A Tight Lower Bound for Edge-Disjoint Paths on Planar DAGs
AA Tight Lower Bound for E
DGE -D ISJOINT P ATHS on Planar DAGs *Rajesh Chitnis
School of Computer Science, University of Birmingham, UK. [email protected]
Abstract
Given a graph G and a set T = (cid:8) ( s i , t i ) : 1 ≤ i ≤ k (cid:9) of k pairs, the V ERTEX -D ISJOINT P ATHS (resp.E
DGE -D ISJOINT P ATHS ) problems asks to determine whether there exist pairwise vertex-disjoint(resp. edge-disjoint) paths P , P , . . . , P k in G such that P i connects s i to t i for each 1 ≤ i ≤ k . Unliketheir undirected counterparts which are FPT (parameterized by k ) from Graph Minor theory, both theedge-disjoint and vertex-disjoint versions in directed graphs were shown by Fortune et al. (TCS ’80)to be NP-hard for k =
2. This strong hardness for D
ISJOINT P ATHS on general directed graphs led tothe study of parameterized complexity on special graph classes, e.g., when the underlying undirectedgraph is planar. For V
ERTEX -D ISJOINT P ATHS on planar directed graphs, Schrijver (SICOMP’94) designed an n O ( k ) time algorithm which was later improved upon by Cygan et al. (FOCS ’13)who designed an FPT algorithm running in 2 O ( k ) · n O ( ) time. To the best of our knowledge, theparameterized complexity of E DGE -D ISJOINT P ATHS on planar directed graphs is unknown.We resolve this gap by showing that E DGE -D ISJOINT P ATHS is W[1]-hard parameterizedby the number k of terminal pairs, even when the input graph is a planar directed acyclic graph(DAG). This answers a question of Slivkins (ESA ’03, SIDMA ’10). Moreover, under the ExponentialTime Hypothesis (ETH), we show that there is no f ( k ) · n o ( k ) algorithm for E DGE -D ISJOINT P ATHS on planar DAGs, where k is the number of terminal pairs, n is the number of vertices and f isany computable function. Our hardness holds even if both the maximum in-degree and maximumout-degree of the graph are at most 2.We now place our result in the context of previously known algorithms and hardness for E DGE -D ISJOINT P ATHS on special classes of directed graphs:•
Implications for E
DGE -D ISJOINT P ATHS on DAGs : Our result shows that the n O ( k ) al-gorithm of Fortune et al. (TCS ’80) for E DGE -D ISJOINT P ATHS on DAGs is asymptoticallytight, even if we add an extra restriction of planarity. The previous best lower bound (also underETH) for E
DGE -D ISJOINT P ATHS on DAGs was f ( k ) · n o ( k / log k ) by Amiri et al. (MFCS ’16,IPL ’19) which improved upon the f ( k ) · n o ( √ k ) lower bound implicit in Slivkins (ESA ’03,SIDMA ’10).• Implications for E
DGE -D ISJOINT P ATHS on planar directed graphs : As a special case ofour result, we obtain that E
DGE -D ISJOINT P ATHS on planar directed graphs is W[1]-hardparameterized by the number k of terminal pairs. This answers a question of Cygan et al.(FOCS ’13) and Schrijver (pp. 417-444, Building Bridges II, ’19), and completes the landscape(see Table 2) of the parameterized complexity status of edge and vertex versions of the D ISJOINT P ATHS problem on planar directed and planar undirected graphs.
The D
ISJOINT P ATHS problem is one of the most fundamental problems in graph theory: given a graphand a set of k terminal pairs, the question is to determine whether there exists a collection of k pairwise * A preliminary version of this paper appeared in CIAC 2021. A directed graph is planar if its underlying undirected graph is planar. a r X i v : . [ c s . D S ] J a n isjoint paths where each path connects one of the given terminal pairs. There are four natural variants ofthis problem depending on whether we consider undirected or directed graphs and the edge-disjoint orvertex-disjoint requirement. In undirected graphs, the edge-disjoint version is reducible to the vertex-disjoint version in polynomial time by considering the line graph. In directed graphs, the edge-disjointversion and vertex-disjoint version are known to be equivalent in terms of designing exact algorithms.Besides its theoretical importance, the D ISJOINT P ATHS problem has found applications in VLSI design,routing, etc. The interested reader is referred to the surveys [20] and [42, Chapter 9] for more details.The case when the number of terminal pairs k are bounded is of special interest: given a graph with n vertices and k terminal pairs the goal is to try to design either FPT algorithms, i.e., algorithms whoserunning time is f ( k ) · n O ( ) for some computable function f , or XP algorithms, i.e., algorithms whoserunning time is n g ( k ) for some computable function g . We now discuss some of the known results onexact algorithms for different variants of the D ISJOINT P ATHS problem before stating our result.
Prior work on exact algorithms for D
ISJOINT P ATHS on undirected graphs:
The NP-hardness for E
DGE -D ISJOINT P ATHS and V
ERTEX -D ISJOINT P ATHS on undirected graphs wasshown by Even et al. [16]. Solving the V
ERTEX -D ISJOINT P ATHS problem on undirected graphs isan important subroutine in checking whether a fixed graph H is a minor of a graph G . Hence, a corealgorithmic result of the seminal work of Robertson and Seymour was their FPT algorithm [40] forV ERTEX -D ISJOINT P ATHS (and hence also E
DGE -D ISJOINT P ATHS ) on general undirected graphs whichruns in O ( g ( k ) · n ) time for some function g . The cubic dependence on the input size was improvedto quadratic by Kawarabayashi et al. [28] who designed an algorithm running in O ( h ( k ) · n ) time forsome function h . Both the functions g and h are quite large (at least quintuple exponential as per [2]).This naturally led to the search for faster FPT algorithms on planar graphs: Adler et al. [2] designedan algorithm for V ERTEX -D ISJOINT P ATHS on planar graphs which runs in 2 O ( k ) · n O ( ) time. Veryrecently, this was improved to an single-exponential time FPT algorithm which runs in 2 O ( k ) · n O ( ) timeby Lokshtanov et al. [32].There are two more variants of the D ISJOINT P ATHS problem: the half-integral version where eachvertex/edge can belong to at most two paths, and the parity version where the length of each pathis required to respect a given parity (even or odd) condition. FPT algorithms are known for each ofthe following versions of V
ERTEX -D ISJOINT P ATHS on general undirected graphs: the half-integralversion [24, 31], the half-integral version with parity [25] and finally just the parity version (withouthalf-integral) [27].
Prior work on exact algorithms for D
ISJOINT P ATHS on directed graphs:
Unlike undirected graphs where both E
DGE -D ISJOINT P ATHS and V
ERTEX -D ISJOINT P ATHS are FPTparameterized by k , the D ISJOINT P ATHS problem becomes significantly harder for directed graphs:Fortune et al. [19] showed that both E
DGE -D ISJOINT P ATHS and V
ERTEX -D ISJOINT P ATHS on generaldirected graphs are NP-hard even for k =
2. For general directed graphs, Giannopoulou et al. [21] recentlydesigned an XP algorithm for the half-integral version of D
ISJOINT P ATHS : here the goal is to either finda set of k paths P , P , . . . , P k such that P i is an s i (cid:32) t i path for each i ∈ [ k ] and each vertex in the graphappears in at most two of the paths, or conclude that the given instance has no solution with pairwisedisjoint paths. This algorithm improves upon an older XP algorithm of Kawarabayashi et al. [26] for thequarter-integral case in general digraphs.The D ISJOINT P ATHS problem has also been extensively studied on special subclasses of digraphs:• D ISJOINT P ATHS on DAGs : It is easy to show that V
ERTEX -D ISJOINT P ATHS and E
DGE -D ISJOINT P ATHS are equivalent on the class of directed acyclic graphs (DAGs). Fortune et al. [19] This paper focuses on exact algorithms for the D
ISJOINT P ATHS problem so we do not discuss here the results regarding(in)approximability. n O ( k ) algorithm for E DGE -D ISJOINT P ATHS on DAGs. Slivkins [44] showed W[1]-hardness for E
DGE -D ISJOINT P ATHS on DAGs and a f ( k ) · n o ( √ k ) lower bound (for any computablefunction f ) under the Exponential Time Hypothesis [22, 23] (ETH) follows from that reduction.Amiri et al. [3] improved the lower bound to f ( k ) · n o ( k / log k ) thus showing that the algorithm ofFortune et al. [19] is almost-tight.• D ISJOINT P ATHS on directed planar graphs : Schrijver [41] designed an n O ( k ) algorithm forV ERTEX -D ISJOINT P ATHS on directed planar graphs. This was improved upon by Cygan et al. [12]who designed an FPT algorithm running in 2 O ( k ) · n O ( ) time. As pointed out by Cygan et al. [12],their FPT algorithm for V ERTEX -D ISJOINT P ATHS on directed planar graphs does not work forthe E
DGE -D ISJOINT P ATHS problem. The status of parameterized complexity (parameterized by k ) of E DGE -D ISJOINT P ATHS on directed planar graphs remained an open question. Table 1 givesa summary of known results for exact algorithms for D
ISJOINT P ATHS on (subclasses of) directedgraphs.
