### 数学代写|组合优化代写Combinatorial optimization代考|k-edge-connected Spanning Subgraph Polyhedron

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## 数学代写|组合优化代写Combinatorial optimization代考|k-edge-connected Spanning Subgraph Polyhedron

The dominant of a polyhedron $P$ is $\operatorname{dom}(P)={x: x=y+z$, for $y \in P$ and $z \geq \mathbf{0}}$. Note that $P_{k}(G)$ is the dominant of the convex hull of all $k$-edgeconnected spanning subgraphs of $G$ that have each edge taken at most $k$ times. Since the dominant of a polyhedron is a polyhedron, $P_{k}(G)$ is a polyhedron even though it is the convex hull of an infinite number of points.

From now on, $k \geq 2$. Didi Biha and Mahjoub (1996) gave a complete description of $P_{k}(G)$ for all $k$, when $G$ is series-parallel.

Theorem 2. Let $G$ be a series-parallel graph and $k$ be a positive integer. Then, when $k$ is even, $P_{k}(G)$ is described by:
(1) $\left{\begin{array}{l}x(D) \geq k \text { for all cuts } D \text { of } G, \ x \geq 0\end{array}\right.$
and when $k$ is odd, $P_{k}(G)$ is described by:
(2) $\left{\begin{array}{l}x(M) \geq \frac{k+1}{2} d_{M}-1 \text { for all multicuts } M \text { of } G, \ x \geq 0 .\end{array}\right.$
The incidence vector of a family $F$ of $E$ is the vector $\chi^{F}$ of $\mathbb{Z}^{E}$ such that $e$ ‘s coordinate is the multiplicity of $e$ in $F$ for all $e$ in $E$. Since there is a bijection between families and their incidence vectors, we will often use the same terminology for both. Since the incidence vector of a multicut $\delta\left(V_{1}, \ldots, V_{d_{M}}\right)$ is the half-sum of the incidence vectors of the bonds $\delta\left(V_{1}\right), \ldots, \delta\left(V_{d_{M}}\right)$, we can deduce an alternative description of $P_{2 h}(G)$.

Corollary 1. Let $G$ be a series-parallel graph and $k$ be a positive even integer. Then $P_{k}(G)$ is described by:
$$\left{\begin{array}{l} x(M) \geq \frac{k}{2} d_{M} \text { for all multicuts } M \text { of } G, \ x \geq \mathbf{0} . \end{array}\right.$$
We call constraints (2a) and (3a) partition constraints. A multicut $M$ is tight for a point of $P_{k}(G)$ if this point satisfies with equality the partition constraint (2a) (resp. (3a)) associated with $M$ when $k$ is odd (resp. even). Moreover, $M$ is tight for a face $F$ of $P_{k}(G)$ if it is tight for all the points of $F$.

The following results give some insight on the structure of tight multicuts.
Theorem 3 (Didi Biha and Mahjoub (1996)). Let $k>1$ be odd, let $x$ be a point of $P_{k}(G)$, and let $M=\delta\left(V_{1}, \ldots, V_{d_{M}}\right)$ be a multicut tight for $x$. Then, the following hold:
(i) if $d_{M} \geq 3$, then $x\left(\delta\left(V_{i}\right) \cap \delta\left(V_{j}\right)\right) \leq \frac{k+1}{2}$ for all $i \neq j \in\left{1, \ldots, d_{M}\right}$.
(ii) $G \backslash V_{i}$ is connected for all $i=1, \ldots, d_{M}$.
Observation 4. Let $M$ be a multicut of $G$ strictly containing $\delta(v)={f, g}$. If $M$ is tight for a point of $P_{k}(G)$, then both $M \backslash f$ and $M \backslash g$ are multicuts of $G$ of order $d_{M}-1$.

Chopra (1994) gave sufficient conditions for an inequality to be facet defining. The following proposition is a direct consequence of (Chopra 1994, Theorem 2.4).

## 数学代写|组合优化代写Combinatorial optimization代考|Box-TDIness of Pk

In this section we show that, for $k \geq 2, P_{k}(G)$ is a box-TDI polyhedron if and only if $G$ is series-parallel.

When $k \geq 2, P_{k}(G)$ is not box-TDI for all graphs as stated by the following lemma.

Lemma 1. For $k \geq 2$, if $G=(V, E)$ contains a $K_{4}$-minor, then $P_{k}(G)$ is not box-TDI.

Proof. When $k$ is odd, Proposition 2 shows that there exists a facet-defining inequality that is described by a non equimodular matrix. Thus, $P_{k}(G)$ is not box-TDI by Statement (ii) of Theorem 1 .

We now prove the case when $k$ is even. Since $G$ is connected and has a $K_{4^{-}}$ minor, there exists a partition $\left{V_{1}, \ldots, V_{4}\right}$ of $V$ such that $G\left[V_{i}\right]$ is connected and $\delta\left(V_{i}, V_{j}\right) \neq \emptyset$ for all $i<j \in{1, \ldots, 4}$. We prove that the matrix $T$ whose three rows are $\chi^{\delta\left(V_{i}\right)}$ for $i=1,2,3$ is a face-defining matrix for $P_{k}(G)$ which is not equimodular. This will end the proof by Statement (ii) of Theorem 1 .

Let $e_{i j}$ be an edge in $\delta\left(V_{i}, V_{j}\right)$ for all $i<j \in{1, \ldots, 4}$. The submatrix of $T$ formed by the columns associated with edges $e_{i j}$ is the following:

The matrix $T$ is not equimodular as the first three columns form a matrix of determinant $-2$ whereas the last three ones have determinant 1 .

