An exponential lower bound for cut sparsifiers in planar graphs

Given an edge-weighted graph $G$ with a set $Q$ of $k$ terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the sizes of minimum cuts between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph $G$ being either an arbitrary graph or coming from a specific graph class. In this note we show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with $k$ terminals that require $2^{k-2}$ edges in any mimicking network. This nearly matches an upper bound of $O(k 2^{2k})$ of Krauthgamer and Rika [SODA 2013, arXiv:1702.05951] and is in sharp contrast with the $O(k^2)$ upper bound under the assumption that all terminals lie on a single face [Goranci, Henzinger, Peng, arXiv:1702.01136]. As a side result we show a hard instance for the double-exponential upper bounds given by Hagerup, Katajainen, Nishimura, and Ragde~[JCSS 1998] and by Khan and Raghavendra~[IPL 2014].

Up to today, the only improvement upon the above upper bound for general graphs is due to Khan and Raghavendra [6], which is also double-exponential. In 2013, Krauthgamer and Rika [9] observed that the aforementioned preprocessing step can be adjusted to yield a mimicking network of size O(k 2 2 2k ) for planar graphs. Furthermore, they introduced a framework for proving lower bounds, and showed that there are (non-planar) graphs, for which any mimicking network has 2 Ω(k) edges; a slightly stronger lower bound of 2 (k−1)/2 has been shown by Khan and Raghavendra [6]. On the other hand, for planar graphs the lower bound of [9] is Ω(k 2 ). Furthermore, the planar graph lower bound applies even in the special case when all the terminals lie on the same face.
Very recently, two improvements upon these results for planar graphs have been announced. In a sequel paper, Krauthgamer and Rika [10] improve the polynomial factor in the upper bound for planar graphs to O(k2 2k ) and show that the exponential dependency actually adheres only to the number of faces containing terminals: if the terminals lie on γ faces, one can obtain a mimicking network of size O(γ 5 2 2γ k 4 ). In a different work, Goranci, Henzinger, and Peng [4] showed a tight O(k 2 ) upper bound for mimicking networks for planar graph with all terminals on a single face.
Our results. We complement these results by showing an exponential lower bound for mimicking networks in planar graphs. Theorem 1.1. For every integer k ≥ 3, there exists a planar graph G with a set Q of k terminals and edge cost function under which every mimicking network for G has at least 2 k−2 edges.
This nearly matches the upper bound of O(k2 2k ) of Krauthgamer and Rika [10] and is in sharp contrast with the polynomial bounds when the terminals lie on a constant number of faces [4,10].
As a side result, we also show a hard instance for mimicking networks in general graphs.
Theorem 1.2. For every integer k ≥ 1, there exists a graph G with a set Q of 3k + 1 terminals and 2 2 Ω(k) vertices such that no two vertices can be identified without strictly increasing the size of some minimum S-separating cut.
The example of Theorem 1.2, obtained by essentially reiterating the construction of Krauthgamer and Rika [9], shows that the doubly exponential bound is natural for the preprocessing step of Hagerup et al [5], and one needs different techniques to improve upon it.
Related work. Apart from the aforementioned work on mimicking networks [4,5,6,9,10], there has been substatial work on preserving cuts and flows approximately, see e.g. [1,3,11]. If one wants to construct mimicking networks for vertex cuts in unweighted graphs with deletable terminals (or with small integral weights), the representative sets approach of Kratsch and Wahlström [7] provides a mimicking network with O(k 3 ) vertices, improving upon a previous quasipolynomial bound of Chuzhoy [2].
We prove Theorem 1.1 in Section 2 and show the example of Theorem 1.2 in Section 3.

Exponential lower bound for planar graphs
In this section we present the main result of the paper. We provide a construction that proves that there are planar graphs on k terminals whose mimicking networks are of size Ω(2 k ).
In order to present the desired graph, for the sake of simplicity, we describe its dual graph (G, c). We let Q = {f n , f s , f 1 , f 2 , . . . , f k−2 } be the set of faces in G corresponding to terminals in the primal graph G * . 1 There are two special terminal faces f n and f s , referred to as the north face and the south face. The remaining faces of Q are referred to as equator faces.