Graph class Problem type Algorithm Lower Bound
General graphs Vertex-disjoint = edge-disjoint ???? NP-hard for k = n O ( k ) [19] f ( k ) · n o ( √ k ) [44] f ( k ) · n o ( k / log k ) [3] f ( k ) · n o ( k ) [this paper] Planar graphs Vertex-disjoint n O ( k ) [41] ????2 O ( k ) · n O ( ) [12]Edge-disjoint ???? f ( k ) · n o ( k ) [this paper] Planar DAGs Vertex-disjoint 2 O ( k ) · n O ( ) [12] ????Edge-disjoint n O ( k ) [19] f ( k ) · n o ( k ) [this paper] Table 1:
The landscape of parameterized complexity results for D
ISJOINT P ATHS on directed graphs. All lowerbounds are under the Exponential Time Hypothesis (ETH). To the best of our knowledge, the entries marked with???? have no known non-trivial results.
Our result:
We resolve this open question by showing a slightly stronger result: the E
DGE -D ISJOINT P ATHS problemis W[1]-hard parameterized by k when the input graph is a planar DAG whose max in-degree and maxout-degree are both at most 2. First we define the E DGE -D ISJOINT P ATHS problem formally below, andthen state our result: E DGE -D ISJOINT P ATHS
Input : A directed graph G = ( V , E ) , and a set T ⊆ V × V of k terminal pairs given by (cid:8) ( s i , t i ) : 1 ≤ i ≤ k (cid:9) . Question : Do there exist k pairwise edge-disjoint paths P , P , . . . , P k such that P i is an s i (cid:32) t i pathfor each 1 ≤ i ≤ k ? Parameter : k Theorem 1.1.
The E DGE -D ISJOINT P ATHS problem on planar DAGs is W[1]-hard parameterized by thenumber k of terminal pairs. Moreover, under ETH, the E DGE -D ISJOINT P ATHS problem on planar DAGscannot be solved f ( k ) · n o ( k ) time where f is any computable function, n is the number of vertices and kis the number of terminal pairs. The hardness holds even if both the maximum in-degree and maximumout-degree of the graph are at most . We note that [3] considers a more general version than D
ISJOINT P ATHS which allows congestion n -variable m -clause 3-SAT cannot besolved in 2 o ( n ) · ( n + m ) O ( ) time [22, 23]. Prior to our result, only the NP-completeness of E DGE -D ISJOINT P ATHS on planar DAGs was known [45]. The reduction used in Theorem 1.1 is heavilyinspired by some known reductions: in particular, the planar DAG structure (Figure 2) is from [6, 7]and the splitting operation (Figure 3 and Definition 2.4) is from [4, 5]. We view the simplicity of ourreduction as evidence of success of the (now) established methodology of showing W[1]-hardness (andETH-based hardness) for planar graph problems using G
RID -T ILING and its variants.
Placing Theorem 1.1 in the context of prior work:
Theorem 1.1 answers a question of Slivkins [44] regarding the parameterized complexity of E
DGE -D ISJOINT P ATHS on planar DAGs. As a special case of Theorem 1.1, one obtains that E
DGE -D ISJOINT P ATHS on planar directed graphs is W[1]-hard parameterized by the number k of terminal pairs: thisanswers a question of Cygan et al. [12] and Schrijver [43]. The W[1]-hardness result of Theorem 1.1completes the landscape (see Table 2) of parameterized complexity of edge-disjoint and vertex-disjointversions of the D ISJOINT P ATHS problem on planar directed and planar undirected graphs. Theorem 1.1also shows that the n O ( k ) algorithm of Fortune et al. [19] for E DGE -D ISJOINT P ATHS on DAGs isasymptotically optimal, even if we add an extra restriction of planarity to the mix. Theorem 1.1 addsanother problem (E
DGE -D ISJOINT P ATHS on DAGs) to the relatively small list of problems for which itis provably known that the planar version has the same asymptotic complexity as the problem on generalgraphs: the only such other problems we are aware of are [5, 7, 38]. This is in contrast to the fact that forseveral problems [1, 14, 17, 18, 29, 30, 33, 34, 36, 38, 39]. the planar version is easier by (roughly) asquare root factor in the exponent as compared to general graphs, and there are lower bounds indicatingthat this improvement is essentially the best possible [35].
Graph class Problem type Parameterized Complexity parameterized by k Planar undirected Vertex-disjoint FPT [2, 28, 32, 40]Edge-disjointPlanar directed Vertex-disjoint FPT [12]Edge-disjoint W[1]-hard [this paper]
Table 2:
The landscape of parameterized complexity results for the four different versions (edge-disjoint vsvertex-disjoint & directed vs undirected) of D
ISJOINT P ATHS on planar graphs.
Organization of the paper:
In Section 2.1 we describe the construction of the instance ( G , T ) of E DGE -D ISJOINT P ATHS . The twodirections of the reduction are shown in Section 2.2 and Section 2.3 respectively. Finally, Section 2.4contains the proof of Theorem 1.1. We conclude with some open questions in Section 3.
Notation:
All graphs considered in this paper are directed and do not have self-loops or multiple edges. Weuse (mostly) standard graph theory notation [15]. The set { , , , . . . , M } is denoted by [ M ] for each M ∈ N . A directed edge (resp. path) from s to t is denoted by s → t (resp. s (cid:32) t ). We use the non-standard notation (to avoid having to consider different cases in our proofs): s (cid:32) s does not representa self-loop but rather is to be viewed as “just staying put” at the vertex s . If A , B ⊆ V ( G ) then wesay that there is an A (cid:32) B path if and only if there exists two vertices a ∈ A , b ∈ B such that thereis an a (cid:32) b path. For A ⊆ V ( G ) we define N + G ( A ) = (cid:8) x / ∈ A : ∃ y ∈ A such that ( y , x ) ∈ E ( G ) (cid:9) and N − G ( A ) = (cid:8) x / ∈ A : ∃ y ∈ A such that ( x , y ) ∈ E ( G ) (cid:9) . For A ⊆ V ( G ) we define G [ A ] to be the graphinduced on the vertex set A , i.e., G [ A ] : = ( A , E A ) where E A : = E ( G ) ∩ ( A × A ) .4 W[1]-hardness of E
DGE -D ISJOINT P ATHS on Planar DAGs
To obtain W[1]-hardness for E
DGE -D ISJOINT P ATHS on planar DAGs, we reduce from the G
RID -T ILING - ≤ problem [37] which is defined below: G RID -T ILING - ≤ Input : Integers k , N , and a collection S of k sets given by (cid:8) S x , y ⊆ [ N ] × [ N ] : 1 ≤ x , y ≤ k (cid:9) . Question : For each 1 ≤ x , y ≤ k does there exist a pair γ x , y ∈ S x , y such that• if γ x , y = ( a , b ) and γ x + , y = ( a (cid:48) , b (cid:48) ) then b ≤ b (cid:48) , and• if γ x , y = ( a , b ) and γ x , y + = ( a (cid:48) , b (cid:48) ) then a ≤ a (cid:48) (1,1)(1,3)(4,2) (1,5)(5,2)(3,5) (1,1)(4,5)(3,3)(2,1)(4,1) (1,3)(4,2) (4,4)(3,2)(3,1) (1,2) (3,3) (1,1) (2,3) (4,3) (3,5) 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 Figure 1:
An instance of G
RID -T ILING - ≤ with k = , N = Figure 1 gives an illustration of an instance of G
RID -T ILING - ≤ along with a solution. It is known [13,Theorem 14.30] that G RID -T ILING - ≤ is W[1]-hard parameterized by k , and under the Exponential TimeHypothesis (ETH) has no f ( k ) · N o ( k ) algorithm for any computable function f . We will exploit thisresult by reducing an instance ( k , N , S ) of G RID -T ILING - ≤ in poly ( N , k ) time to an instance ( G , T ) ofE DGE -D ISJOINT P ATHS such that G is a planar DAG, number of vertices in G is | V ( G ) | = O ( N k ) and number of terminal pairs is |T | = k . Remark 2.1.
