During the past decade, there has been a flurry of works investigating the complexity of solving exactly optimization problems on planar graphs, leading to what was coined as the “square root phenomenon” by the third author [m-srppg-13]: many problems turn out to be easier on planar graphs, and the improvement compared to the general case is captured exactly by a square root. For instance, problems solvable in time in general graphs can be solved in time in planar graphs, and similarly, in a parameterized setting, FPT problems admitting -time algorithms or W-hard problems admitting -time algorithms can often be sped up to and , respectively, when restricted to planar graphs. We have many examples where matching upper bounds (algorithms) and lower bounds (complexity reductions) show that indeed the best possible running time for the problems has this form. On the side of upper bounds, the improvement often stems from the fact that planar graphs have (recursive) planar separators of size , and the theory of bidimensionality provides an elegant framework for a similar speedup in the parameterized setting for some problems [dfht-spabgg-05]. However, in many cases these algorithms rely on highly problem-specific arguments [c-mpbgg-17, mpp-spast-18, km-spktc-12, MarxP15, DBLP:conf/soda/ChitnisHM14, DBLP:conf/fsttcs/LokshtanovSW12, DBLP:conf/focs/FominLMPPS16]. The lower bounds are conditional to the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi, and Zane [IPZ98] and follow from careful reductions from problems displaying this phenomenon, e.g., Planar 3-Coloring, -Clique, or Grid Tiling. We refer to the recent book [cfklm-pa-15] for precise results along these lines.
While the theme of generalizing algorithms from planar graphs to surface-embedded graphs has attracted a lot of attention, and has flourished into an established field mixing algorithmic and topological techniques (see [c-ctgs-18]), the same cannot be said at all of the lower bounds. Actually, up to our knowledge, there are very few works explicitly establishing algorithmic lower bounds based on the genus of the surfaces on which a graph is embedded, or even just hardness results when parameterized by the genus—the only ones we are aware of are the exhaustive treatise [mp-eyawk-14] of the third author and Pilipczuk on Subgraph Isomorphism, where some of the hardness results feature the genus of the graph, the lower bounds of Curticapean and the third author [cm-tclbcp-16] on the problem of counting perfect matchings and the work of Chen et al. [ckpsx-gcccg-07].
In this work, we address this surprising gap by providing lower bounds conditioned on ETH for two fundamental yet seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. In both cases, our lower bounds match the best known algorithms up to a logarithmic factor in the exponent. We believe that the tools that we develop in this paper could pave the way towards establishing lower bounds for other problems on surface-embedded graphs.
The shortest cut graph problem.
A cut graph of an edge-weighted graph cellularly embedded on a surface is a subgraph of that has a unique face, which is a disk. Computing a shortest cut graph is a fundamental problem in algorithm design, as it is often easier to work with a planar graph than with a graph embedded on a surface of positive genus, since the large toolbox that has been designed for planar graphs becomes available. Furthermore, making a graph planar is useful for various purposes in computer graphics and mesh processing, see, e.g., [whds-reti-04]. Be it for a practical or a theoretical goal, a natural measure of the distortion induced by the cutting step is the length of the topological decomposition.
Thus, the last decade has witnessed a lot of effort on how to obtain efficient algorithms for the problems of computing short topological decompositions, see for example the survey [c-ctgs-18]. For the shortest cut graph problem, Erickson and Har-Peled [eh-ocsd-04] showed that the problem is NP-hard when the genus is considered part of the input and gave an exact algorithm running in time , where is the size of the input graph and the genus of the surface, together with an -approximation running in time . The first and fourth authors [cm-fptas-15] gave a -approximation algorithm running in time , where is some explicit computable function. Whether it is possible to improve upon the exact algorithm of Erickson and Har-Peled by designing an FPT algorithm for the problem, namely an exact algorithm running in time , has been raised by these authors [eh-ocsd-04, Conclusion] and has remained an open question over the last 17 years.
In this paper, we solve this question by proving that the result of Erickson and Har-Peled cannot be significantly improved. We indeed show a lower bound of (for the associated decision problem, even in the unweighted case) assuming the Exponential Time Hypothesis (ETH) of Impagliazzo et al. [IPZ98] (see Definition 2.1), and also prove that the problem is W-hard:
Let us consider the Shortest Cut Graph problem: Given an unweighted graph with vertices cellularly embedded on an orientable surface of genus , and an integer , decide whether admits a cut graph of length at most .
This problem is W-hard when parameterized by .
Assuming ETH, there exists a universal constant such that for any fixed integer , there is no algorithm solving all the Shortest Cut Graph instances of genus at most in time .
(In the second item, the constraint is just here to ensure that is well-defined.)
The multiway cut problem.