A set S ⊂ Q is important if f n ∈ S and f s / ∈ S. Note that there are 2 k−2 important sets; in what follows we care only about minimum cuts in the primal graph for separations between important sets and their complements. For an important set S, we define its signature as a bit vector χ(S) ∈ [2] |Q|−2 whose i'th position is defined as χ(S)[i] = 1 iff f i ∈ S. Graph G will be composed of 2 k−2 cycles referred to as important cycles, each corresponding to an important subset S ⊂ Q. A cycle corresponding to S is referred to as C χ(S) and it separates S from S. Topologically, we draw the equator faces on a straight horrizontal line that we call the equator. We put the north face f n above the equator and the south face f s below the equator. For any important S ⊂ Q, in the plane drawing of G the corresponding cycle C χ(S) is a curve that goes to the south of f i if f i ∈ S and otherwise to the north of f i . We formally define important cycles later on, see Definition 2.1.
We now describe in detail the construction of G. We start with a graph H that is almost a tree, and then embed H in the plane with a number of edge crossings, introducing a new vertex on every edge crossing. The graph H consists of a binary tree of height k − 2 with root v and an extra vertex w that is adjacent to the root v and every of the 2 k−2 leaves of the tree. In what follows, the vertices of H are called branching vertices, contrary to crossing vertices that will be introduced at edge crossings in the plane embedding of H.
To describe the plane embedding of H, we need to introduce some notation of the vertices of H.
We emphasize here that the new bit in the index is added as the first symbol. Every next layer is twice the size of the previous one, hence |L i | = |E H i | = 2 i . Finally the vertices of L k−2 are all of degree 2. Each of them is connected to a vertex in L k−3 via an edge in E H k−2 and to the vertex w via an edge in E H k−1 . We now describe the drawing of H, that we later make planar by adding crossing vertices, in order to obtain the graph G. As we mentioned before, we want to draw equator faces f 1 , . . . f k−2 in that order from left to right on a horizontal line (referred to as an equator). Consider equator face f i and vertex layer L i for some i > 0. Imagine a vertical line through f i perpendicular to the equator, and let us refer to it as an i'th meridian. We allign the vertices of L i along the i'th meridian, from the north to the south. We start with the vertex of L i with the (lexicographically) lowest index, and continue drawing vertices of L i more and more to the south while the indices increase. Moreover, the first half of L i is drawn to the north of f i , and the second half to the south of f i . Every edge of H, except for e, is drawn as a straight line segment connecting its endpoints. The edge e is a curve encapsulating the northface f n and separating it from f s -the outer face of G.
The crossing vertices are added whenever the line segments cross. This way the edges of H are subdivided and the resulting graph is denoted by G. This completes the description of the structure and the planar drawing of G. We refer to Figure 1 for an illustration of the graph G. The set E i consists of all edges of G that are parts of the (subdivided) edges of E H i from H, see Figure 2. We are also ready to define important cycles formally. We now move on to describing how weights are assigned to the edges of G. The costs of the edges in G admit k − 2 values: Figure 2 for an illustration). As we mentioned before, e ba is subdivided by crossing vertices into a number of edges. If b = 0, then edge e ba is subdivided by 2 dec(a) crossing vertices into dec(a) + 1 edges: , v ba }. Among those edges e dec(a)+1 ba is assigned cost C, and the remaining edges subdividing e ba are assigned cost c i . Analogically, if b = 1, then edge e ba is subdivided by 2 i − 1 − dec(a) crossing vertices into 2 i − dec(a) edges: , v ba }. Again, we let 2 For a bit vector a, dec(a) denotes the integral value of a read as a number in binary. have cost C, and the remaining edges subdividing e ba are assigned cost c i . The cost assignment within an edge layer is presented in Figure 2.
This finishes the description of the dual graph G. We now consider the primal graph G * with the set of terminals Q * consisting of the k vertices of G * corresponding to the faces Q of G. In the remainder of this section we show that there is a cost function on the edges of G * , under which any mimicking network for G * contains at least 2 k−2 edges. This cost function is in fact a small perturbation of the edge costs implied by the dual graph G.