Our definition of G
RID -T ILING - ≤ above is slightly different than the one given in [13,Theorem 14.30]: there the constraints are first coordinate of γ x , y is ≤ first coordinate of γ x + , y and secondcoordinate of γ x , y is ≤ second coordinate of γ x , y + . By rotating the axis by 90 ◦ , i.e., swapping the indices,our version of G RID -T ILING - ≤ is equivalent to that from [13, Theorem 14.30]. ( G , T ) of E DGE -D ISJOINT P ATHS
Consider an instance ( N , k , S ) of G RID -T ILING - ≤ . We now build an instance ( G , T ) of E DGE -D ISJOINT P ATHS as follows: first in Section 2.1.1 we describe the construction of an intermediate graph G (Figure 2). The splitting operation is defined in Section 2.1.2, and the graph G is obtained from G bysplitting each (black) grid vertex. G Given integers k and N , we build a directed graph G as follows (refer to Figure 2):1. Origin : The origin is marked at the bottom left corner of Figure 2. This is defined just so we canview the naming of the vertices as per the usual X − Y coordinate system: increasing horizontallytowards the right, and vertically towards the top.2. Grid (black) vertices and edges : For each 1 ≤ i , j ≤ k we introduce a (directed) N × N grid G i , j where the column numbers increase from 1 to N as we go from left to right, and the row numbersincrease from 1 to N as we go from bottom to top. For each 1 ≤ q , (cid:96) ≤ N the unique vertex which5 c c d d d a a a b b b O r i g i n Figure 2:
The graph G constructed for the input k = N = G for the E DGE -D ISJOINT P ATHS instance is obtained from G by the splitting operation(Definition 2.4) as described in Section 2.1.2.
6s the intersection of the q th column and (cid:96) th row of G i , j is denoted by w q ,(cid:96) i , j . The vertex set and edgeset of G i , j is defined formally as:• V ( G i , j ) = (cid:8) w q ,(cid:96) i , j : 1 ≤ q , (cid:96) ≤ N (cid:9) • E ( G i , j ) = (cid:16) (cid:83) ( q ,(cid:96) ) ∈ [ N ] × [ N − ] w q ,(cid:96) i , j → w q ,(cid:96) + i , j (cid:17) ∪ (cid:16) (cid:83) ( q ,(cid:96) ) ∈ [ N − ] × [ N ] w q ,(cid:96) i , j → w q + ,(cid:96) i , j (cid:17) All vertices and edges of G i , j are shown in Figure 2 using black color. Note that each horizontaledge of the grid G i , j is oriented to the right, and each vertical edge is oriented towards the top. Wewill later (Definition 2.4) modify the grid G i , j to represent the set S i , j .For each 1 ≤ i , j ≤ k we define the set of boundary vertices of the grid G i , j as follows: Left ( G i , j ) : = (cid:8) w ,(cid:96) i , j : (cid:96) ∈ [ N ] (cid:9) ; Right ( G i , j ) : = (cid:8) w N ,(cid:96) i , j : (cid:96) ∈ [ N ] (cid:9) Top ( G i , j ) : = (cid:8) w (cid:96), Ni , j : (cid:96) ∈ [ N ] (cid:9) ; Bottom ( G i , j ) : = (cid:8) w (cid:96), i , j : (cid:96) ∈ [ N ] (cid:9) (1)3. Arranging the k different N × N grids { G i , j } ≤ i , j ≤ k into a large k × k grid : We place the grids G i , j into a big k × k grid of grids left to right according to growing i and from bottom to topaccording to growing j (see the naming of the sets in Figure 1 in blue color). In particular,the grid G , is at bottom left corner of the construction, the grid G k , k at the top right corner, and so on.4. Blue vertices and red edges for horizontal connections : For each ( i , j ) ∈ [ k − ] × [ k ] we add aset of vertices H i + , ji , j : = (cid:8) h i + , ji , j ( (cid:96) ) : (cid:96) ∈ [ N ] (cid:9) shown in Figure 2 using blue color. We also add thefollowing three sets of edges (shown in Figure 2 using red color):• a directed path of N − Path ( H i + , ji , j ) : = (cid:8) h i + , ji , j ( (cid:96) ) → h i + , ji , j ( (cid:96) + ) : (cid:96) ∈ [ N − ] (cid:9) • a directed perfect matching from Right ( G i , j ) to H i + , ji , j given by Matching (cid:0) G i , j , H i + , ji , j (cid:1) : = (cid:8) w N ,(cid:96) i , j → h i + , ji , j ( (cid:96) ) : (cid:96) ∈ [ N ] (cid:9) • a directed perfect matching from H i + , ji , j to Left ( G i + , j ) given by Matching (cid:0) H i + , ji , j , G i + , j (cid:1) : = (cid:8) h i + , ji , j ( (cid:96) ) → w ,(cid:96) i + , j : (cid:96) ∈ [ N ] (cid:9) Blue vertices and red edges for vertical connections : For each ( i , j ) ∈ [ k ] × [ k − ] we add a setof vertices V i , j + i , j : = (cid:8) v i , j + i , j ( (cid:96) ) : (cid:96) ∈ [ N ] (cid:9) shown in Figure 2 using blue color. We also add thefollowing three sets of edges (shown in Figure 2 using red color):• a directed path of N − Path ( V i , j + i , j ) : = (cid:8) v i , j + i , j ( (cid:96) ) → v i , j + i , j ( (cid:96) + ) : (cid:96) ∈ [ N − ] (cid:9) • a directed perfect matching from Top ( G i , j ) to V i , j + i , j given by Matching (cid:0) G i , j , V i , j + i , j (cid:1) : = (cid:8) w (cid:96), Ni , j → v i , j + i , j ( (cid:96) ) : (cid:96) ∈ [ N ] (cid:9) • a directed perfect matching from V i , j + i , j to Bottom ( G i , j + ) given by Matching (cid:0) V i , j + i , j , G i , j + (cid:1) : = (cid:8) v i , j + i , j ( (cid:96) ) → w (cid:96), i , j + : (cid:96) ∈ [ N ] (cid:9) Green (terminal) vertices and magenta edges : For each i ∈ [ k ] we add the following four sets of(terminal) vertices (shown in Figure 2 using green color) A : = (cid:8) a i : i ∈ [ k ] (cid:9) ; B : = (cid:8) b i : i ∈ [ k ] (cid:9) C : = (cid:8) c i : i ∈ [ k ] (cid:9) ; D : = (cid:8) d i : i ∈ [ k ] (cid:9) (2)For each i ∈ [ k ] we add the edges (shown in Figure 2 using magenta color) Source ( A ) : = (cid:8) a i → w (cid:96), i , : (cid:96) ∈ [ N ] (cid:9) ; Sink ( B ) : = (cid:8) w (cid:96), Ni , N → b i : (cid:96) ∈ [ N ] (cid:9) (3)For each j ∈ [ k ] we add the edges (shown in Figure 2 using magenta color) Source ( C ) : = (cid:8) c j → w ,(cid:96) , j : (cid:96) ∈ [ N ] (cid:9) ; Sink ( D ) : = (cid:8) w N ,(cid:96) N , j → d j : (cid:96) ∈ [ N ] (cid:9) (4)This completes the construction of the graph G (see Figure 2). Claim 2.2. G is a planar DAG Proof.
