Construction

Consider an instance I = (k, n,{S_i,j}1≤i,j≤k) of Grid Tiling. We now build an instance (G,T) of Steiner Orientationas follows (refer to Fig.1).

1 Or even an FPT algorithm.

2 Thek-Cliqueproblem asks whether there is a clique of size≥k.

68 R. Chitnis and A. E. Feldmann

Fig. 1. The instance of Steiner Orientation created from an instance of Grid Tiling (before the splitting operation). At this point, the only undirected edges are the green edges. For clarity, we do not show the (directed) perfect matching (which we denote by yellow edges) given byd^j_i →a^j_i andh^j_i →e^j_i for each i∈[k], j ∈[n]. The gadgetG_i,j is highlighted by a dotted rectangle. (Color figure online)

– We first fix the Origin as marked in black³.

– The “horizontal right” direction is viewed as the positive X axis and the

“vertical upward” is viewed as the positiveY axis.

– Black Grid Edges:For each 1≤i, j≤kwe introduce then×ngridG_i,j to correspond to the setS_i,jof theGrid Tilinginstance. In Fig.1we highlight the gadgetG_i,j by a dotted rectangle.

• The bottom left vertex of gadgetG_i,j is denoted byv_i,j^1,1.

3 This is the unique vertex which has incoming edge fromb¹₁ and an outgoing edge tog¹1.

A Tight Lower Bound for Steiner Orientation 69

• Each row of G_i,j is horizontal and the number of the row increases as we go vertically upwards. Similarly, each column of G_i,j is vertical and the number of the column increases as we go horizontally rightwards. For each 1≤h, ≤n the unique vertex which is the intersection of theh^th column and ^throw is denoted byv_i,j^h,.

Orient each horizontal edge of the grid G_i,j to the right, and each vertical edge to the bottom.

– We now define four special sets of vertices for the gadgetG_i,j given by

• Left(G_i,j) ={v^1,_i,j : ∈[n]}

• Right(G_i,j) ={v^n,_i,j : ∈[n]}

• Top(G_i,j) ={v^,n_i,j : ∈[n]}

• Bottom(G_i,j) ={v_i,j^,1 : ∈[n]}

– Horizontal Orange Inter-Grid Edges:For each 1≤i≤k−1,1≤j≤k

• Add the directed perfect matching from vertices of Right(G_i,j) to Left(G_i+1,j) given by the set of edges{v^n,_i,j →v_i+1,j^1, : ∈[n]}.

– VerticalOrange Inter-Grid Edges: For each 2≤j≤k,1≤i≤k

• Add the directed perfect matching from vertices of Bottom(G_i,j) to Top(G_i,j−1) given by the set of edges{v_i,j^,1→v_i,j−1^,n : ∈[n]}.

– We introduce 8k·nredvertices given by

• A:={a^j_i |i∈[k], j∈[n]}

• B:={b^j_i |i∈[k], j∈[n]}

• C:={c^j_i |i∈[k], j∈[n]}

• D:={d^j_i |i∈[k], j∈[n]}

• E:={e^j_i |i∈[k], j∈[n]}

• F :={f_i^j |i∈[k], j∈[n]}

• G:={g^j_i | i∈[k], j∈[n]}

• H :={h^j_i |i∈[k], j∈[n]} – Blue Edges:

• For each i∈[k], j∈[n]

∗add the directed edgeh^j_i →g_i^j,

∗add the directed edgev_i,^j,1₁ →g^j_i,

∗add the directed edgef_i^j →e^j_i,

∗add the directed edgef_i^j →v^j,n_i,k.

• For each i∈[k], j∈[n]

∗add the directed edged^j_i →c^j_i,

∗add the directed edgev_k,i^n,j→c^j_i,

∗add the directed edgeb^j_i →a^j_i,

∗add the directed edgeb^j_i →v₁¹_,i^,j.