The second result of our paper concerns the Multiway Cut problem (also known as the Multiterminal Cut problem). Given an edge-weighted graph together with a subset of vertices called terminals, a multiway cut is a set of edges whose removal disconnects all pairs of terminals. Computing a minimum-weight multicut is a classic problem that generalizes the minimum cut problem and some closely related variants have been actively studied since as early as 1969 [Hu63]. On general graphs, while the problem is polynomial-time solvable for , it becomes NP-hard for any fixed , see [djpsy-cmc-94]. In the case of planar graphs, it remains NP-hard if is arbitrarily large, but can be solved in time , where is the number of vertices and edges of the graph [km-spktc-12], and a lower bound of was proved (conditionally on ETH) by the third author [m-tlbpmc-12]. A generalization to higher-genus graphs was recently obtained by the second author [c-mpbgg-17] who devised an algorithm running in time in graphs of genus , for some function (actually, for the more general Multicut problem). If one allows some approximation, this can be significantly improved: three of the authors recently provided a -approximation algorithm running in time [ccm-nlasm-18].
We prove a lower bound of for the associated decision problem, even in the unweighted case, which almost matches the aforementioned best known upper bound, and generalizes the lower bound of the third author [m-tlbpmc-12] for the planar case. Actually, we prove a lower bound that holds for any value of the integers and as long as . The precise theorem is the following, where we use to denote so that the quantities are well-defined for and :
Let us consider the Multiway Cut problem: Given an unweighted graph , a set of vertices, and an integer , decide whether there exists a multiway cut of of value at most .
Assuming ETH, there exists a universal constant such that for any fixed choice of integers and , there is no algorithm that decides all the Multiway Cut instances for which is embeddable on the orientable surface of genus and , in time .
Note that taking in this theorem yields lower bounds for the Planar Multiway Cut problem, and recovers, up to a logarithmic factor, the lower bounds obtained by the third author [m-tlbpmc-12] for that problem. In the opposite regime, we also prove W-hardness with respect to the genus for instances with terminals, see Proposition 4.1. We remark that corresponds to the minimum cut problem, which is polynomial-time solvable, so a lower bound on is necessary. While the last remaining case, for , is known to be NP-hard [djpsy-cmc-94], our techniques do not seem to encompass it, and we leave its parameterized complexity with respect to the genus as an open problem.
Remark: Parameterized lower bounds in the literature often have the form “assuming ETH, there is no algorithm to solve problem X, for any function ”, where is some specific dependency on the parameter. The lower bounds that we prove in Theorems 1.1 and 1.2 are instead of the form “assuming ETH, there exists a universal constant such that for any fixed , there is no algorithm to solve problem X”. The latter lower bounds imply the former: indeed, for a fixed . Our results are stronger, concerning instances for any fixed . Moreover, lower bounds with two parameters are difficult to state with notation. The statement of Theorem 1.2 handles every combination of the two parameters in a completely formal way.
Main ideas of the proof.
What is a good starting problem to prove hardness results for surface-embedded graphs? For planar graphs, the Grid Tiling problem of the third author [m-opgas-07] has now emerged as a convenient, almost universal, tool to establish parameterized hardness results and precise lower bounds based on ETH. A similar approach, based on constraint satisfaction problems (CSPs) on -dimensional grids, was used by the third author and Sidiropoulos [ms-lbldwo-14] to obtain lower bounds for geometric problems on low-dimensional Euclidean inputs (see also [bbkmz-fetalb-18] for a similar framework for geometric intersection graphs). However, these techniques do not apply directly for the problems that we consider. Indeed, the bounds implied by these approaches are governed by the treewidths of the underlying graphs and are of the type or respectively, where is the parameter of interest and the dimension of the grid in the latter case. In contrast, here, we are looking for bounds of the form (while this is not apparent from looking at Theorem 1.2, this also turns out to be the main regime of interest for the Multiway Cut problem).
Our first contribution, in Section 3, is to introduce a new hard problem for embedded graphs, which is versatile enough to be used as a starting point to obtain lower bounds for both the Shortest Cut Graph and the Multiway Cut problem (and hopefully others). It is a variant of the Grid Tiling problem which we call 4-Regular Graph Tiling; in a precise sense, it generalizes the Grid Tiling problem to allow for embedded -regular graphs different from the planar grid to be used as the structure graph of the problem. We show that a CSP instance with binary constraints can be simulated by a 4-Regular Graph Tiling instance with parameter . A result of the third author [m-cybt-10] shows that, assuming the ETH, such CSP instances cannot be solved in time , giving a similar lower bound for 4-Regular Graph Tiling (Theorem 3.1).
We then establish in Sections 4 and 5 the lower bounds for the Shortest Cut Graph and “one half” of the lower bound for Multiway Cut, namely, for the regime where the genus dominates the number of terminals. Both reductions proceed from 4-Regular Graph Tiling and use as a building block an intricate set of cross gadgets originally designed by the third author [m-tlbpmc-12] for his hardness proof of the Planar Multiway Cut problem. While it does not come as a surprise that these gadgets are useful for more general non-planar Multiway Cut instances, it turns out that via basic planar duality, they also provide exactly the needed technical tool for establishing the hardness of Shortest Cut Graph.