Figure 2 gives a planar embedding of G . It is easy to verify from the construction of G describedat the start of Section 2.1.1 (see also Figure 2) that G is a DAG.7 q ,(cid:96) i , j west ( w q ,(cid:96) i , j ) east ( w q ,(cid:96) i , j ) south ( w q ,(cid:96) i , j ) north ( w q ,(cid:96) i , j ) Splitting Operation w q ,(cid:96) i , j , TR w q ,(cid:96) i , j , LB west ( w q ,(cid:96) i , j ) east ( w q ,(cid:96) i , j ) south ( w q ,(cid:96) i , j ) north ( w q ,(cid:96) i , j ) Figure 3:
The splitting operation for the vertex w q ,(cid:96) i , j when ( q , (cid:96) ) / ∈ S i , j . The idea behind this splitting is if we wantedge-disjoint paths then we can go either left-to-right or bottom-to-top but not in both directions. On the otherhand, if ( q , (cid:96) ) ∈ S i , j then the picture on the right-hand side (after the splitting operation) would look exactly likethat on the left-hand side. G from G via the splitting operation Observe (see Figure 2) that every (black) grid vertex in G has in-degree two and out-degree two.Moreover, the two in-neighbors and two out-neighbors do not appear alternately. For each (black) gridvertex z ∈ G we set up the notation: Definition 2.3. (four neighbors of each grid vertex in G ) For each (black) grid vertex z ∈ G wedefine the following four vertices• west ( z ) is the vertex to the left of z (as seen by the reader) which has an edge incoming into z • south ( z ) is the vertex below z (as seen by the reader) which has an edge incoming into z • east ( z ) is the vertex to the right of z (as seen by the reader) which has an edge outgoing from z • north ( z ) is the vertex above z (as seen by the reader) which has an edge outgoing from z We now define the splitting operation which allows us to obtain the graph G from the graph G constructed in Section 2.1.1. Definition 2.4. (splitting operation)
For each i , j ∈ [ k ] and each q , (cid:96) ∈ [ N ] • If ( q , (cid:96) ) / ∈ S i , j , then we split the vertex w q ,(cid:96) i , j into two distinct vertices w q ,(cid:96) i , j , LB and w q ,(cid:96) i , j , TR and addthe edge w q ,(cid:96) i , j , LB → w q ,(cid:96) i , j , TR (denoted by the dotted edge in Figure 3). The 4 edges (see Definition 2.3)incident on w q ,(cid:96) i , j are now changed as follows (see Figure 3): – Replace the edge west ( w q ,(cid:96) i , j ) → w q ,(cid:96) i , j by the edge west ( w q ,(cid:96) i , j ) → w q ,(cid:96) i , j , LB – Replace the edge south ( w q ,(cid:96) i , j ) → w q ,(cid:96) i , j by the edge south ( w q ,(cid:96) i , j ) → w q ,(cid:96) i , j , LB – Replace the edge w q ,(cid:96) i , j → east ( w q ,(cid:96) i , j ) by the edge w q ,(cid:96) i , j , TR → east ( w q ,(cid:96) i , j ) – Replace the edge w q ,(cid:96) i , j → north ( w q ,(cid:96) i , j ) by the edge w q ,(cid:96) i , j , TR → north ( w q ,(cid:96) i , j ) • Otherwise, if ( q , (cid:96) ) ∈ S i , j then the vertex w q ,(cid:96) i , j is not split , and we define w q ,(cid:96) i , j , LB = w q ,(cid:96) i , j = w q ,(cid:96) i , j , TR .Note that the four edges (Definition 2.3) incident on w q ,(cid:96) i , j are unchanged. Remark 2.5.
To avoid case distinctions in the forthcoming proof of correctness of the reduction, we willuse the following non-standard notation: the edge s (cid:32) s does not represent a self-loop but rather is to beviewed as “just staying put” at the vertex s . Note that this does not affect edge-disjointness.We are now ready to define the graph G and the set T of terminal pairs: Definition 2.6.
The graph G is obtained by applying the splitting operation ( Definition 2.4) to each(black) grid vertex of G , i.e., the set of vertices given by (cid:83) ≤ i , j ≤ k V ( G i , j ) . The set of terminal pairs is T : = (cid:8) ( a i , b i ) : i ∈ [ k ] (cid:9) ∪ (cid:8) ( c j , d j ) : j ∈ [ k ] (cid:9) G we have• All vertices in G except A ∪ C have out-degree at most 2• All vertices in G except B ∪ D have in-degree at most 2We will later show (see last paragraph in the proof of Theorem 1.1) how to edit G such that each vertexhas both in-degree and out-degree at most 2. The next claim shows that G is also both planar and acyclic(like G ). Claim 2.7. G is a planar DAG Proof.
In Claim 2.2, we have shown that G is a planar DAG. By Definition 2.6, G is obtained from G by applying the splitting operation (Definition 2.4) on every (black) grid vertex, i.e., every vertex fromthe set (cid:83) ≤ i , j ≤ k V ( G i , j ) .By Definition 2.3, every vertex of G that is split has exactly two in-neighbors and two out-neighborsin G . Hence, it is easy to see (Figure 3) that the splitting operation (Definition 2.4) does not destroyplanarity when we construct G from G . Since G is a DAG, replacing each split (black) grid vertex w in G by w LB followed by w TR in the topological order of G gives a topological order for G . Hence, G is a planar DAG.We now set up notation for the grids in G : Definition 2.8.
For each i , j ∈ [ k ] , we define G split i , j to be the graph obtained by applying the splittingoperation (Definition 2.4) to each vertex of G i , j . For each i , j ∈ [ k ] and each q , (cid:96) ∈ [ N ] we define split ( w q ,(cid:96) i , j ) : = (cid:8) w q ,(cid:96) i , j , LB , w q ,(cid:96) i , j , TR (cid:9) . DGE -D ISJOINT P ATHS ⇒ Solution for G
RID -T ILING - ≤ In this section, we show that if the instance ( G , T ) of E DGE -D ISJOINT P ATHS has a solution then theinstance ( k , N , S ) of G RID -T ILING - ≤ also has a solution.Suppose that the instance ( G , T ) of E DGE -D ISJOINT P ATHS has a solution, i.e., there is a collectionof 2 k pairwise edge-disjoint paths (cid:8) P , P , . . . , P k , Q , Q , . . . , Q k (cid:9) in G such that P i is an a i (cid:32) b i path ∀ i ∈ [ k ] Q j is an c j (cid:32) d j path ∀ j ∈ [ k ] (5)To streamline the arguments of this section, we define the following subsets of vertices of G : Definition 2.9. (horizontal & vertical levels)
For each j ∈ [ k ] , we define the following set of vertices:H ORIZONTAL ( j ) = { c j , d j } ∪ (cid:16) k (cid:91) i = V ( G split i , j ) (cid:17) ∪ (cid:16) k − (cid:91) i = H i + , ji , j (cid:17) For each i ∈ [ k ] , we define the following set of vertices:V ERTICAL ( i ) = { a i , b i } ∪ (cid:16) k (cid:91) j = V ( G split i , j ) (cid:17) ∪ (cid:16) k − (cid:91) j = V i , j + i , j (cid:17) From Definition 2.9, it is easy to verify that V
ERTICAL ( i ) ∩ V ERTICAL ( i (cid:48) ) = /0 = H ORIZONTAL ( i ) ∩ H ORIZONTAL ( i (cid:48) ) for every 1 ≤ i (cid:54) = i (cid:48) ≤ k . Definition 2.10. (boundary vertices in G ) For each 1 ≤ i , j ≤ k we define the set of boundary verticesof the grid G split i , j in the graph G as follows: Left ( G split i , j ) : = (cid:8) w ,(cid:96) i , j , LB : (cid:96) ∈ [ N ] (cid:9) ; Right ( G split i , j ) : = (cid:8) w N ,(cid:96) i , j , TR : (cid:96) ∈ [ N ] (cid:9) Top ( G split i , j ) : = (cid:8) w (cid:96), Ni , j , TR : (cid:96) ∈ [ N ] (cid:9) ; Bottom ( G split i , j ) : = (cid:8) w (cid:96), i , j , LB : (cid:96) ∈ [ N ] (cid:9) (6)9 emma 2.11. For each i ∈ [ k ] the path P i satisfies the following two structural properties:• every edge of the path P i has both end-points in V ERTICAL ( i ) • P i contains an Bottom ( G split i , j ) (cid:32) Top ( G split i , j ) path for each j ∈ [ k ] . Proof.
