About the Complexity of Two-Stage Stochastic IPs

Abstract

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form of multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $\max \{ c^T x \mid \mathcal {A} x = b, l \le x \le u, x \in \mathbb {Z}^{s + nt} \}$ where the constraint matrix $\mathcal {A} \in \mathbb {Z}^{r n \times s +nt}$ consists roughly of n repetitions of a block matrix $A \in \mathbb {Z}^{r \times s}$ on the vertical line and n repetitions of a matrix $B \in \mathbb {Z}^{r \times t}$ on the diagonal.

In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [16] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $\varDelta$, where $\varDelta$ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about the intersection of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $f(r,s,\varDelta ) \cdot poly(n,t)$, where f is a doubly exponential function.

This work was mostly done during the authors time at EPFL. The project was supported by the Swiss National Science Foundation (SNSF) within the project Convexity, geometry of numbers, and the complexity of integer programming (Nr.163071).

A full version of the paper is available at https://arxiv.org/abs/1901.01135.

Continuation of the Proof of Lemma 2

In the proof we need the notion of a cone which is simply the relaxation of an integer cone. For a generating set $B \subset \mathbb {Z}_{\ge 0}^d$, a cone is defined by

$$\begin{aligned} cone(B) = \{ \sum _{b \in B} \lambda _b b \mid \lambda \in \mathbb {R}_{\ge 0}^B \}. \end{aligned}$$

Proof

Claim 1: If for all i we have $\left\| x^{(i)}\right\| _1 > d \cdot O((d \varDelta )^{d (\ell -1)})$ then there exist non-zero vectors $y^{(1)}, \ldots , y^{(\ell )} \in \mathbb {Z}_{\ge 0}^d$ with $y^{(1)} \le x^{(1)}, \ldots , y^{(\ell )} \le x^{(\ell )}$ and $\left\| y^{(i)}\right\| _1 \le d \cdot O((d \varDelta )^{d (\ell -1)})$ such that $B^{(1)}y^{(1)} = \ldots = B^{(\ell )} y^{(\ell )}$.

Note that all basic feasible solutions $x^{(i)} \in \mathbb {R}^{d}_{\ge 0}$ have to be of similar size. Since $B x^{(i)} = b$ holds for all $1 \le i \le \ell$ we know that $\left\| x^{(i)}\right\| _1$ and $\left\| x^{(j)}\right\| _1$ can only differ by a factor of $d \varDelta$ for all $1 \le i,j \le \ell$. Hence all basic feasible solutions $x^{(i)}$ have to be either small or all have to be large. This claim considers the case that the size of all $x^{(i)}$ is large.

Proof of the Claim: Note that $B^{(i)} x^{(i)} = b$ and hence $b \in cone(B^{(i)})$. In the following, our goal is to find a non-zero point $q \in \mathbb {Z}_{\ge 0}^d$ such that $q = B^{(1)} y^{(1)} = \ldots = B^{(\ell )} y^{(\ell )}$ for some vectors $y^{(1)}, \ldots , y^{(\ell )} \in \mathbb {Z}_{\ge 0}^d$. However, this means that q has to be in the integer cone $int.cone(B^{(i)})$ for every $1 \le i \le \ell$ and therefore in the intersection of all the integer cones, i.e. $q \in \bigcap _{i=1}^n int.cone(B^{(i)})$. By Lemma 4 there exists a set of generating elements $\hat{B}$ such that

• $int.cone(\hat{B}) = \bigcap _{i=1}^n int.cone(B^{(i)})$ and $int.cone(\hat{B}) \ne \{ 0 \}$ as $b \in cone(\hat{B})$ and

• each generating vector $p \in \hat{B}$ can be represented by $p = B^{(i)} \lambda$ for some $\lambda \in \mathbb {Z}_{\ge 0}^d$ with $\left\| \lambda \right\| _1 \le O((d \varDelta )^{d (\ell -1)})$ for each basis $B^{(i)}$.

As $b \in cone(\hat{B})$ there exists a vector $\hat{x} \in \mathbb {R}_{\ge 0}^{\hat{B}}$ with $\hat{B} \hat{x} = b$. Our goal is to show that there exists a non-zero vector $q \in \hat{B}$ with $\hat{x}_q \ge 1$. In this case b can be simply written by $b = q + q'$ for some $q' \in cone(\hat{B})$. As q and $q'$ are contained in the intersection of all cones, there exists for each generating set $B^{(j)}$ a vectors $y^{(j)} \in \mathbb {Z}_{\ge 0}^{B^{(j)}}$ and $z^{(j)} \in \mathbb {R}_{\ge 0}^{B^{(j)}}$ such that $B^{(j)} y^{(j)} = q$ and $B^{(j)} z^{(j)} = q'$. Hence $x^{(j)} = y^{(j)} + z^{(j)}$ and we finally obtain that $x^{(j)} \ge y^{(j)}$ for $y^{(j)} \in \mathbb {Z}_{\ge 0}^{B^{(j)}}$ which shows the claim.

Therefore it only remains to prove the existence of the point q with $\hat{x}_q \ge 1$. By Lemma 4, each vector $p \in \hat{B}$ can be represented, by $p = B^{(i)} x^{(p)}$ for some $x^{(p)} \in \mathbb {Z}_{\ge 0}^{B^{(i)}}$ with $\left\| x^{(p)}\right\| _1 \le O((d \varDelta )^{d (\ell -1)})$ for every basis $B^{(i)}$.