In order to establish the “second half” of the lower bound in Theorem 1.2, in the regime where the number of terminals dominates the genus, we use a similar strategy in Section 6 but bypass the use of the 4-Regular Graph Tiling problem. Instead, we rely directly on the aforementioned theorem of the third author on the parameterized hardness of CSPs, which we apply not to a family of expanders, but to blow-ups of expanders, i.e., expanders where vertices are replaced by grids of a well-chosen size. This size is prescribed exactly by the tradeoff between the genus and the number of terminals, as described with the two integers and in Theorem 1.2. The key property of these blow-ups is that their treewidth is and thus the lower bound on the complexity of CSPs with these blow-ups as primal graphs yields exactly the target lower bound. The reduction from CSPs to Multiway Cut is carried out in Proposition 6.1 and also relies on cross gadgets.
Note that while Theorem 1.2 does not use an embedded graph as an input, we can find an embedding of a graph on a surface with minimum possible genus in time [kmr-sltaeg-08, m-ltaega-99]. Thus, the same hardness result holds in the embedded case and the question is not about whether we are given the embedding or not.
Graphs on surfaces.
For extensive background on graphs on surfaces, we refer to the classic textbook of Mohar and Thomassen [mt-gs-01]. Throughout this article, we only consider surfaces that are compact, connected, and orientable. By the classification theorem of surfaces, each such surface is homeomorphic to a sphere with handles attached and disks removed; is called the genus of the surface and its number of boundaries. A path, or curve, is a continuous map from to . A path is simple if it is injective.
An embedding of on is a crossing-free drawing of on , i.e., the images of the vertices are pairwise distinct and the image of each edge is a simple path intersecting the image of no other vertex or edge, except possibly at its endpoints. When embedding a graph on a surface with boundaries, we adopt the convention that while vertices can be mapped to a boundary, interiors of edges can not. The genus of a graph is the smallest genus of a surface into which it can be embedded. A face of the embedding is a connected component of the complement of the graph. A cellular embedding is an embedding of a graph where every face is a topological disk. By a slight abuse of language, we will often identify an abstract graph with its embedding. If is a graph embedded on , the surface obtained by cutting along is the disjoint union of the faces of , it is an (a priori disconnected) surface with boundary.
To a graph cellularly embedded on , one can naturally associate a dual graph embedded on , whose vertices are the faces of and two such vertices are connected by an edge for every edge their dual faces share; crosses and no other edge of .
The Exponential Time Hypothesis.
Our lower bounds are conditioned on the Exponential Time Hypothesis (ETH), which was conjectured in [IPZ98].
Conjecture 2.1 (Exponential Time Hypothesis [Ipz98]).
There exists a positive real value such that 3-CNF-SAT, parameterized by , has no -time algorithm (where denotes the number of variables and denotes the number of clauses).
We refer to the survey [lms-lbbeth-13] for background and discussion of this conjecture.
Expanders and their treewidth.
For a graph , we denote by
the second largest eigenvalue of its adjacency graph. A family of-regular expanders is an infinite family of -regular graphs such that for some constant . A family of graphs is dense if for any , there exists a graph in with vertices (where the hides a universal constant).
There exists a dense family of bipartite four-regular expanders.
This lemma can be proved using a well-known simple probabilistic argument showing that random bipartite regular graphs are expanders, or with more intricate explicit constructions. We refer to the survey of Hoory, Linial, and Wigderson [hlw-ega-06] or the groundbreaking recent works of Marcus, Spielman, and Srivastava [mss-if1br-15].
The treewidth of a graph is a parameter measuring quantitatively how close it is to a tree. Since we will use this parameter in a black-box manner and not rely precisely on its definition, we do not include it here and refer to graph theory textbooks, e.g., Diestel [d-gt-00, Chapter 12].
Every -regular graph satisfies .
Proof of Lemma 2.3.
We have , where is the (outer) vertex expansion of , by Grohe and the third author [grohe2009tree, Proposition 1]. Moreover, , where denotes the edge expansion of . Finally, the “easy direction” of the Cheeger inequality gives , see for example Chung [c-sgt-97, Chapter 2]. ∎
Constraint satisfaction problems.
A binary constraint satisfaction problem is a triple where
is a set of variables,
is a domain of values,
is a set of constraints, , which are all pairs , where is a pair of variables called the scope, and is a subset of called the relation.
All the constraint satisfaction problems (CSPs) in this paper will be binary, and thus we will omit the adjective binary. A solution to a constraint satisfaction problem instance is a function such that for each constraint with , the pair is a member of . An algorithm decides a CSP instance if it outputs true if and only if that instance admits a solution.