For this proof, define H , j , j : = { c j } and H k + , jk , j : = { d j } for each j ∈ [ k ] .Fix any i ∗ ∈ [ k ] . Note that P i ∗ is an a i ∗ (cid:32) b i ∗ path and hence starts and ends at a vertex inV ERTICAL ( i ∗ ) . We now prove the first part of lemma by showing two claims which state that P i ∗ cannot contain any vertex of N + G (cid:0) V ERTICAL ( i ∗ ) (cid:1) and N − G (cid:0) V ERTICAL ( i ∗ ) (cid:1) respectively. Claim 2.12. P i ∗ does not contain any vertex of N + G (cid:0) V ERTICAL ( i ∗ ) (cid:1) . Proof.
The structure of G implies that• N + G (cid:0) V ERTICAL ( i ) (cid:1) = (cid:83) kj = H i + , ji , j for each i ∈ [ k ] • N + G (cid:0) (cid:83) kj = H i + , ji , j (cid:1) ⊆ V ERTICAL ( i + ) for each 0 ≤ i ≤ k − N + G (cid:0) (cid:83) kj = H k + , jk , j (cid:1) = /0 since each vertex of D is a sink in G Hence, if P i ∗ contains a vertex from N + G (cid:0) V ERTICAL ( i ∗ ) (cid:1) then it cannot ever return back to V ERTICAL ( i ∗ ) which contradicts the fact that the last vertex of P i ∗ is b i ∗ ∈ V ERTICAL ( i ∗ ) . Claim 2.13. P i ∗ does not contain any vertex of N − G (cid:0) V ERTICAL ( i ∗ ) (cid:1) . Proof.
The structure of G implies that• N − G (cid:0) V ERTICAL ( i ) (cid:1) = (cid:83) kj = H i , ji − , j for each i ∈ [ k ] • N − G (cid:0) (cid:83) kj = H i + , ji , j (cid:1) ⊆ V ERTICAL ( i ) for each 1 ≤ i ≤ k • N − G (cid:0) (cid:83) kj = H , j , j (cid:1) = /0 since each vertex of C is a source in G Hence, if P i ∗ contains a vertex from N − G (cid:0) V ERTICAL ( i ∗ ) (cid:1) then P i ∗ cannot have started at a vertex ofV ERTICAL ( i ∗ ) which contradicts the fact that the first vertex of P i ∗ is a i ∗ ∈ V ERTICAL ( i ∗ ) .This concludes the proof of the first part of the lemma. We now show the second part of the lemma.We define V i ∗ , i ∗ , : = { a i ∗ } and V i ∗ , k + i ∗ , k : = { b i ∗ } . The structure of G implies that• N + G [ V ERTICAL ( i ∗ )] (cid:0) G split i ∗ , j (cid:1) = V i ∗ , j + i ∗ , j and N − G [ V ERTICAL ( i ∗ )] (cid:0) G split i ∗ , j (cid:1) = V i ∗ , ji ∗ , j − for each j ∈ [ k ] • N + G [ V ERTICAL ( i ∗ )] (cid:0) V i ∗ , j + i ∗ , j (cid:1) = Bottom (cid:0) G split i ∗ , j + (cid:1) for each 0 ≤ j ≤ k − N − G [ V ERTICAL ( i ∗ )] (cid:0) V i ∗ , j + i ∗ , j (cid:1) = Top (cid:0) G split i ∗ , j (cid:1) for each 1 ≤ j ≤ k These three relations, combined with the first part of the lemma which states that P i ∗ lies within G [ V ERTICAL ( i ∗ )] , implies that P i ∗ contains an Bottom ( G split i ∗ , j ) (cid:32) Top ( G split i ∗ , j ) path for each j ∈ [ k ] .This concludes the proof of Lemma 2.11.The proof of the next lemma is very similar to that of Lemma 2.11, and we skip repeating the details. Lemma 2.14.
For each j ∈ [ k ] the path Q j satisfies the following two structural properties:• every edge of the path Q j has both end-points in H ORIZONTAL ( j ) • Q j contains an Left ( G split i , j ) (cid:32) Right ( G split i , j ) path for each i ∈ [ k ] Lemma 2.15.
For any ( i , j ) ∈ [ k ] × [ k ] , let P (cid:48) , Q (cid:48) be any Bottom ( G split i , j ) (cid:32) Top ( G split i , j ) , Left ( G split i , j ) (cid:32) Right ( G split i , j ) paths in G respectively. If P (cid:48) and Q (cid:48) are edge-disjoint then there exists ( µ , δ ) ∈ S i , j suchthat the vertex w µ , δ i , j , LB = w µ , δ i , j = w µ , δ i , j , TR = belongs to both P (cid:48) and Q (cid:48) Proof.
Let P (cid:48)(cid:48) , Q (cid:48)(cid:48) be the paths obtained from P (cid:48) , Q (cid:48) by contracting all the dotted edges on P (cid:48) , Q (cid:48) respect-ively. By the construction of G (Definition 2.6) and the splitting operation (Definition 2.4), it follows that P (cid:48)(cid:48) , Q (cid:48)(cid:48) are Bottom ( G i , j ) (cid:32) Top ( G i , j ) , Left ( G i , j ) (cid:32) Right ( G i , j ) paths in G respectively. Hence, thereexist x , x ∈ [ N ] such that P (cid:48)(cid:48) is a w x , i , j → w x , Ni , j path and y , y ∈ [ N ] such that Q (cid:48)(cid:48) is a w , y i , j → w N , y i , j path. We now show that P (cid:48)(cid:48) and Q (cid:48)(cid:48) must intersect in G Claim 2.16. P (cid:48)(cid:48) and Q (cid:48)(cid:48) have a common vertex in G roof. For each x ∈ [ N ] such that x ≤ x ≤ x define P (cid:48)(cid:48) ( x ) = (cid:8) y ∈ [ N ] : w x , yi , j ∈ P (cid:48)(cid:48) (cid:9) . For each x ∈ [ N ] suchthat x ≤ x ≤ x define Q (cid:48)(cid:48) ( x ) = (cid:8) y ∈ [ N ] : w x , yi , j ∈ Q (cid:48)(cid:48) (cid:9) . We will prove the claim by showing that thereexists x ∗ , y ∗ ∈ [ N ] such that y ∗ ∈ (cid:0) P (cid:48)(cid:48) ( x ∗ ) ∩ Q (cid:48)(cid:48) ( x ∗ ) (cid:1) . By the orientation of the edges in G i , j , it follows thatmax P (cid:48)(cid:48) ( z ) = min P (cid:48)(cid:48) ( z + ) and max Q (cid:48)(cid:48) ( z ) = min Q (cid:48)(cid:48) ( z + ) ∀ x ≤ z < x If 1 ≤ u ≤ z ≤ N then max P (cid:48)(cid:48) ( u ) ≤ min P (cid:48)(cid:48) ( z ) and max Q (cid:48)(cid:48) ( u ) ≤ min Q (cid:48)(cid:48) ( z ) (7)By definition of Q (cid:48)(cid:48) , we have y ∈ Q (cid:48)(cid:48) ( ) and hence y ≥ y ≥ y ∈ Q (cid:48)(cid:48) ( x ) . If (cid:0) P (cid:48)(cid:48) ( x ) ∩ Q (cid:48)(cid:48) ( x ) (cid:1) (cid:54) = /0 then we are done. Otherwise, we have that min Q (cid:48)(cid:48) ( x ) > max P (cid:48)(cid:48) ( x ) since 1 ∈ P (cid:48)(cid:48) ( x ) . Nowif (cid:0) P (cid:48)(cid:48) ( x + ) ∩ Q (cid:48)(cid:48) ( x + ) (cid:1) (cid:54) = /0 then we are done. Otherwise, we have min Q (cid:48)(cid:48) ( x + ) > max P (cid:48)(cid:48) ( x + ) since min Q (cid:48)(cid:48) ( x + ) = max Q (cid:48)(cid:48) ( x ) . Continuing this way, we must find an x ∗ ∈ N such that x ≤ x ∗ ≤ x and (cid:0) P (cid:48)(cid:48) ( x ∗ ) ∩ Q (cid:48)(cid:48) ( x ∗ ) (cid:1) (cid:54) = /0: this is because N ∈ P (cid:48)(cid:48) ( x ) and hence min Q (cid:48)(cid:48) ( x ) ≤ N = max P (cid:48)(cid:48) ( x ) . Since (cid:0) P (cid:48)(cid:48) ( x ∗ ) ∩ Q (cid:48)(cid:48) ( x ∗ ) (cid:1) (cid:54) = /0 let y ∗ ∈ (cid:0) P (cid:48)(cid:48) ( x ∗ ) ∩ Q (cid:48)(cid:48) ( x ∗ ) (cid:1) , i.e., the vertex w x ∗ , y ∗ i , j belongs to both P (cid:48)(cid:48) and Q (cid:48)(cid:48) .By Claim 2.16, the paths P (cid:48)(cid:48) , Q (cid:48)(cid:48) have a common vertex in G . Let this vertex be w µ , δ i , j . Viewing thepaths P (cid:48)(cid:48) , Q (cid:48)(cid:48) in G , i.e., “un-contracting” the dotted edges (Definition 2.4), it follows that both P (cid:48) and Q (cid:48) share the dotted edge w µ , δ , LB i , j → w µ , δ i , j , TR . Since P (cid:48) and Q (cid:48) are given to be edge-disjoint, this implies thatthe edge w µ , δ , LB i , j → w µ , δ i , j , TR cannot exist in G , i.e., ( µ , δ ) ∈ S i , j and the vertex w µ , δ i , j , LB = w µ , δ i , j = w µ , δ i , j , TR belongs to both P (cid:48) and Q (cid:48) (recall Definition 2.4). This concludes the proof of Lemma 2.15. Lemma 2.17.