In order to accoplish this, we use the framework introduced in [9]. In what follows, mincut G,c (S, S ) stands for the minimum cut separating S from S in a graph G with cost function c. Below we provide the definition of the cutset-edge incidence matrix and the Main Technical Lemma from [9].  Recall that G * is the dual graph to the graph G that we constructed. By slightly abusing the notation, we will use the cost function c defined on the dual edges also on the corresponding primal edges. Let Q * = {f n * , f s * , f 1 * , . . . f k−2 * } be the set of terminals in G * corresponding to f n , f s , f 1 , . . . f k−2 respectively. We want to apply Lemma 2.3 to G * and Q * . For that we need to show that the cuts in G * corresponding to important sets are unique and that rank(A G * ,c ) is high.
As an itermediate step let us argue that the following holds.
Claim 2.4. There is k edge disjoint simple paths in G * from f n * to f s * : π 0 , π 1 , . . . , π k−2 , π k−1 . Each π i is composed entirely of edges dual to the edges of E i whose cost equals C. For i ∈ {1 . . . k −2}, π i contains vertex f i * . Let π n i be the prefix of π i from f n * to f i * and π s i be the suffix from f i * to f s * . The number of edges on π i is 2 i , and the number of edges on π n i and π s i is 2 i−1 . Proof. The primal graph G * together with paths π 0 , π 1 . . . π k−2 , π k−1 is pictured in Figure 3. The paths π k−2 , π k−1 visit the same vertices in the same manner, so for the sake of clarity only one of these paths is shown in the picture. This proof contains a detailed description of these paths and how they emerge from in the dual graph G.
Consider a layer L i . Recall that for any ba ∈ Again, c(e 2 i −dec(a) ba ) = C. Consider the edges of E i incident to vertices in L i . If we order these edges lexicographically by their lower index, then each consecutive pair of edges shares a common face. Moreover, the first edge e 1 00...0 is incident to f n and the last edge e 1 11...1 is incident to f s . This gives a path π i from f n to f s through f i in the primal graph where all the edges on π i have cost C. Path π k−1 is given by the edges of E k−1 in a similar fashion and path π 0 is composed of a single edge dual to e.
We move on to proving that the condition in Lemma 2.3 holds. We extend the notion of important sets S ⊆ Q to sets S * ⊆ Q * in the natural manner. Lemma 2.5. For every important S * ⊂ Q * , the minimum cut separating S * from S * is unique and corresponds to cycle C χ(S) in G.
Proof. Let C be the set of edges of G corresponding to some minimum cut between S * and S * in G * . Let S ⊆ Q be the set of faces of G corresponding to the set S * . We start by observing that the edges of G * corresponding to C χ(S) form a cut between S * and S * . Consequently, the total weight of edges of C is at most the total weight of the edges of C χ(S) . By Claim 2.4, C contains at least k edges of cost C, at least one edge of cost C per edge layer (it needs to hit an edge in every path π 0 , . . . π k−1 ). Note that C χ(S) contains exactly k edges of cost C. We assign the weights in a way that C is larger than all other edges in the graph taken together. This implies that C contains exactly one edge of cost C in every edge layer E i . In particular, C contains the edge e = {v, w}.
Furthermore, the fact that f i * lies on π i implies that the edge of weight C in E i ∩ C lies on π n i if f i * / ∈ S and lies on π s i otherwise. Consequently, in G * − C there is one connected component containing all vertices of S * and one connected component containing all vertices of S * . By the minimality of C, we infer that G * − C contains no other connected components apart from the aforementioned two components. By planarity, since any minimum cut in a planar graph corresponds to a collection of cycles in its dual, this implies that C is a single cycle in G.
Let e i be the unique edge of E i ∩ C of weight C and let e i be the unique edge of E i ∩ C χ(S) of weight C. We inductively prove that e i = e i and that the subpath of C between e i and e i+1 is the same as on C χ(S) . For the base of the induction, note that e 0 = e 0 = e.