The primal graph of a CSP instance is a graph with vertex set such that distinct vertices are adjacent if and only if there is a constraint whose scope contains both and .
The starting points for the reductions in this paper are the following two theorems, which state in a precise sense that the treewidth of the primal graph of a binary CSP establishes a lower bound on the best algorithm to decide it.
Theorem 2.4 ([gss-wecqt-01, g-chcsps-07]).
Let be an arbitrary class of graphs with unbounded treewidth. Let us consider the problem of deciding the binary CSP instances whose primal graph, , lies in . This problem is W-hard parameterized by the treewidth.
Theorem 2.5 ([m-cybt-10]).
Assuming ETH, there exists a universal constant such that for any fixed primal graph such that , there is no algorithm deciding the binary CSP instances whose primal graph is in time .
The first theorem is due to Grohe et al. [gss-wecqt-01] (see also Grohe [g-chcsps-07]). The second one follows from the work of the third author [m-cybt-10]. Since this statement differs from the main theorem of [m-cybt-10], we now explain how to prove it.
We first need to recall some definitions and results from [m-cybt-10]. A graph is a minor of if can be obtained from by a sequence of vertex deletions, edge deletions, and edge contractions. Equivalently, a graph is a minor of if there is a minor mapping from to , which is a function satisfying the following properties: (1) is a connected vertex set in for every , (2) for every , and (3) if , then there is an edge of intersecting both and . Given a graph and an integer , we denote by the graph obtained by replacing every vertex with a clique of size and replacing every edge with a complete bipartite graph on vertices. The main combinatorial result of [m-cybt-10] is the following embedding theorem:
Theorem 2.6 ([m-cybt-10]).
There are computable functions , , and a universal constant such that for every , if is a graph with and is a graph with and no isolated vertices, then is a minor of for . Furthermore, such a minor mapping can be found in time .
We will also need the following (fairly straightforward) reductions from [m-cybt-10]:
Given an instance of 3SAT with variables and clauses, it is possible to construct in polynomial time an equivalent CSP instance with variables, binary constraints, and domain size .
Assume that is a minor of . Given a binary CSP instance with primal graph and a minor mapping from to , it is possible to construct in polynomial time an equivalent instance with primal graph and the same domain.
Given a binary CSP instance with primal graph (where has no isolated vertices), it is possible to construct (in time polynomial in the size of the output) an equivalent instance with primal graph and .
With these tools at hand, we are ready to prove Theorem 2.5.
Proof of Theorem 2.5.
Let be the universal constant from Conjecture 2.1 and let be the universal constant from Theorem 2.6. We define . Suppose that some graph violates the statement of the theorem and there is an algorithm deciding CSP instances whose primal graph is in time , where . We show that algorithm can be used to solve 3SAT in a running time that violates Conjecture 2.1.
Consider an instance of 3SAT with variables and clauses. Using Lemma 2.7, we construct an instance of CSP with variables, binary constraints, and domain size 3. Let be the primal graph of . If , then we can solve in time. Otherwise, by Theorem 2.6, graph is a minor of for , and the minor mapping can be found in time . Therefore, by Lemma 2.8, can be turned into an instance with primal graph and domain size 3, which, by Lemma 2.9, can be turned into an instance with primal graph and domain size . The assumed algorithm can solve in time
violating Conjecture 2.1. ∎
We rely extensively on the following intricate family of gadgets introduced by the third author in his proof of hardness of Planar Multiway Cut [m-tlbpmc-12], which we call cross gadgets. Let be an integer. The gadgets always have the form of a planar graph embedded on a disk, with distinguished vertices on its boundary, which are, in clockwise order, denoted by
The embedding is chosen so that the boundary of the disk intersects the graph precisely in this set of distinguished vertices; the interior of the edges lie in the interior of the disk. We consider the vertices , and as terminals in that gadget, and thus a multiway cut of the gadget is a subset of the edges of such that has at least four components, and each of the terminals is in a distinct component. We say that a multiway cut of the gadget represents the pair (where, as usual, denotes the set ) if has exactly four components that partition the distinguished vertices into the following classes:
We remark that, as in the original article [m-tlbpmc-12], the notation is in matrix form. We will use the same convention throughout this paper, especially in Section 3.
The boundary of a cross gadget and a multiway cut representing a pair are pictured on Figure 1, left. The properties that we require are summarized in the following lemma:
Lemma 2.10 ([m-tlbpmc-12, Lemma 2]).
Given a subset , we can construct in polynomial time a planar gadget with unweighted edges and vertices, and an integer such that the following properties hold:
For every , the gadget has a multiway cut of weight representing .
Every multiway cut of has weight at least .
If a multiway cut of has weight , then it represents some .
