total dual integrality, where a system $A x \geq b$ is box-totally dual integral (box-TDI) if $A x \geq b, \ell \leq x \leq u$ is TDI for all rational vectors $\ell$ and $u$ (possibly with infinite components). General properties of such systems can be found in (Cook, 1986) and Chapter $22.4$ of (Schrijver, 1998). Note that, although every rational polyhedron ${x: A x \geq b}$ can be described by a TDI system, not every polyhedron can be described by a box-TDI system. A polyhedron which can be described by a box-TDI system is called a boxTDI polyhedron. As proved by Cook (1986), every TDI system describing such a polyhedron is actually box-TDI.

Recently, several new box-TDI systems were exhibited. Chen et al. (2008) characterized box-Mengerian matroid ports. Ding et al. (2017) characterized the graphs for which the TDI system of Cunningham and Marsh (1978) describing the matching polytope is actually box-TDI. Ding et al. (2018) introduced new subclasses of box-perfect graphs. Cornaz et al. (2019) provided several box-TDI systems in series-parallel graphs. For these graphs, Barbato et al. (2020) gave the box-TDI system for the flow cone having integer coefficients and the minimum number of constraints. Chen et al. (2009) provided a box-TDI system describing the 2-edge-connected spanning subgraph polyhedron for the same class of graphs.
In this paper, we are interested in integrality properties of systems related to $k$-edge-connected spanning subgraphs. Given a positive integer $k$, a $k$-edgeconnected spanning subgraph of a connected graph $G=(V, E)$ is a connected graph $H=(V, F)$, with $F$ a family of elements of $E$, that remains connected after the removal of any $k-1$ edges.

These objects model a kind of failure resistance of telecommunication networks. More precisely, they represent networks which remain connected when $k-1$ links fail. The underlying network design problem is the $k$-edge-connected spanning subgraph problem ( $k$-ECSSP): given a graph $G$, and positive edge costs, find a $k$-edge-connected spanning subgraph of $G$ of minimum cost. Special cases of this problem are related to classic combinatorial optimization problems. The 2-ECSSP is a well-studied relaxation of the traveling salesman problem (Erickson et al. 1987) and the 1-ECSSP is nothing but the well-known minimum spanning tree problem. While this latter is polynomial-time solvable, the $k$-ECSSP is NPhard for every fixed $k \geq 2$ (Garey and Johnson, 1979).

Different algorithms have been devised in order to deal with the $k$-ECSSP. Notable examples are branch-and-cut procedures (Cornaz et al. 2014), approximation algorithms (Gabow et al. 2009), Cutting plane algorithms Grötschel et al. (1992), and heuristics (Clarke and Anandalingam, 1995). Winter (1986) introduced a linear-time algorithm solving the 2-ECSSP on series-parallel graphs. Most of these algorithms rely on polyhedral considerations.

The $k$-edge-connected spanning subgraph polyhedron of $G$, hereafter denoted by $P_{k}(G)$, is the convex hull of all the $k$-edge-connected spanning subgraphs of $G$. Cornuéjols et al. (1985) gave a system describing $P_{2}(G)$ for series-parallel graphs.

## 数学代写|组合优化代写Combinatorial optimization代考|Definitions and Preliminary Results

Let $G=(V, E)$ be a loopless undirected graph. The graph $G$ is 2-connected if it remains connected whenever a vertex is removed. A 2-connected graph is called trivial if it is composed of a single edge. The graph obtained from two disjoint graphs by identifying two vertices, one of each graph, is called a 1 -sum. A subset of edges of $G$ is called a circuit if it induces a connected graph in which every vertex has degree 2 . Given a subset $U$ of $V$, the cut $\delta(U)$ is the set of edges having exactly one endpoint in $U$. A bond is a minimal nonempty cut. Given a partition $\left{V_{1}, \ldots, V_{n}\right}$ of $V$, the set of edges having endpoints in two distinct $V_{i}$ ‘s is called multicut and is denoted by $\delta\left(V_{1}, \ldots, V_{n}\right)$. We denote respectively by $\mathcal{M}{G}$ and $\mathcal{B}{G}$ the set of multicuts and the set of bonds of $G$. For every multicut $M$, there exists a unique partition $\left{V_{1}, \ldots, V_{d_{M}}\right}$ of vertices of $V$ such that $M=\delta\left(V_{1}, \ldots, V_{d_{M}}\right)$, and $G\left[V_{i}\right]$ – the graph induced by the vertices of $V_{i}$-is connected for all $i=1, \ldots, d_{M}$; we say that $d_{M}$ is the order of $M$.

We denote the symmetric difference of two sets $S$ and $T$ by $S \Delta T$. It is well-known that the symmetric difference of two cuts is a cut.

We denote by $K_{n}$ the complete graph on $n$ vertices, that is, the simple graph with $n$ vertices and one edge between each pair of distinct vertices.

A graph is series-parallel if its 2-connected components can be constructed from an edge by repeatedly adding edges parallel to an existing one, and subdividing edges, that is, replacing an edge by a path of length two. Duffin (1965) showed that series-parallel graphs are those having no $K_{4}$-minor. By construction, simple nontrivial 2-connected series-parallel graphs have at least one vertex of degree 2 .