The instance ( k , N , S ) of G RID -T ILING - ≤ has a solution. Proof.
Fix any ( i , j ) ∈ [ k ] × [ k ] . By Lemma 2.11, P i contains an Bottom ( G split i , j ) (cid:32) Top ( G split i , j ) pathsay P i , j . By Lemma 2.14, Q j contains an Left ( G split i , j ) (cid:32) Right ( G split i , j ) path say Q i , j . Since P i and Q j are edge-disjoint (Equation 5), it follows that the paths P i , j and Q i , j are also edge-disjoint.Applying Lemma 2.15 to the paths P i , j and Q i , j we get that there exists ( µ i , j , δ i , j ) ∈ [ N ] × [ N ] such that ( µ i , j , δ i , j ) ∈ S i , j and the vertex w µ i , j , δ i , j i , j , LB = w µ i , j , δ i , j i , j = w µ i , j , δ i , j i , j , TR belongs to P i , j (and hence also to P i ) and Q i , j (and hence also to Q j ).We now claim that the values (cid:8) ( µ i , j , δ i , j ) : ( i , j ) ∈ [ k ] × [ k ] (cid:9) form a solution for the instance ( k , N , S ) ofG RID -T ILING - ≤ . In the last paragraph, we have already shown that ( µ i , j , δ i , j ) ∈ S i , j for each ( i , j ) ∈ [ k ] × [ k ] . For each ( i , j ) ∈ [ k − ] × [ k ] both the vertices w µ i , j , δ i , j i , j , LB = w µ i , j , δ i , j i , j , TR and w µ i + , j , δ i + , j i + , j , LB = w µ i + , j , δ i + , j i + , j , TR belongto the path Q j which is contained in G [ H ORIZONTAL ( j )] (Lemma 2.14). Hence, by the orientationof the edges in G , it follows that δ i , j ≤ δ i + , j . Similarly, it can be shown that µ i , j ≤ µ i , j + for each ( i , j ) ∈ [ k ] × [ k − ] . RID -T ILING - ≤ ⇒ Solution for E
DGE -D ISJOINT P ATHS
In this section, we show that if the instance ( k , N , S ) of G RID -T ILING - ≤ has a solution then the instance ( G , T ) of E DGE -D ISJOINT P ATHS also has a solution.Suppose that the instance ( k , N , S ) of G RID -T ILING - ≤ has a solution given by the pairs (cid:8) ( α i , j , β i , j ) : i , j ∈ [ k ] (cid:9) . Hence, we have (cid:0) α i , j , β i , j (cid:1) ∈ S i , j for each ( i , j ) ∈ [ k ] × [ k ] α i , j ≤ α i , j + for each ( i , j ) ∈ [ k ] × [ k − ] β i , j ≤ β i + , j for each ( i , j ) ∈ [ k − ] × [ k ] (8) Definition 2.18. (row-paths and column-paths in G ) For each ( i , j ) ∈ [ k ] × [ k ] and (cid:96) ∈ [ N ] we define• RowPath (cid:96) ( G split i , j ) to be the w ,(cid:96) i , j , LB (cid:32) w N ,(cid:96) i , j , TR path in G [ G split i , j ] consisting of the following edges(in order): for each r ∈ [ N − ] – w r ,(cid:96) i , j , LB → w r ,(cid:96) i , j , TR and w r ,(cid:96) i , j , TR → w r + ,(cid:96) i , j , LB followed finally by the edge w N ,(cid:96) i , j , LB → w N ,(cid:96) i , j , TR ColumnPath (cid:96) ( G split i , j ) to be the w (cid:96), i , j , LB (cid:32) w (cid:96), Ni , j , TR path in G consisting of the following edges (inorder): for each r ∈ [ N − ] – w (cid:96), ri , j , LB → w (cid:96), ri , j , TR and w (cid:96), ri , j , TR → w (cid:96), r + i , j , LB followed finally by the edge w (cid:96), Ni , j , LB → w (cid:96), Ni , j , TR Using the special types of paths from Definition 2.18, we can now show the following lemma:
Lemma 2.19.
The instance ( G , T ) of E DGE -D ISJOINT P ATHS has a solution.Proof.
We build a collection of 2 k paths P : = (cid:8) R , R , . . . , R k , T , T , . . . , T k (cid:9) and show that it forms asolution for the instance ( G , T ) of E DGE -D ISJOINT P ATHS . First, we describe this collection of pathsbelow: - Description of the set of paths { R , R , . . . , R k } :For each i ∈ [ k ] , we build the path R i as follows:• Start with the edge a i → w α i , , i , , LB • For each j ∈ [ k − ] use the w α i , j , i , j , LB (cid:32) w α i , j + , i , j + , LB path obtained by concatenating – the w α i , j , i , j , LB (cid:32) w α i , j , Ni , j , TR path ColumnPath α i , j ( G split i , j ) from Definition 2.18 – the w α i , j , Ni , j , TR (cid:32) w α i , j + , i , j + , LB path w α i , j , Ni , j , TR → v i , j + i , j ( α i , j ) → · · · · · · → v i , j + i , j ( α i , j + ) → w α i , j + , i , j + , LB which exists since Equation 8 implies α i , j ≤ α i , j + .• Now, we have reached the vertex w α i , k , i , k , LB . Use the w α i , k , i , k , LB (cid:32) w α i , k , Ni , k , TR path ColumnPath α i , k ( G split i , k ) from Definition 2.18 to reach the vertex w α i , k , Ni , k , TR .• Finally, use the edge w α i , k , Ni , k , TR → b i to reach b i . - Description of the set of paths { T , T , . . . , T k } :For each j ∈ [ k ] , we build the path T j as follows:• Start with the edge c j → w , β , j , j , LB • For each i ∈ [ k − ] use the w , β i , j i , j , LB (cid:32) w , β i + , j i + , j , LB path obtained by concatenating – the w , β i , j i , j , LB (cid:32) w N , β i , j i , j , TR path RowPath β i , j ( G split i , j ) from Definition 2.18 – the w N , β i , j i , j , TR (cid:32) w , β i + , j i + , j , LB path w N , β i , j i , j , TR → h i + , ji , j ( β i , j ) → · · · · · · → h i + , ji , j ( β i + , j ) → w , β i + , j i + , j , LB which exists since Equation 8 implies β i , j ≤ β i + , j .• Now, we have reached the vertex w , β k , j k , j , LB . Use the w , β k , j k , j , LB (cid:32) w N , β k , j k , j , TR path RowPath β k , j ( G split k , j ) from Definition 2.18 to reach the vertex w N , β k , j k , j , TR .• Finally, use the edge w N , β k , j k , j , TR → d j to reach d j .By Definition 2.9, it follows that every edge of the path R i has both endpoints in V ERTICAL ( i ) for every i ∈ [ k ] . Since V ERTICAL ( i ) ∩ V ERTICAL ( i (cid:48) ) = /0 for every 1 ≤ i (cid:54) = i (cid:48) (cid:54) = k , it follows that thecollection of paths { R , R , . . . , R k } are pairwise edge-disjoint.By Definition 2.9, it follows that every edge of the path T j has both endpoints in H ORIZONTAL ( j ) forevery j ∈ [ k ] . Since H ORIZONTAL ( j ) ∩ H ORIZONTAL ( j (cid:48) ) = /0 for every 1 ≤ j (cid:54) = j (cid:48) (cid:54) = k , it follows thatthe collection of paths { T , T , . . . , T k } are pairwise edge-disjoint.Fix any ( i , j ) ∈ [ k ] × [ k ] . We now conclude the proof of this lemma by showing that R i and T j areedge-disjoint. By the construction of G (Figure 2 and Figure 3) and definitions of the paths R i and T j ,it follows that the only common edge between R i and T j could be w α i , j , β i , j i , j , LB → w α i , j , β i , j i , j , TR . By Equation 8,we have that ( α i , j , β i , j ) ∈ S i , j . Hence, by the splitting operation (Definition 2.4), we have that w α i , j , β i , j i , j , LB = w α i , j , β i , j i , j = w α i , j , β i , j i , j , TR , i.e., the only possible common edge w α i , j , β i , j i , j , LB → w α i , j , β i , j i , j , TR between R i and T j is not anedge in G . Hence, R i and T j are edge-disjoint. 12 .4 Proof of Theorem 1.1 Finally we are ready to prove our main theorem (Theorem 1.1) which is restated below:
Theorem 1.1.