Consider an index i > 0 and the face f i . If f i ∈ S, i.e., f i belongs to the north side, then e i lies south of f i , that is, lies on π s i . Otherwise, if f i / ∈ S, then e i lies north of f i , that is, lies on π n i . Let v a and v ba be the vertices of C χ(S) that lie on L i−1 and L i , respectively. By the inductive assumption, v a is an endpoint of e i−1 = e i−1 that lies on C. Let e i = xv bc , where v bc ∈ L i and let e i = x v ba . Since C is a cycle in G that contains exactly one edge on each path π i , we infer that C contains a path between v a and v bc that consists of e i and a number of edges of E i of weight c i . A direct check shows that the subpath from v a to v ba on C χ(S) is the unique such path with minimum number of edges of weight c i . Since the weight c i is larger than the total weight of all edges of smaller weight, from the minimality of C we infer that v ba = v bc and C and C χ(S) coincide on the path from v a to b ba .
Consequently, C and C χ(S) coincide on the path from the edge e = vw to the vertex v←−− χ(S) ∈ L k−2 .
From the minimality of C we infer that also the edge {w, v←−− χ(S) } lies on the cycle C and, hence, C = C χ(S) . This completes the proof.
Proof. Recall Definition 2.1 and the fact that C χ(S) is defined for every important S ⊆ Q. This means that the only edge in E k−1 that belongs to C χ(S) is the edge adjacent to v←−− χ(S) . Let us consider the part of adjacency matrix where rows correspond to the cuts corresponding to C χ(S) for important S ⊂ Q and where columns correspond to the edges in E k−1 of weight C. Let us order the cuts according to ←−− χ(S) and the edges by the index of the adjacent vertex in L k−2 (lexicographically). Then this part of A G,c is identity matrix. Hence, rank(A G,c ) ≥ 2 k−2 .
Lemma 2.5 and Claim 2.6 provide the conditions necessary for Lemma 2.3 to apply. This proves our main result stated in Theorem 1.1.

Doubly exponential example
In this section we show an example graph for which the compression technique introduced by Hagerup et al [5] does indeed produce a mimicking network on 2 2 Ω(k) vertices. Our example relies on doubly exponential edge costs. Note that an example with single exponential costs can be compressed into a mimicking network of size single exponential in k using the techniques of [7].
Hagerup et al [5] introduce the concept of mimicking networks and give a simple method to obtain a mimicking network with O(2 2 k ) vertices. It may seem that this upperbound is crude, thus the question arises wheather so large networks are really needed. We give here an affirmative answer to this question, if the only allowed operation is merging vertices in G. Before we go on, let us recall the technique of Hagerup et al. Let G be a weighted graph and Q be the set of terminals. Observe that a minimum cut separating S ⊂ Q from S = Q \ S, when removed from G, divides the vertices of G into two sides: the side of S and the side of S. The side is defined for each vertex, as all connected components obtained by removing the minimum cut contain a terminal. Now if two vertices u and v are on the same side of the minimum cut between S and S for every S ⊂ Q, then they can be merged without changing the size of any minimum S-separating cut. As a result there is at most 2 2 k vertices in the graph. After this brief introduction we move on to describing our example.
Our construction builds up on the example provided in [9] in proof of Theorem 1.2. Without loss of generality, assume that k is divisible by 3 and that l := k 2 3 k is even. Their graph is a complete bipartite graph G = (Q, U, E), where one side of the graph consists of the k terminals Q = {q 1 , . . . , q k }, and the other side of the graph consists of l = k 2 3 k non-terminals U = {u S 1 , . . . , u S l }, with S 1 , . . . , S l denoting the different subsets of terminals of size 2/3k. The costs of the edges of G are as follows: every non-terminal u S i is connected by edges of cost 1 to every terminal in S i , and by edges of cost 2 + to every terminal in S i = Q \ S i , for = 1/k. We modify the cost function defined in this example by multiplying each edge cost by a constant α that we define later. We also need to be more careful with . We set = 3 k + 6 k 2 . In addition to that we build a third layer of m = l l/2 vertices W = {w Z 1 , . . . , w Zm }, where Z 1 , . . . , Z m denote different subsets of U of size l/2. There is a complete bipartite graph between U and W . An edge {u S i , w Z j } has cost 0 if u S i ∈ Z j and has cost 1 otherwise. We add one more vertex to the graph which we refer to as x and connect it with edges of cost l/2 − 1 to each vertex in W . Let us refer to the resulting graph as G . We let Q = Q ∪ {x} be the corresponding terminal set; a set S ⊆ Q is important if x / ∈ S and |S| = 2 3 k.