Proposition 1. For a simple nontrivial 2-connected series-parallel graph, at least one of the following holds:
(a) two vertices of degree 2 are adjacent,
(b) a vertex of degree 2 belongs to a circuit of length 3,
(c) two vertices of degree 2 belong to a same circuit of length 4 .
Proof. We proceed by induction on the number of edges. The base case is $K_{3}$ for which (a) holds.

Let $G$ be a simple 2-connected series-parallel graph such that for every simple, 2-connected series-parallel graph with fewer edges at least one among (a), (b), and (c) holds. Since $G$ is simple, it can be built from a graph $H$ by subdividing an edge $e$ into a path $f, g$. Let $v$ be the vertex of degree 2 added with this operation. By the induction hypothesis, either $H$ is not simple, or one among (a), (b), and (c) holds for $H$.

Let first suppose that $H$ is not simple, then, by $G$ being simple, $e$ is parallel to exactly one edge $e_{0}$. Hence, $e_{0}, f, g$ is a circuit of $G$ length 3 containing $v$, hence (b) holds for $G$.

From now on, suppose that $H$ is simple. If (a) holds for $H$, then it holds for $G$.

Suppose that (b) holds for $H$, that is, in $H$ there exists a circuit $C$ of length 3 containing a vertex $w$ of degree 2 . Without loss of generality, we suppose that

$e \in C$, as otherwise (b) holds for $G$. By subdividing $e$, we obtain a circuit of length 4 containing $v$ and $w$, and hence (c) holds for $G$.

At last, suppose that (c) holds for $H$, that is, $H$ has a circuit $C$ of length 4 containing two vertices of degree 2. Without loss of generality, we suppose that $e \in C$, as otherwise (c) holds for $G$. By subdividing $e$, we obtain a circuit of length 5 containing three vertices of degree 2 . Then, at least two of them are adjacent, and so (a) holds for $G$.

## 数学代写|组合优化代写Combinatorial optimization代考|Box-Total Dual Integrality

Let $A \in \mathbb{R}^{m \times n}$ be a full row rank matrix. This matrix is equimodular if all its $m \times m$ non-zero determinants have the same absolute value. The matrix $A$ is face-defining for a face $F$ of a polyhedron $P \subseteq \mathbb{R}^{n}$ if aff $(F)=\left{x \in \mathbb{R}^{n}: A x=b\right}$ for some $b \in \mathbb{R}^{m}$. Such matrices are the face-defining matrices of $P$.

Theorem 1 (Chervet et al. (2020)). Let P be a polyhedron, then the following statements are equivalent:
(i) $P$ is box-TDI.
(ii) Every face-defining matrix of $P$ is equimodular.
(iii) Every face of $P$ has an equimodular face-defining matrix.
The equivalence of conditions (ii) and (iii) stems from the following observation.
Observation 1 (Chervet et al. (2020)). Let $F$ be a face of a polyhedron. If a face-defining matrix of $F$ is equimodular, then so are all face-defining matrices of $F$.

Observation 2. Let $A \in \mathbb{R}^{I \times J}$ be a full row rank matrix, $j \in J$, c be a column of $A$, and $\mathbf{v} \in \mathbb{R}^{I}$. If $A$ is equimodular, then so are:
(i) $[A \mathbf{c}]$,
(ii) $\left[\begin{array}{c}A \ \pm \chi^{j}\end{array}\right]$ if it is full row rank,
(iii) $\left[\begin{array}{cc}A & \mathbf{v} \ \mathbf{0}^{\top} & \pm 1\end{array}\right]$
and (iv) $\left[\begin{array}{cc}A & 0 \ \pm \chi^{j} & \pm 1\end{array}\right]$

Observation 3 (Chervet et al. (2020)). Let $P \subseteq \mathbb{R}^{n}$ be a polyhedron and let $F={x \in P: B x=b}$ be a face of $P$. If $B$ has full row rank and $n-\operatorname{dim}(F)$ rows, then $B$ is face-defining for $F$.

## 数学代写|组合优化代写Combinatorial optimization代考|Box-TDIness in Series-Parallel Graphs

70 年代后期引入的完全对偶积分系统与组合优化中的最小-最大关系密切相关（Schrijver，1998 年）。线性不等式的合理系统一种X≥b如果线性规划对偶中的最大化问题是完全对偶积分 (TDI)：
\min \left{c^{\top} x: A x \geq b\right}=\max \left{b^{\top} y: A^{\top} y=c, y \geq \mathbf {0}\右}\min \left{c^{\top} x: A x \geq b\right}=\max \left{b^{\top} y: A^{\top} y=c, y \geq \mathbf {0}\右}

## 数学代写|组合优化代写Combinatorial optimization代考|Definitions and Preliminary Results

（a）两个 2 次顶点相邻，
（b）一个 2 次顶点属于长度为 3 的回路，
(c) 两个度数为 2 的顶点属于长度为 4 的同一回路。

## 数学代写|组合优化代写Combinatorial optimization代考|Box-Total Dual Integrality

(i)磷是box-TDI。
(ii) 每个人脸定义矩阵磷是等模的。
(iii) 每一张脸磷具有等模人脸定义矩阵。

(i)[一种C],
(ii)[一种 ±χj]如果它是全行排名，
（iii）[一种在 0⊤±1]
(iv)[一种0 ±χj±1]

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