Geometric random edge

Abstract

We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs $\max \{c^Tx :x \in \mathbb R^n, \, Ax\leqslant b\}$, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by $n-1$ other rows of A is at least $\delta$. This property is a geometric generalization of A being integral and each sub-determinant of A being bounded by $\Delta$ in absolute value. In this case $\delta \geqslant 1/(\Delta ^2 n)$. In particular, linear programs defined by totally unimodular matrices are captured in this framework. Here $\delta \geqslant 1/ n$ and Dyer and Frieze previously described a strongly polynomial-time randomized simplex algorithm for linear programs with A totally unimodular. The expected number of pivots of the simplex algorithm is polynomial in the dimension and $1/\delta$ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al., using the ideas of Dyer and Frieze.

Appendix

Proof of Lemma 2

We denote the normal cones of B and $B'$ by

$$\begin{aligned} {C} = \left\{ \sum _{i\in B} \lambda _i a_i:\lambda _i\geqslant 0 \right\} \quad \text {and} \quad {C}' = \left\{ \sum _{j \in B'} \mu _j a_j :\mu _j \geqslant 0\right\} . \end{aligned}$$

By a gift-wrapping technique, we construct a hyperplane $(h^T x = 0)$, $h \in \mathbb R^n {\setminus } \{0\}$ such that the following conditions hold.

• (i) The hyperplane separates the interiors of C and $C'$.

• (ii) The row $a_k$ does not lie on the hyperplane.

• (iii) The hyperplane is spanned by $n-1$ rows of A.

Once we construct $(h^Tx = 0)$ we can argue as follows. The distance of $\mu _k \cdot a_k$ to the hyperplane $(h^Tx = 0)$ is at least $\mu _k\cdot \delta$. Since $c'$ is the sum of $\mu _k \cdot a_k$ and a vector that is on the same side of the hyperplane as $a_k$ it follows that the distance of $c'$ to this hyperplane is also at least $\mu _k \cdot \delta$. Since c lies on the opposite side of the hyperplane, the distance of c and $c'$ is at least $\mu _k \cdot \delta$.

We start with a hyperplane $(h^Tx = 0)$ strictly separating the interiors of C and $C'$. The conditions (i, ii) are satisfied. Suppose that (iii) is not satisfied and let $\ell < n-1$ be the maximum number of linearly independent rows of A that are contained in $(h^Tx = 0)$.

We tilt the hyperplane by moving its normal vector h along a chosen equator of the ball of radius $\Vert h\Vert$ to augment this number. Since $\ell < n-1$ there exists an equator leaving the rows of A that are in contained in $(h^T = 0)$ invariant under each rotation.

However, as soon as the hyperplane contains a new row of A we stop. If this new row of A is not $a_k$ then still, conditions (i, ii) hold and the hyperplane now contains $\ell +1$ linearly independent rows of A.

If this new row is $a_k$, then we redo the tilting operation but this time by moving h in the opposite direction on the chosen equator. Since there are n linearly independent rows of A without the row $a_k$ this tilting will stop at a new row of A which is not $a_k$ and we end the first tilting operation.

This tilting operation has to be repeated at most $n-1 - |B \cap B'|$ times to obtain the desired hyperplane. $\square$

1.1 Phase 1

We now describe an approach to determine an initial basic feasible solution or to assert that the linear program (1) is infeasible. Furthermore, we justify the assumption that the set of feasible solutions is a bounded polytope. This phase 1 is different from the usual textbook method since the linear programs that we need to solve have to comply with the $\delta$ -distance property.

To find an initial basic feasible solution, we start by identifying n linearly independent linear inequalities $\widetilde{a}_1^Tx \leqslant \widetilde{b}_1, \dots ,\widetilde{a}_n^Tx \leqslant \widetilde{b}_n$ of $Ax \leqslant b$. Then we determine a ball that contains all feasible solutions. This standard technique is for example described in [13]. Using the normal-vectors $\widetilde{a}_1,\dots ,\widetilde{a}_n$ we next determine values $\beta _i, \gamma _i \in \mathbb R$, $i=1,\dots ,n$ such that this ball is contained in the parallelepiped $Z = \{ x \in \mathbb R^n :\beta _i \leqslant \widetilde{a}_i^Tx \leqslant \gamma _i, \, i=1,\dots ,n\}$. We start with a basic feasible solution $x_0^*$ of this parallelepiped and then proceed in m iterations. In iteration i, we determine a basic feasible solution $x^*_i$ of the polytope

$$\begin{aligned} P_i = Z \cap \left\{ x \in \mathbb R^n :a_j^Tx \leqslant b_j, \,1 \leqslant j \leqslant i \right\} \end{aligned}$$

using the basic feasible solution $x^*_{i-1}$ from the previous iteration by solving the linear program

$$\begin{aligned} \min \left\{ a_i^Tx :x \in P_{i-1} \right\} . \end{aligned}$$

If the optimum value of this linear program is larger than $b_i$, we assert that the linear program (1) is infeasible. Otherwise $x^*_i$ is the basic feasible solution from this iteration.

Finally, we justify the assumption that $P = \{x \in \mathbb R^n :Ax \leqslant b\}$ is bounded as follows. Instead of solving the linear program (1), we solve the linear program $\max \{ c^Tx :x \in P \cap Z\}$ with the initial basic feasible solution $x^*_m$. If the optimum solution is not feasible for (1) then we assert that (1) is unbounded.