Abstract
Uniquesink 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 ddimensional 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.
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Acknowledgements
We would like to thank Bernd Gärtner for introducing the problem at the 2015 Gremo Workshop on Open Problems (GWOP), June 1–5, 2015, Feldis (GR), Switzerland. This work was supported in part by the National Natural Science Foundation of China Grants No. 61433014, 61832003, 61761136014, 61872334, 61502449, the 973 Program of China Grant No. 2016YFB1000201.
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The problems studied in this paper have been considered before by a subset of the authors in [2].
A Proof of the Lower Bound of Proposition 5
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 \(s' :=(2k  1, (k1)\mathbf {1}_{d1})\) and the sink is in \(t' :=(k, \mathbf {0}_{d1})\). 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 uniquesink property and (ii) the Holt–Klee property.
Ad (i). Our construction is actually a semidirect product \(G = G' \times (G''_x)_{x \in V(G)}\), where \(G'\) is an \((d  1)\)dimensional uniform (hence uniquesink) orientation, and \(G''_x\) is the onedimensional 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 onedimensional, this implies the uniquesink 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 vertexdisjoint paths from the source to the sink. The source of our grid is
whereas a look at Fig. 10 reveals that the sink is either of the two vertices
depending on the choice of orientation for the type3 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 vertexdisjoint s–tpaths. 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 vertexdisjoint s–tpaths 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–tpaths \(P_0, \ldots , P_{k1}\). 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 vertexdisjoint. Furthermore they are internally vertexdisjoint 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
this proves the theorem.
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 setup 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 pseudohyperplanes instead.
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Barba, L., Milatz, M., Nummenpalo, J. et al. The Complexity of Optimization on Grids. Algorithmica 81, 3494–3518 (2019). https://doi.org/10.1007/s00453019005874
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Keywords
 Uniquesink orientation
 Optimization
 Linear programming