Note that in [m-tlbpmc-12], the third author uses weights to define the gadgets, but as he explains at the end of the introduction, the weights are polynomially large integers and thus can be emulated with parallel unweighted edges.
In the following, we will also use the dual of the graph as one of our gadgets, yielding a dual cross gadget (see Figure 1). Its properties mirror exactly the ones of cross gadgets in a dual setting. In the dual setting, the gadget still has the form of a planar graph embedded on a disk , with distinguished faces incident to its boundary, which are, in clockwise order, denoted by
In , the vertices dual to boundary faces of lie on the boundary on the disk instead of the interior, see Figure 1, right. As above, the boundary of the disk intersects the graph precisely in the distinguished vertices.
A dual multiway cut is a set of edges such that cutting the disk along yields at least four connected components, and the four terminal faces end up in distinct components. We say that a dual multiway cut represents a pair if cutting the disk yields exactly four connected components that partition the distinguished faces into the following classes:
For convenience, we restate the content of Lemma 2.10 in the dual setting in a separate lemma.
Lemma 2.11 ([m-tlbpmc-12, Dual version of Lemma 2]).
Given a subset , we can construct in polynomial time a planar gadget with unweighted edges and vertices, and an integer such that the following properties hold:
For every , the gadget has a dual multiway cut of weight representing .
Every dual multiway cut of has weight at least .
If a dual multiway cut of has weight , then it represents some .
3 The -regular graph tiling problem
We call the two conditions above the compatibility conditions of the 4-Regular Graph Tiling instance. The graph in the input is allowed to have parallel edges. It is easy to see that the Grid Tiling problem [m-opgas-07] is a special case of 4-Regular Graph Tiling.
In this section, we prove a larger lower bound for this more general problem: we prove an lower bound, conditionally to ETH, for 4-Regular Graph Tiling, even when the problem is restricted to bipartite instances and when fixing . We also show that it is W-hard when parameterized by the integer (even for bipartite instances). Precisely:
The 4-Regular Graph Tiling problem restricted to instances whose underlying graph is bipartite, parameterized by the integer , is W-hard.
Assuming ETH, there exists a universal constant such that for any fixed integer , there is no algorithm that decides all the 4-Regular Graph Tiling instances whose underlying graph is bipartite and has at most vertices, in time .
The analogous result for Grid Tiling by the third author [m-opgas-07] embeds the -Clique problem in a grid. Here we start from a hardness result for 4-regular binary CSPs that follows from Theorem 2.5 and directly represent the problem as a 4-Regular Graph Tiling instance by locally replacing each variable and each binary constraint in an appropriate way.
In the proof, we will use the well-known fact that a -regular bipartite graph can be properly edge-colored with colors. This is proved by induction on : The case is trivial; in general, take a perfect matching of , which exists by Hall’s marriage theorem; color the edges with color ; the subgraph of made of the uncolored edges satisfies the induction hypothesis with , so it admits a proper edge-coloring with colors; thus has a proper edge-coloring with colors. This also implies that computing such a proper edge-coloring takes polynomial time.
The proof of the theorem proceeds by a reduction from the binary CSP instances involved in Theorems 2.4 and 2.5. Starting from a binary CSP instance whose primal graph is , a -regular bipartite graph, we define an instance of 4-Regular Graph Tiling, , in the following way.
We set and .
We find a proper edge coloring of with colors, as indicated above.
Denoting by and the two subsets of vertices of corresponding to the bipartition of , for each vertex of , we create four vertices in which we connect in a cycle in this order using two and two edges. Similarly, for each vertex of , we create four vertices in which we connect in a cycle in this order using two and two edges.
For each edge labeled with a color , where and , we create one vertex in , which is connected to via two edges, one labeled and one labeled , and to via two edges, one labeled and one labeled .
For each vertex or of coming from a vertex of , the corresponding subset or is set to be .
For each vertex of coming from an edge of , where and , the corresponding subset is set to be the relation corresponding to .
See Figure 2 for an illustration of this reduction. We claim that the graph
is bipartite: The bipartition is obtained by picking for one side the odd-numberedand vertices and the vertices for labeled by an even color, and for the other side the even-numbered and vertices and the vertices for labeled by an odd color. It follows from the construction that this is a bipartition.
We claim that this instance of 4-Regular Graph Tiling is satisfiable if and only if is satisfiable. Indeed, if is satisfiable, the truth assignment for can be used to find the values for the in the following way. For a vertex or of coming from a vertex of , the value or can be chosen to be . For a vertex of coming from an edge of where and , the value of can be chosen to be . The compatibility conditions are trivially fulfilled. In the other direction, the values for the four vertices of coming from a vertex of are identical and of the form . Choosing as the truth assignment for in yields a solution to the CSP .
