The Complexity of Optimization on Grids

Abstract

Unique-sink orientations of grids are models for linear optimization problems. Vertices of the grid represent possible solutions to the optimization problem; edge orientations indicate improving directions. The computational goal is to find the unique sink, representing the optimal solution. We study the query complexity of this model, where we consider two natural types of queries, vertex queries and edge queries. We describe a deterministic algorithm showing that the vertex query complexity of d-dimensional grids is $O(n^{ \lceil d/2 \rceil })$. For the edge query complexity we obtain nearly the same bound, incurring only an $n^{o(1)}$ overhead. Our algorithms rely on structural results with potential further applications in optimization theory.

A Proof of the Lower Bound of Proposition 5

To prove the lower bound, we adapt Ziegler’s construction in § 7.4 of [3] to our setting. Here we will define the edge orientations directly; the reader may refer to the remark below to gain some intuition why we choose the edge orientations exactly in this way. Fix some number k and consider the grid G with $n_1 = 2k$ and $n_i = k$ for all $i = 2, 3, \ldots , d$. Here we let $[n_i] = \{ 0, \ldots , n_i - 1 \}$ by slightly abusing the notation, so that the coordinates start from zero. The grid consists of an “upper part”, containing the vertices whose first coordinate is in the range $0, \ldots , k - 1$, and a “lower part”, containing the vertices whose first coordinate is in the range $k, \ldots , 2k - 1$. We begin by directing the edges within the upper part, and the edges within the lower part, using in each case a uniform orientation as shown in Fig. 10. (Note that the figure interprets the coordinates in matrix order; that is, the first coordinate is the row index and the second coordinate is the column index.) Specifically, for the upper part we choose the uniform orientation with source equal to $s :=\mathbf {1}_d$ and sink equal to $t :=\mathbf {0}_d$; for the lower part, the source is in $s' :=(2k - 1, (k-1)\mathbf {1}_{d-1})$ and the sink is in $t' :=(k, \mathbf {0}_{d-1})$. It remains to direct the edges between the upper and the lower part, that is, the edges uv where u and v differ only in their first coordinate and furthermore $u_1 = \{0, \ldots , k - 1\}$, $v_1= \{k, \ldots , 2k - 1\}$. We use the following rule:

• (1) If $u_1 + \cdots + u_d > v_1 - k$, then we direct the edge from u to v.

• (2) If $u_1 + \cdots + u_d < v_1 - k$, then we direct the edge from v to u.

• (3) If $u_1 + \cdots + u_d = v_1 - k$, then we direct the edge arbitrarily (and we claim that we obtain an admissible orientation if we do so).

For the edge in (3) we always have two choices, and there are $\varTheta (k^d)$ edges of type (3). This shows that we have constructed $2^{\varTheta (k^d)}$ many different orientations of the grid G. Since G has grid size $n = (d + 1)k$, this shows that we have constructed as many distinct orientations as we claimed. It remains to argue that our orientations are really admissible, that is, that they satisfy (i) the unique-sink property and (ii) the Holt–Klee property.

Ad (i). Our construction is actually a semi-direct product $G = G' \times (G''_x)_{x \in V(G)}$, where $G'$ is an $(d - 1)$-dimensional uniform (hence unique-sink) orientation, and $G''_x$ is the one-dimensional orientation specified by the rules (1)–(3) above, $x = (u_2, \ldots , u_d)$. It is easy to see that every $G''_x$ is acyclic; since it is one-dimensional, this implies the unique-sink property for each $G''_x$ and, hence, for the entire grid G.

Ad (ii). Since the argument is essentially the same for every subgrid, we prove only that the entire grid G possesses $n - d$ internally vertex-disjoint paths from the source to the sink. The source of our grid is

$$\begin{aligned} s = (k - 1, k - 1, k - 1, \ldots , k - 1), \end{aligned}$$

whereas a look at Fig. 10 reveals that the sink is either of the two vertices

$$\begin{aligned} t = (0, 0, 0, \ldots , 0) \quad t' = (k, 0, 0, \ldots , 0), \end{aligned}$$

depending on the choice of orientation for the type-3 edge $tt'$. Since, again, the argument is very similar in both cases, let us assume without loss of generality that t is the sink of G. We need to construct a set $\mathcal {P}$ of $n - d$ internally vertex-disjoint s–t-paths. In the case that we are focusing on, both the sink and the source lie in the “upper part” of G, a uniformly oriented grid of size $n - k$; so there are clearly $n - k - d$ internally vertex-disjoint s–t-paths within the upper part. Let $\mathcal {P}'$ denote the set of these paths. The set $\mathcal {P}'$ contains already nearly all of the paths that we are looking for; we are left to find k further s–t-paths $P_0, \ldots , P_{k-1}$. The starting vertex of each $P_i$ is, of course, the source s. The next vertex on the path shall be $(i + k, k - 1, k - 1, \ldots , k - 1)$, which lies in the lower part. For the next few vertices we fix the first coordinate to be i, and we continue our path with a Hamming path towards $(i + k, 0, 0, \ldots , 0)$. Finally we go one additional step from $(i + k, 0, 0, \ldots , 0)$ to t. Clearly all these paths are internally vertex-disjoint. Furthermore they are internally vertex-disjoint with the paths from $\mathcal {P}'$, because the internal vertices of $\mathcal {P}'$ all lie in the upper part, whereas the internal vertices of $P_i$ lie in the lower part. Letting

$$\begin{aligned} \mathcal {P} :=\mathcal {P}' \cup \{P_0, \ldots , P_{k - 1}\}, \end{aligned}$$

this proves the theorem.

Fig. 10

The partially oriented grid G for the proof of Proposition 5, here for $d = 2$ and $k = 3$. The lower and the upper part each receive a uniform orientation as shown here. The orange edges are the edges of type (3) and may be directed freely. The edges of type (1) and (2) are not shown in the figure (Color figure online)

Remark 1

(Relation to Ziegler’s construction) Our construction of admissible grid orientations is closely related to Ziegler’s construction of oriented matroids [3, Theorem 7.4.2]. To highlight the connection we explain, in Fig. 11, how our grid orientations can be obtained from the topological set-up there. For the purpose of arguing through pictures we restrict ourselves here to the case $d = 2$, where our grid orientation can be obtained from an arrangement of pseudolines in the projective plane. (Due to a shift in terminologies, this would be the case $d = 3$ for people who work in oriented matroids.) For $d = 2$ we would be looking at an arrangement of n pseudo-hyperplanes instead.

Fig. 11

The arrangement of pseudolines underlying the grid orientation seen in Fig. 10. For each element of the value set $[n_1] \cup [n_2]$ there is one pseudoline. The pseudolines are divided into two groups: the (roughly) horizontal pseudolines 0, 1, 2, 3, 4, 5 corresponding to the elements of $[n_1]$, and the vertical pseudolines 0, 1, 2 corresponding to the elements of $[n_2]$. Every crossing of a horizontal pseudoline i with a vertical pseudoline j corresponds to vertex (i, j) of the grid, and the arrow labelled “OPT” informally prescribes the orientations of the edges. At every little detour made by the “horizontal” (actually, diagonal) pseudolines 0, 1, 2, there is a choice of passing over or under the nearby crossing, corresponding to the choice of orientation for the corresponding orange edge in Fig. 10