The E DGE -D ISJOINT P ATHS problem on planar DAGs is W[1]-hard parameterized by thenumber k of terminal pairs. Moreover, under ETH, the E DGE -D ISJOINT P ATHS problem on planar DAGscannot be solved f ( k ) · n o ( k ) time where f is any computable function, n is the number of vertices and kis the number of terminal pairs. The hardness holds even if both the maximum in-degree and maximumout-degree of the graph are at most .Proof. Given an instance ( k , N , S ) of G RID -T ILING - ≤ , we use the construction from Section 2.1 to buildan instance ( G , T ) of E DGE -D ISJOINT P ATHS such that G is a planar DAG (Claim 2.7). It is easy tosee that n = | V ( G ) | = O ( N k ) and G can be constructed in poly ( N , k ) time.It is known [13, Theorem 14.30] that G RID -T ILING - ≤ is W[1]-hard parameterized by k , and underETH cannot be solved in f ( k ) · N o ( k ) time for any computable function f . Combining the two directionsfrom Section 2.2 and Section 2.3, we get a parameterized reduction from G RID -T ILING - ≤ to an instanceof E DGE -D ISJOINT P ATHS which is a planar DAG and has |T | = k terminal pairs. Hence, it followsthat E DGE -D ISJOINT P ATHS on planar DAGs is W[1]-hard parameterized by number k of terminal pairs,and under ETH cannot be solved in f ( k ) · n o ( k ) time for any computable function f .Finally we show how to edit G , without affecting the correctness of the reduction, so that both themax out-degree and max in-degree are at most 2. We present the argument for reducing the out-degree:the argument for reducing the in-degree is analogous. Note that the only vertices in G with out-degree > A ∪ C . For each c j ∈ C we replace the directed star whose edges are from c j to each vertex of Left ( G , j ) with a directed binary tree whose root is c i , leaves are the set of vertices Left ( G , j ) and eachedge is directed away from the root. It is easy to see that in this directed binary tree the set of pathsfrom c j to the different leaves (i.e.,vertices of Left ( G , j ) ) are pairwise edge-disjoint, and we have onlyincreased the number of vertices by O ( k ) while maintaining both planarity and (directed) acyclicity. Wedo a similar transformation for each a i ∈ A . It is easy to see that this editing adds O ( k ) new vertices andtakes poly ( k ) time, and therefore it is still true that n = | V ( G ) | = O ( N k ) and G can be constructed inpoly ( N , k ) time. In this paper we have shown that E
DGE -D ISJOINT P ATHS on planar DAGs is W[1]-hard parameterizedby k , and has no f ( k ) · n o ( k ) algorithm under the Exponential Time Hypothesis (ETH) for any computablefunction f . The hardness holds even if both the maximum in-degree and maximum out-degree of thegraph are at most 2. Our result answers a question of Slivkins [44] regarding the parameterized complexityof E DGE -D ISJOINT P ATHS on planar DAGS, and a question of Cygan et al. [12] and Schrijver [43]regarding the parameterized complexity of E
DGE -D ISJOINT P ATHS on planar directed graphs.We now propose some open questions related to the complexity of the D
ISJOINT P ATHS problem:• What is the correct parameterized complexity of E
DGE -D ISJOINT P ATHS on planar graphs para-meterized by k ? Can we design an XP algorithm, or is the problem NP-hard even for k = O ( ) likethe general version? Note that to prove the latter result, one would need to have directed cyclesinvolved in the reduction since there is n O ( k ) algorithm of Fortune et al. [19] for E DGE -D ISJOINT P ATHS on DAGs.• Is the half-integral version of E DGE -D ISJOINT P ATHS
FPT on directed planar graphs or DAGs? Itis easy to see that our W[1]-hardness reduction does not work for this problem.• Given our W[1]-hardness result, can we obtain FPT (in)approximability results for the E
DGE -D ISJOINT P ATHS problem on planar DAGs? To the best of our knowledge, there are no known(non-trivial) FPT (in)approximability results for any variants of the D
ISJOINT P ATHS problem.This question might be worth considering even for those versions of the D
ISJOINT P ATHS problem Each edge can belong to at most two of the paths
ISJOINT P ATHS problem might be relevant.
Acknowledgements
We thank the anonymous reviewers of CIAC 2021 for their helpful comments. In particular, one ofthe reviewers suggested the strengthening of Theorem 1.1 for the case when the input graph has bothin-degree and out-degree at most 2.
References [1] Pierre Aboulker, Nick Brettell, Fr´ed´eric Havet, D´aniel Marx, and Nicolas Trotignon. Coloringgraphs with constraints on connectivity.
Journal of Graph Theory , 85(4):814–838, 2017. doi:10.1002/jgt.22109. URL https://doi.org/10.1002/jgt.22109.[2] Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh,and Dimitrios M. Thilikos. Irrelevant vertices for the planar Disjoint Paths Problem.
J. Comb.Theory, Ser. B , 122:815–843, 2017. URL https://doi.org/10.1016/j.jctb.2016.10.001.[3] Saeed Akhoondian Amiri, Stephan Kreutzer, D´aniel Marx, and Roman Rabinovich. Routing withcongestion in acyclic digraphs.
Inf. Process. Lett. , 151, 2019. URL https://doi.org/10.1016/j.ipl.2019.105836.[4] Rajesh Chitnis and Andreas Emil Feldmann. A Tight Lower Bound for Steiner Orientation. In
CSR2018 , pages 65–77. URL https://doi.org/10.1007/978-3-319-90530-3 7.[5] Rajesh Chitnis, Andreas Emil Feldmann, and Ondrej Such´y. A Tight Lower Bound for PlanarSteiner Orientation.
Algorithmica , 81(8):3200–3216, 2019. URL https://doi.org/10.1007/s00453-019-00580-x.[6] Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and D´aniel Marx. Tight Bounds for PlanarStrongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions). In
SODA2014 , pages 1782–1801, 2014. URL https://doi.org/10.1137/1.9781611973402.129.[7] Rajesh Hemant Chitnis, Andreas Emil Feldmann, Mohammad Taghi Hajiaghayi, and D´aniel Marx.Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (andExtensions).