We thus have a linear-time reduction from binary CSP, restricted to instances whose primal graph has vertices, is four-regular and is bipartite, to instances of 4-Regular Graph Tiling on a bipartite graph with vertices. Combined with Theorem 2.4 applied to the infinite family of four-regular bipartite expanders output by Lemma 2.2 and Lemma 2.3 relating their treewidth to their number of vertices, this proves the first item of the theorem. For the second item, we fix an integer ; by Lemma 2.2, there exists a constant so that if , there exists a four-regular bipartite expander with expansion constant , and with at least and at most vertices. We set to be equal to , where is the constant of Theorem 2.5. If is smaller than , such an expander may not exist in , but since , the trivial linear lower bound for the -Regular Graph Tiling problem, which holds for any , is enough to conclude. If is at least , observing that the polynomial-time reduction blows up the number of vertices by , we have that an algorithm deciding all the -Regular Graph Tiling bipartite instances with at most vertices in time would decide binary CSP instances whose primal graph is in time
Remark: It might seem more natural to use a definition of 4-Regular Graph Tiling where half-edges are labeled by and , so that every edge contains either and , or and labels. This fits more the intuition that the top side of a vertex should be attached to the bottom side of the next vertex. It follows from roughly the same proof that the same hardness result also holds for that variant. However, it seems that both the bipartiteness and the unusual labeling are required for the reduction in Section 4.
4 Multiway cut with four terminals
In this section, we prove the following proposition, which will yield Theorem 1.2 in the regime where the genus dominates the number of terminals.
The Multiway Cut problem when restricted to instances in which and is embeddable on the surface of genus is W-hard parameterized by .
Assuming ETH, there exists a universal constant such that for any fixed integer , there is no algorithm that decides all the Multiway Cut instances for which is embeddable on the surface of genus and , in time .
The idea is to reduce 4-Regular Graph Tiling instances of Theorem 3.1 to the instances of Multiway Cut specified by the proposition. Consider an instance of 4-Regular Graph Tiling where the underlying graph is bipartite and has at most vertices ( being arbitrary for now). In polynomial time, we transform it into an equivalent instance of Multiway Cut as follows.
To each vertex of corresponds a cross gadget where and the subset is chosen to be .
For each edge of labeled , we identify the vertices of the side of the cross gadget to the corresponding vertices of the side of the cross gadget . Similarly for the edges labeled , , and for which the vertices on the , , and sides, respectively, are identified. Note that only vertices, and not edges, are identified.
The four corner vertices , and of all the cross gadgets are identified in four vertices , and , where the four terminals are placed.
Note that since the sides are consistently matched in this last step, the four terminals remain distinct after this identification.
We claim that this instance admits a multiway cut of weight at most (where is the integer from Lemma 2.10) if and only if the 4-Regular Graph Tiling instance is satisfiable. Assume first that the 4-Regular Graph Tiling instance is satisfiable. For each vertex of , one can use the value to choose, using Lemma 2.10(1), a multiway cut in representing . We claim that the construction ensures that taking the union of all these sets of edges forms a multiway cut separating the four terminals in . Indeed, after removing the multiway cuts, the four terminals lie in four different components in each of the cross gadgets. This remains the case after identifying the four sides: consider two cross gadgets that have two sides identified; let be a vertex on that common side; then, by the compatibility conditions in the definition of 4-Regular Graph Tiling, is connected, in the first gadget, to a terminal (, , , or ) if and only if it is connected to the corresponding terminal in the second gadget. The multiway cut has weight at most , since it is the union of edge sets of weight at most .
For the other direction, we first observe that if the instance admits a multiway cut of weight at most , then each of the cross gadgets must admit a multiway cut (otherwise the four terminals would not be disconnected). By Lemma 2.10(2), each of these multiway cuts has weight exactly . Therefore, by Lemma 2.10(3), each of them represents some , which will be used as the value for the 4-Regular Graph Tiling instance. Furthermore, we claim that the multiway cuts need to match along identified sides, i.e., if a multiway cut represents the pair , then a multiway cut in a cross gadget adjacent along an edge labeled or needs to represent a pair for some , and similarly a multiway cut in a cross gadget adjacent along an edge labeled or needs to represent a pair for some , for otherwise the four terminals are not separated. Indeed, if, say, a multiway cut representing the pair is connected along an edge labeled to a multiway cut representing the pair for , there is a path connecting the terminals and , as pictured in Figure 3, contradicting the fact that we have a multiway cut. Therefore, the compatibility conditions of the 4-Regular Graph Tiling instance are satisfied.
The genus of the graph is .
Proof of Claim 4.2.