To show that $T$ is face-defining, we exhibit $|E|-2$ affinely independent points of $P_{k}(G)$ satisfying the partition constraint (3a) associated with the multicut $\delta\left(V_{i}\right)$, that is, $x\left(\delta\left(V_{i}\right)\right)=k$, for $i=1,2,3$.

Let $D_{1}=\left{e_{12}, e_{14}, e_{23}, e_{34}\right}, D_{2}=\left{e_{12}, e_{13}, e_{24}, e_{34}\right}, D_{3}=\left{e_{13}, e_{14}, e_{23}, e_{24}\right}$ and $D_{4}=\left{e_{14}, e_{24}, e_{34}\right}$. First, we define the points $S_{j}=\sum_{i=1}^{4} k \chi^{E\left[V_{i}\right]}+\frac{k}{2} \chi^{D_{j}}$, for $j=1,2,3$, and $S_{4}=\sum_{i=1}^{4} k \chi^{E\left[V_{i}\right]}+k \chi^{D_{4}}$. Note that they are affinely independent.
Now, for each edge $e \notin\left{e_{12}, e_{13}, e_{14}, e_{23}, e_{24}, e_{34}\right}$, we construct the point $S_{c}$ as follows. When $e \in E\left[V_{i}\right]$ for some $i=1, \ldots, 4$, we define $S_{c}=S_{4}+\chi^{e}$. Adding the point $S_{e}$ maintains affine independence as $S_{e}$ is the only point not satisfying $x_{e}=k$. When $e \in \delta\left(V_{i}, V_{j}\right)$ for some $i, j$, we define $S_{e}=S_{\ell}-\chi^{e_{i j}}+\chi^{e}$, where $S_{\ell}$ is $S_{1}$ if $e \in \delta\left(V_{1}, V_{4}\right) \cup \delta\left(V_{2}, V_{3}\right)$ and $S_{2}$ otherwise. Affine independence comes because $S_{e}$ is the only point involving $e$.

## 数学代写|组合优化代写Combinatorial optimization代考|TDI Systems for Pk

Let $G$ be a series-parallel graph. In this section, we study the total dual integrality of systems describing $P_{k}(G)$. Due to length limitation, some of the proofs of the results below are omitted. They can be found in the appendix.

The following result characterizes series-parallel graphs in terms of TDIness of System (2).

Theorem 6. For $k>1$ odd and integer, System (2) is TDI if and only if $G$ is series-parallel.

Proof (sketch). We first prove that if $G$ is not series-parallel, then System (2) is not TDI. Indeed, every TDI system with integer right-hand side describes an integer polyhedron (Edmonds and Giles, 1977 ), but, when $G$ has a $K_{4}$-minor, System (2) describes a noninteger polyhedron (Chopra, 1994).

Let us sketch the other direction of the proof, that is, when the graph is series-parallel. We proceed by contradiction and consider a minimal counterexample $G$. First, we show that $G$ is simple and 2-connected. Then, we show that $G$ contains none of the following configurations.

Since the red vertices in the figure above have degree 2 in $G$, this contradicts Proposition $1 .$

For $k>1$, by Theorem $5, P_{k}(G)$ is a box-TDI polyhedron if and only if $G$ is series-parallel. Together with (Cook 1986, Corollary 2.5), this implies the following.

Corollary 2. For $k>1$ odd and integer, System (2) is box-TDI if and only if $G$ is series-parallel.The following theorem gives a TDI system for $P_{k}(G)$ when $G$ is series-parallel and $k$ is even.

## 数学代写|组合优化代写Combinatorial optimization代考|k-edge-connected Spanning Subgraph Polyhedron

(1) $\left{X(D)≥ķ 对于所有削减 D 的 G, X≥0\对。一种nd在H和nķ一世s这dd,P_{k}(G)一世sd和sCr一世b和db是:(2)\剩下{X(米)≥ķ+12d米−1 对于所有多切 米 的 G, X≥0.\对。吨H和一世nC一世d和nC和在和C吨这r这F一种F一种米一世l是F这F和一世s吨H和在和C吨这r\chi^{F}这F\mathbb{Z}^{E}s在CH吨H一种吨和‘sC这这rd一世n一种吨和一世s吨H和米在l吨一世pl一世C一世吨是这F和一世nFF这r一种ll和一世n和.小号一世nC和吨H和r和一世s一种b一世j和C吨一世这nb和吨在和和nF一种米一世l一世和s一种nd吨H和一世r一世nC一世d和nC和在和C吨这rs,在和在一世ll这F吨和n在s和吨H和s一种米和吨和r米一世n这l这G是F这rb这吨H.小号一世nC和吨H和一世nC一世d和nC和在和C吨这r这F一种米在l吨一世C在吨\delta\left(V_{1}, \ldots, V_{d_{M}}\right)一世s吨H和H一种lF−s在米这F吨H和一世nC一世d和nC和在和C吨这rs这F吨H和b这nds\delta\left(V_{1}\right), \ldots, \delta\left(V_{d_{M}}\right),在和C一种nd和d在C和一种n一种l吨和rn一种吨一世在和d和sCr一世p吨一世这n这FP_{2 h}(G)$。

$$\left{X(米)≥ķ2d米 对于所有多切 米 的 G, X≥0.\对。$$

（i）如果d米≥3， 然后X(d(在一世)∩d(在j))≤ķ+12对全部i \neq j \in\left{1, \ldots, d_{M}\right}i \neq j \in\left{1, \ldots, d_{M}\right}.
(二)G∖在一世为所有人连接一世=1,…,d米.

Chopra (1994) 给出了定义一个不等式的充分条件。以下命题是 (Chopra 1994, Theorem 2.4) 的直接结果。

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