SIAM J. Comput. , 49(2):318–364, 2020. URL https://doi.org/10.1137/18M122371X.[8] Julia Chuzhoy, David H. K. Kim, and Shi Li. Improved approximation for node-disjoint paths inplanar graphs. In
STOC 2016 , pages 556–569, . URL https://doi.org/10.1145/2897518.2897538.[9] Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. New hardness results for routing on disjointpaths. In
STOC 2017 , pages 86–99, . URL https://doi.org/10.1145/3055399.3055411.[10] Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. Almost polynomial hardness of node-disjointpaths in grids. In
STOC 2018 , pages 1220–1233, . URL https://doi.org/10.1145/3188745.3188772.[11] Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. Improved Approximation for Node-DisjointPaths in Grids with Sources on the Boundary. In
ICALP 2018 , volume 107, pages 38:1–38:14, 2018.URL https://doi.org/10.4230/LIPIcs.ICALP.2018.38.[12] Marek Cygan, D´aniel Marx, Marcin Pilipczuk, and Michal Pilipczuk. The Planar Directed k -Vertex-Disjoint Paths Problem Is Fixed-Parameter Tractable. In FOCS 2013 , pages 197–206. URLhttps://doi.org/10.1109/FOCS.2013.29. 1413] Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, D´aniel Marx, Marcin Pilipczuk,Michal Pilipczuk, and Saket Saurabh.
Parameterized Algorithms . Springer, 2015. ISBN 978-3-319-21274-6. URL https://doi.org/10.1007/978-3-319-21275-3.[14] Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos.Subexponential parameterized algorithms on bounded-genus graphs and H -minor-free graphs. J.ACM , 52(6):866–893, 2005. URL https://doi.org/10.1145/1101821.1101823.[15] Reinhard Diestel.
Graph Theory, 4th Edition . Volume 173 of Graduate Texts in Mathematics.Springer, 2012. ISBN 978-3-642-14278-9. URL https://doi.org/10.1007/978-3-662-53622-3.[16] Shimon Even, Alon Itai, and Adi Shamir. On the Complexity of Timetable and Multi-CommodityFlow Problems. In
FOCS 1975 , pages 184–193. URL https://doi.org/10.1109/SFCS.1975.21.[17] Fedor V. Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Subexpo-nential Algorithms for Rectilinear Steiner Tree and Arborescence Problems. In
SoCG 2016 , pages39:1–39:15, . URL https://doi.org/10.4230/LIPIcs.SoCG.2016.39.[18] Fedor V. Fomin, Daniel Lokshtanov, D´aniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and SaketSaurabh. Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs viaLow Treewidth Pattern Covering. In
FOCS 2016 , pages 515–524, . URL https://doi.org/10.1109/FOCS.2016.62.[19] Steven Fortune, John E. Hopcroft, and James Wyllie. The Directed Subgraph HomeomorphismProblem.
Theor. Comput. Sci. , 10:111–121, 1980. URL https://doi.org/10.1016/0304-3975(80)90009-2.[20] Andr´as Frank. Packing paths, circuits, and cuts - a survey,. In Alexander Schrijver, LaszloLovasz, Bernhard Korte, Hans Jurgen Promel, and R. L. Graham, editors,
Paths, Flows and VLSI-Layouts , volume 148 of
LIPIcs , pages 49–100. Springer-Verlag, 1990. ISBN 0387526854. URLhttps://dl.acm.org/doi/book/10.5555/574821.[21] Archontia C. Giannopoulou, Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon. Thecanonical directed tree decomposition and its applications to the directed disjoint paths problem.
CoRR , abs/2009.13184, 2020. URL https://arxiv.org/abs/2009.13184.[22] Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k -SAT. J. Comput. Syst. Sci. ,62(2):367–375, 2001. URL https://doi.org/10.1006/jcss.2000.1727.[23] Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which Problems Have StronglyExponential Complexity?
J. Comput. Syst. Sci. , 63(4):512–530, 2001. URL https://doi.org/10.1006/jcss.2001.1774.[24] Ken-ichi Kawarabayashi and Bruce A. Reed. A nearly linear time algorithm for the half integraldisjoint paths packing. In
SODA 2008 , pages 446–454, . URL http://dl.acm.org/citation.cfm?id=1347082.1347131.[25] Ken-ichi Kawarabayashi and Bruce A. Reed. A nearly linear time algorithm for the half integralparity disjoint paths packing problem. In
SODA 2009 , pages 1183–1192, . URL http://dl.acm.org/citation.cfm?id=1496770.1496898.[26] Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Stephan Kreutzer. An excluded half-integral gridtheorem for digraphs and the directed disjoint paths problem. In
STOC 2014 , pages 70–78, . URLhttps://doi.org/10.1145/2591796.2591876.[27] Ken-ichi Kawarabayashi, Bruce A. Reed, and Paul Wollan. The Graph Minor Algorithm with ParityConditions. In
FOCS 2011 , pages 27–36, . URL https://doi.org/10.1109/FOCS.2011.52.1528] Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem inquadratic time.
J. Comb. Theory, Ser. B , 102(2):424–435, 2012. URL https://doi.org/10.1016/j.jctb.2011.07.004.[29] Philip N. Klein and D´aniel Marx. Solving Planar k -Terminal Cut in O ( n c √ k ) Time. In
ICALP 2012 ,pages 569–580, . URL https://doi.org/10.1007/978-3-642-31594-7 48.[30] Philip N. Klein and D´aniel Marx. A subexponential parameterized algorithm for Subset TSP onplanar graphs. In
SODA 2014 , pages 1812–1830, . URL https://doi.org/10.1137/1.9781611973402.131.[31] Jon M. Kleinberg. Decision Algorithms for Unsplittable Flow and the Half-Disjoint Paths Problem.In
STOC 1998 , pages 530–539. ACM, 1998. URL https://doi.org/10.1145/276698.276867.[32] Daniel Lokshtanov, Pranabendu Misra, Michal Pilipczuk, Saket Saurabh, and Meirav Zehavi. Anexponential time parameterized algorithm for planar disjoint paths. In
STOC 2020 , pages 1307–1316,. URL https://doi.org/10.1145/3357713.3384250.[33] Daniel Lokshtanov, Saket Saurabh, and Magnus Wahlstr¨om. Subexponential Parameterized OddCycle Transversal on Planar Graphs. In
FSTTCS 2012 , pages 424–434, . URL https://doi.org/10.4230/LIPIcs.FSTTCS.2012.424.[34] D´aniel Marx. A Tight Lower Bound for Planar Multiway Cut with Fixed Number of Terminals. In
ICALP 2012 , pages 677–688. URL https://doi.org/10.1007/978-3-642-31594-7 57.[35] D´aniel Marx. The Square Root Phenomenon in Planar Graphs. In
ICALP 2013 , volume 7966,page 28. Springer, 2013. URL https://doi.org/10.1007/978-3-642-39212-2 4.[36] D´aniel Marx and Michal Pilipczuk. Optimal Parameterized Algorithms for Planar Facility LocationProblems Using Voronoi Diagrams. In
ESA 2015 , pages 865–877. URL https://doi.org/10.1007/978-3-662-48350-3 72.[37] D´aniel Marx and Anastasios Sidiropoulos. The limited blessing of low dimensionality: when1 − / d is the best possible exponent for d -dimensional geometric problems. In SOCG 2014 ,page 67. URL https://doi.org/10.1145/2582112.2582124.[38] D´aniel Marx, Marcin Pilipczuk, and Michal Pilipczuk. On Subexponential Parameterized Algorithmsfor Steiner Tree and Directed Subset TSP on Planar Graphs. In
FOCS, 2018 , pages 474–484. URLhttps://doi.org/10.1109/FOCS.2018.00052.[39] Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs. In
STACS 2013 , pages 353–364.URL https://doi.org/10.4230/LIPIcs.STACS.2013.353.[40] Neil Robertson and Paul D. Seymour. Graph Minors XIII. The Disjoint Paths Problem.
J. Comb.Theory, Ser. B , 63(1):65–110, 1995. URL https://doi.org/10.1006/jctb.1995.1006.[41] Alexander Schrijver. Finding k Disjoint Paths in a Directed Planar Graph.
SIAM J. Comput. , 23(4):780–788, 1994. URL https://doi.org/10.1137/S0097539792224061.[42] Alexander Schrijver.
Combinatorial Optimization: Polyhedra and Efficiency k Partially Disjoint Paths in a Directed Planar Graph.
BuildingBridges II. Bolyai Society Mathematical Studies. , 28:417–444, 2019. URL https://doi.org/10.1007/978-3-662-59204-5 13. 1644] Aleksandrs Slivkins. Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs.
SIAM J. Discret. Math. , 24(1):146–157, 2010. URL https://doi.org/10.1137/070697781.[45] Jens Vygen. NP-completeness of Some Edge-disjoint Paths Problems.