We prove here that the genus of the graph is . For this, the fact that is bipartite turns out to be crucial. For Step 1 above, let us embed the cross gadgets corresponding to in the plane. Let be the bipartition of the vertices of . We embed the cross gadgets corresponding to in the plane with the natural orientation (, , , in clockwise order), and the cross gadgets corresponding to with the opposite orientation. Second, let us connect the vertices on the sides of the cross gadgets as in Step 2 above; but for now, just for clarity of exposition, instead of identifying pairs of vertices, let us connect each pair by a new edge. We can add these new edges corresponding to a single edge of by putting them on a ribbon connecting the sides of the cross gadgets (see Figure 4). We emphasize that, because of the orientation chosen to embed the gadgets corresponding to and , the ribbons are drawn “flat” in the plane (though possibly with some overlapping between them), and the vertices in one cross gadget are connected to the corresponding vertices in the other cross gadget (for example, in the case of an edge labeled , vertex in the first gadget is connected to vertex in the second gadget). Thus, since we started with a graph embedded on the plane and added at most “flat” ribbons, we obtain a graph embedded on an orientable surface with genus at most (without boundary, after attaching disks to each boundary component). We now contract every newly added edge, which can only decrease the genus. For Step 3 above, the graph is obtained by identifying four groups of at most vertices of the previous graph (the terminals of all cross gadgets, and similarly for , , and ) into four vertices; these vertex identifications increase the genus by . (To see this, we can for example add edges to connect in a linear way all the vertices to be identified, which increases the genus by , and then contract these new edges.) This proves that is embeddable on a surface of genus . ∎
To summarize: Given an instance of 4-Regular Graph Tiling where the underlying graph is bipartite and has at most vertices, for an arbitrary , we can transform it in polynomial time into an equivalent instance of Multiway Cut with four terminals and whose graph has at most vertices and edges and is embeddable on a surface of genus at most , for some universal constant , where the polynomial is inherited from Lemma 2.10.
Combined with Theorem 3.1(1), this proves the first item. For the second one, for a given choice of , we pick . If , setting where is the degree of the polynomial and combining Theorem 3.1(2) with this reduction proves the second item of the theorem. Otherwise, as in the proof of Theorem 3.1, choosing to be smaller than and using the trivial linear lower bound suffices to conclude. ∎
5 Shortest cut graph
In this section, we prove Theorem 1.1 on the hardness of the Shortest Cut Graph problem.
The idea is to reduce 4-Regular Graph Tiling instances of Theorem 3.1 to instances of Shortest Cut Graph. Let be a bipartite four-regular graph with vertices.
From , we build a surface as follows (see Figure 5): We build one cylindrical tube for each edge of and one sphere minus four disks for each vertex of , attaching them in the natural way to obtain an orientable surface. By Euler’s formula, this surface has genus . Moreover, the graph is naturally embedded in , though not cellularly. In order to have a cellular embedding, and actually a cut graph, we transform as follows. Let be a spanning tree of . Let be the graph obtained from by subdividing each edge not in into two edges, and adding a loop in the middle vertex. Now, embed into in the natural way: Starting from the embedding of into , put each middle vertex on the corresponding cylindrical tube of , and make the corresponding loop go around the tube. The resulting graph is a cut graph of (indeed, it has a single face, because we only add loops in the middle of edges not in the spanning tree ; has vertices and edges (being four-regular); so its unique face is a disk, by Euler’s formula).
Let be the bipartition of the vertices of . We note that the above construction is possible while enforcing an arbitrary cyclic ordering of the edges incident to each vertex of ; we do it in a way that the cyclic ordering of the edges around each vertex in is the standard one (, , , in clockwise order), while the cyclic ordering around each vertex in is reversed (, , , in clockwise order). We now build a graph embedded on the same surface as follows, obtained by replacing each vertex of with a dual cross gadget and by (almost) identifying vertices on the corresponding sides of adjacent gadgets. In detail:
To each vertex of corresponds a dual cross gadget where and the subset is chosen to be . We embed that dual cross gadget with the same orientation as the corresponding vertex of .
For each edge of , we identify the vertices (not the edges) on the side of corresponding to the label of to the vertices on the same side of . By the choice of the rotation systems, and for the same reason as in Figure 4, this identifies the vertices in the gadget associated with to the corresponding vertices in the gadget associated with ; for example, if the label of edge is , the vertex of the first gadget is associated to vertex of the second gadget).
For an edge of not in , we use another dual cross gadget , for which we choose to be the unconstrained relation . We put that gadget on the vertex of . We identify the vertices on the side of corresponding to the label of to the vertices of the same side of , and similarly the vertices on the side of corresponding to the label of to the vertices on the opposite side of . The two opposite sides of which are not yet identified are identified to each other.
The following claim, whose proof is deferred to the end of the section, shows that the reduction works as expected.
The embedded graph admits a cut graph of weight at most if and only if the 4-Regular Graph Tiling instance on is satisfiable.