– Yellow Edges(these are left out in Fig.1): For eachi∈[k], j∈[n]

• Category I: add the directed edged^j_i →a^j_i,

• Category II: add the directed edgeh^j_i →e^j_i. – GreenEdges:For each i∈[k]

70 R. Chitnis and A. E. Feldmann

• add the undirected patha¹_i −a²_i −a³_i −. . . .−aⁿ⁻¹_i −aⁿ_i, and denote this path⁴ byA_i,

• add the undirected pathb¹_i−b²_i−b³_i−. . . .−bⁿ⁻¹_i −bⁿ_i, and denote this path byB_i,

• add the undirected pathc¹_i−c²_i−c³_i−. . . .−cⁿ⁻_i ¹−cⁿ_i, and denote this path byC_i,

• add the undirected path d¹_i −d²_i −d³_i −. . . .−dⁿ⁻_i ¹−dⁿ_i, and denote this path byD_i,

• add the undirected pathe¹_i−e²_i−e³_i−. . . .−eⁿ⁻_i ¹−eⁿ_i, and denote this path byE_i,

• add the undirected pathf_i¹−f_i²−f_i³−. . . .−f_iⁿ⁻¹−f_iⁿ, and denote this path byF_i,

• add the undirected path g¹_i −g²_i −g_i³−. . . .−gⁿ⁻¹_i −g_iⁿ, and denote this path byG_i,

• add the undirected pathh¹_i −h²_i −h³_i −. . . .−hⁿ⁻¹_i −hⁿ_i, and denote this path byH_i.

– For each 1 ≤ i, j ≤ k and each 1 ≤ x, y ≤ n we perform the following operation on the vertexv_i,j^x,y:

• If (x, y)∈S_i,j then we keep the vertexv_i,j^x,y as is.

• Otherwise wesplit the vertex v_i,j^x,y into two vertices v^x,y_i,j,LB and v^x,y_i,j,TR. Note thatv_i,j^x,y had 4 incident edges: two incoming (one each from the left and the top) and two outgoing (one each to the right and the bottom).

We change the edges as follows (see Fig.2):

∗Make the left incoming edge and bottom outgoing edge incident on v^s,t_i,j,LB(denoted by red color in Fig.2).

∗ Make the top incoming edge and right outgoing edge incident on v^s,t_i,j,TR (denoted by blue color in Fig.2).

∗ Add an undirectededge between v^s,t_i,j,LB and v_i,j,TR^s,t (denoted by the dotted edge in Fig.2).

– The setT of terminal pairs are given by

• Type I:(bⁿ_j, a¹_j),(b¹_j, aⁿ_j),(dⁿ_j, c¹_j) and (d¹_j, cⁿ_j) for each j∈[k]

• Type II:(f_jⁿ, e¹_j),(f_j¹, eⁿ_j),(hⁿ_j, g¹_j) and (h¹_j, gⁿ_j) for eachj∈[k]

• Type III:(dⁿ_j, a¹_j) and (d¹_j, aⁿ_j) for eachj ∈[k]

• Type IV:(hⁿ_j, e¹_j) and (h¹_j, eⁿ_j) for eachj∈[k]

• Type V:(b¹_j, cⁿ_j) and (bⁿ_j, c¹_j) for eachj ∈[k]

• Type VI:(f_j¹, gⁿ_j) and (f_jⁿ, g¹_j) for eachj∈[k]

Note that the total number of terminal pairs is 16k.

Remark 1. Note that the graph Gconstructed above can be drawn on the sur- face of a torus without any two edges crossing: removing the yellow edges, the graph is clearly planar (as depicted in Fig.1) and can be drawn on a square polygon. Identifying the right edge R and left edge L of the square such that the lower left corner equals the lower right corner, and also the square’s topT

4 Sometimes we also abuse notation slightly and useA_ito denote this set of vertices.

A Tight Lower Bound for Steiner Orientation 71

Fig. 2.The splitting operation for vertexv_i,j^h,when (h, )∈/S_i,j. The idea behind this splitting is that no matter which way we orient the undirected dotted edge we cannot go both from left to right and from top to bottom. However, if we just want to go from left to right (top to bottom) then it is possible by orienting the dotted edge to the right (left), respectively. (Color figure online)

and bottom B edges such that the upper left corner equals the lower left cor- ner, gives an orientable surface of genus 1 (i.e. a torus). The horizontal yellow edges of Category I can connect through L=R, and the vertical yellow edges of Category II can connect through T =B, without any edges crossing.

Remark 2. For simplicity, we add two “dummy” indices 1 and n which do not belong to any of the sets in theGrid Tilinginstance. Hence no vertices on the boundary of the gridsG_i,j (for any 1≤i, j≤k) are split.

Before proving the correctness of the reduction, we first introduce some nota- tion concerning orientations of the green edges (i.e. potential solutions) of the instance.

Definition 1. For anyi∈[n], a path onnverticesa1−a2−. . . .−a_n is said to be oriented towards (away from)i if every edgea_j−1−a_j is oriented towards (away from) a_j for every j ≤ i and every edge a_j −a_j+1 is oriented towards (away from) a_j for every j≥i, respectively.