Lemma 3.1. Let S i ⊂ Q be important. For α = 2 2 k · 2 k , the vertex w Z j is on the S i -side of the minimum cut between S i and S i if and only if u S i ∈ Z i .
Proof. In [8] it is proven that the unique minimum cut separating S i from S i in G, We refer to [8] for the proof details, but the reason why this holds is the following. The terminals are connected only via vertices of U . Every vertex u S j can either cut the edges E(u S j , S i ) connecting u S j with S i (choosing S i side) or cut the edges E(u S j , S i ) connecting u S j with S i (choosing S i side). For j = i it holds that c(E(U S j , S i )) = 2/3k while c(E(u S j , S i )) = (2 + )k/3, so it is better for u S i to join S i side. Moreover, the difference between these two values is greater than 1 for = 3 k + 6 k 2 . For j = i it holds that c(E(u S j , S i )) ≥ 2/3k + (1 + ) while c(E(u S j , S i )) ≤ k/3(2 + ) − (1 + ). It is easy to verify that for = 3 k + 6 k 2 it is better for u S j to join S i side and that the difference between the two alternative cut values is greater than 1. The bottom line is that u S i picks S i side, whereas Figure 4: Illustration of the construction. The two panels correspond to two cases in the proof, all other u S j vertices pick S i side. If a vertex switches sides, the value of the minimum cut increases by more than 1.
In our example we multiply all the edge weights in this example by α, so the increse in the cut value is more than α. Let us now consider graph G with terminal set Q . Consider the cut between S i and S i = Q \ S i (so S i contains x). Graph G contains G as a subgraph, so to disconnect S i from Q \ S i , each vertex u S j again has to cut either E(u S j , S i ) or E(u S j , Q \ S i ). Set α = 2 2 k · 2 k . Consider the minimum cut in G. The minimum cut in G restricted to G uses the same edges. It does not pay of to flip sides for any vertex in U , as we can never make up for the difference α with no more than |U | · |W | edges of cost 1. Now fix a vertex w Z j ∈ W . We consider two cases: u S i ∈ Z j and u S i / ∈ Z j ; see also Figure 4.
Case 1: u S i ∈ Z j .
As argued above, all vertices of U choose their side according to what is best in G, so u S i is the only vertex in U on the S i side. To join the S i side, w Z j has to cut edges {x, w Z j } and {u S i , w Z j } of total cost l/2 − 1. To join the S i side, w Z j needs to cut l/2 edges of cost 1 to vertices u S i for u S i / ∈ Z j , i = i. Thus, it is less costly if w Z j joints the S i side.
Again all vertices of U choose their side according to what is best in G, so u S i is the only vertex in U on the S i side. To join the S i side, w Z j has to cut edges {x, w Z j } and {u S i , w Z j } of total cost l/2. To join the S i side, w Z j needs to cut l/2 − 1 edges of cost 1 to vertices u S i for u S i / ∈ Z j , i = i. Thus, it is less costly for w Z j to join the S i side.
Lemma 3.1 shows that G does not compress using the technique presented in [5]. To see that let us fix two vertices w Z j and w Z j and let S i ∈ Z j \ Z j . Then, Lemma 3.1 shows that w Z j and w Z j lie on different sides of the minimum cut between S i and S i . Thus, w Z j and w Z j cannot be merged. A similar but simpler arguments show that no other pair of vertices in G can be merged, finishing the proof of Theorem 1.2.