As $B^{(i)} x^{(i)} = b = \sum _{p \in \hat{B}} \hat{x}_p p = \sum _{p \in \hat{B}} \hat{x}_p (B^{(i)} x^{(p)})$, every $x^{(i)}$ can be written by $x^{(i)} = \sum _{p \in \hat{B}} x^{(p)} \hat{x}_p$ and we obtain a bound on $\left\| x^{(i)}\right\| _1$ assuming that every for every $p \in \hat{B}$ we have $\hat{x}_p < 1$.

$$\begin{aligned} \left\| x^{(i)}\right\| _1 \le \sum _{p \in \hat{B}} \left\| x^{(p)} \hat{x}_p\right\| _1 {\mathop {<}\limits ^{\hat{x}_p < 1}} \sum _{p \in \hat{B}} \left\| x^{(p)}\right\| _1 \le d \cdot O((d \varDelta )^{d (\ell -1)}). \end{aligned}$$

The last inequality follows as we can assume by Caratheodory’s theorem [28] that the number of non-zero components of $\hat{x}$ is less or equal than d. Hence if $\left\| x^{(i)}\right\| _1 \ge d \cdot O((d \varDelta )^{d (\ell -1)})$ then there has to exist a vector $q \in \hat{B}$ with $x_q \ge 1$ which proves the claim.

Claim 2: For every vector $\lambda ^{(i)} \in \mathbb {Z}_{\ge 0}^P$ with $\sum _{p \in P} \lambda _p p = b$ there exists a basic feasible solution $x^{(k)}$ of LP (2) with basis $B^{(k)}$ such that $\frac{1}{\ell }x^{(k)} \le \lambda ^{(i)}$ in the sense that $\frac{1}{\ell } x^{(k)}_p \le \lambda ^{(i)}_p$ for every $p \in B^{(k)}$.

Proof of the Claim: The proof of the claim can be easily seen as each multiplicity vector $\lambda ^{(i)}$ is also a solution of the linear program (2). By standard LP theory, we know that each solution of the LP is a convex combination of the basic feasible solutions $x^{(1)}, \ldots , x^{(\ell )}$. Hence, each multiplicity vector $\lambda ^{(i)}$ can be written as a convex combination of $x^{(1)}, \ldots , x^{(\ell )}$, i.e. for each $\lambda ^{(i)}$, there exists a $t \in \mathbb {R}_{\ge 0}^\ell$ with $\left\| t\right\| _1 = 1$ such that $\lambda ^{(i)} = \sum _{j=1}^\ell t_j \bar{x}^{(j)}$, where $\bar{x}^{(j)}_p~=~{\left\{ \begin{array}{ll} x^{(j)}_p &{}\text { if } p \in B^{(j)} \\ 0 &{} \text { otherwise}\end{array}\right. }$. By the pigeonhole principle, there exists for each multiplicity vector $\lambda ^{(i)}$ an index k with $t_k \ge \frac{1}{\ell }$ which proves the claim.

Using the above two claims, we can now prove the claim of the lemma by showing that for each $\lambda ^{(i)}$, there exist a vector $y^{(i)} \le \lambda ^{(i)}$ with bounded 1-norm such that $\sum _{p \in P} y^{(1)}_p p = \ldots = \sum _{p \in P} y^{(n)}_p p$.

First, consider the case that there exists a basic feasible solution $x^{(j)}$ of LP 2 with $\left\| x^{(j)}\right\| _1 \le \ell d \cdot O((d \varDelta )^{d (\ell -1)})$. In this case we have for all $1 \le i \le n$ that $\left\| \lambda ^{(i)}\right\| _1 \le \ell d^2 \varDelta \cdot O((d \varDelta )^{d (\ell -1)})$ as the size of solutions of LP (2) can not differ by a factor of more than $d \varDelta$ (this is because for every $p,p' \in P$ the sizes $\left\| p\right\| _1, \left\| p'\right\| _1$ can not differ by a factor of more than $d \varDelta$).

Now, assume that for all basic feasible solutions $x^{(i)}$ we have $\left\| x^{(i)}\right\| _1 > \ell d \cdot O((d \varDelta )^{d (\ell -1)})$. We can argue by Claim 2 that for each $\lambda ^{(i)}$ (with $1 \le i \le n$) we find one of the basic feasible solutions $x^{(k)}$ ($1 \le k \le \ell$) with $\frac{1}{\ell } x^{(k)} \le \lambda ^{(i)}$. As $\frac{1}{\ell } x^{(i)} \ge d \cdot O((d \varDelta )^{d (\ell -1)})$ for all $1 \le i \le \ell$, we can apply the first claim to vectors $\frac{1}{\ell } x^{(1)}, \ldots , \frac{1}{\ell } x^{(\ell )}$ with $\frac{1}{\ell }b = \frac{1}{\ell } Bx^{(1)} = \ldots = \frac{1}{\ell } B x^{(\ell )}$, we obtain vectors $y^{(1)} \le \frac{1}{\ell }x^{(1)}, \ldots , y^{(\ell )} \le \frac{1}{\ell }x^{(\ell )}$ with $By^{(1)} = \ldots = B y^{(\ell )}$. Hence, we find for each $\lambda ^{(i)}$ a vector $y^{(k)} \in \mathbb {Z}_{\ge 0}^{B^{(k)}}$ with $y^{(k)} \le \lambda ^{(i)}$.

Finally we obtain that

$$\begin{aligned} \left\| y^{(j)}\right\| _1 \le d^2 \varDelta \ell \cdot O((d \varDelta )^{d (\ell -1)}) = (d \varDelta )^{O(d (\varDelta ^{d^2}))} \end{aligned}$$

using that $\ell$ is bounded by $\left( {\begin{array}{c}|P|\\ d\end{array}}\right) \le |P|^d$ and $|P| \le \varDelta ^d$.