To summarize: Given an instance of 4-Regular Graph Tiling where the underlying graph is bipartite and has vertices, for an arbitrary , we can transform it in polynomial time into an equivalent instance of Shortest Cut Graph whose graph has vertices and edges, embedded on a surface of genus . Combined with Theorem 3.1(1), this proves the first item of the theorem. For the second item, for any choice of integer , we choose , and the above reduction, combined with Theorem 3.1(2), finishes the proof for (where is the degree of the polynomial of Lemma 2.11). For , we set and conclude using the trivial linear lower bound. ∎
To conclude the proof, there just remains to prove Claim 5.1.
Proof of Claim 5.1..
We claim that the embedded graph admits a cut graph of size at most if and only if the 4-Regular Graph Tiling instance is satisfiable. (Recall that has vertices.) Let us first assume that the 4-Regular Graph Tiling instance is satisfiable. For each vertex of , one can use the value to choose, using Lemma 2.11(1), a dual multiway cut in representing . By construction, the union of all these dual edges contains a cut graph of . Indeed, the union of all these edges contains a subdivision of the graph described above (possibly after transforming a degree-four vertex into two degree-three vertices connected by an edge), which is a cut graph. Furthermore, this union has weight , since each of the dual multiway cuts has weight . Thus, some subgraph of it, of weight at most , is a cut graph.
The reverse direction requires more effort. First, let us call a cut graph reduced if it has no degree-one vertex. We will use the following fact: Every simple closed curve crossing some reduced cut graph exactly once is non-contractible. Indeed, if is contractible, it bounds a disk; since the cut graph intersects the boundary of the disk exactly once, the part of the cut graph inside the disk is a tree, and thus has a degree-one vertex; this is a contradiction.
Assume that the instance admits a cut graph of weight at most . Note that the edge set of is the disjoint union of the edges of the dual cross gadgets, and one can naturally talk about the restriction of (the set of edges of) to a given dual cross gadget (before the identification of the vertices of the dual cross gadgets). We first prove that the restriction of to each of the dual cross gadgets contains a dual multiway cut. Indeed, assume that there is a path in the gadget connecting two terminal faces in a gadget without crossing . Let be a path with the same endpoints as and that does not enter the gadgets; such a path exists because the union of the gadgets is a thickened version of the graph , which is a cut graph, and which thus has a single face. The closed curve that is the concatenation of and is contractible, because it does not cross . On the other hand, we can slightly modify in the gadget to obtain a reduced cut graph that crosses , and thus also , exactly once (which, by the previous paragraph, implies that is contractible, and thus the contradiction). To prove this fact, there are two cases. If connects two “neighboring” terminals in , e.g., and , then we locally homotope to the side of the gadget, except for the edge that crosses the side, and we draw that latter edge in a way that it crosses exactly once. If connects two “opposite” terminals in , e.g., and , then we first replace the four-valent vertex of in the gadget with two three-valent vertices and , in a way that is connected to the and sides and to , and similarly is connected to the and sides and to . We put in the corner, in the corner, and connect them by an edge crossing exactly once. Thus, in both cases, we have a reduced cut graph that crosses , and thus , exactly once, as desired.
Thus, every dual cross gadget or is a dual multiway cut representing some , by Lemma 2.10(2) and (3). Furthermore, the dual multiway cuts must match on the boundaries, i.e., if a dual multiway cut represents the pair , the multiway cut in a cross gadget adjacent along an edge labeled or needs to represent a pair for some , and similarly the dual multiway cut in a cross gadget adjacent along an edge labeled or needs to represent a pair for some . Indeed, otherwise, there is a path connecting two terminal faces, as pictured in Figure 3, and by the same argument as above, this path can be completed into a non-contractible cycle not crossing , which is a contradiction.
Since each dual cross gadget has to represent some , we use as the value of for the 4-Regular Graph Tiling instance. The compatibility conditions follow from the fact that the dual multiway cuts match on the boundaries; thus, the 4-Regular Graph Tiling instance is satisfiable. ∎
6 Multiway cut with a large number of terminals
The goal of this section is to prove the following proposition, which will yield Theorem 1.2 in the regime where the number of terminals dominates the genus. Recall that denotes .
Assuming ETH, there exists a universal constant such that for any fixed choice of integers and , there is no algorithm that decides all the Multiway Cut instances for which is embeddable on the surface of genus and , in time .
In order to prove this proposition, we will blow up expander graphs by replacing their vertices by grids, a construction similar to one in Gilbert, Hutchinson and Tarjan [ght-stgbg-84]. Given a four-regular graph cellularly embedded on a surface , we define in the following way (see Figure 6). We replace each vertex of with a -grid. The boundary of the grid is thus made of vertices, which we divide into four contiguous segments of vertices starting from a corner, in clockwise order along the boundary of the grid (corners appear in two segments). For each edge of between two adjacent vertices and , we blow up in the following way: we place an edge between the th vertex of a segment of and the st vertex of a segment of . The segments are selected in such a way that the cyclic ordering of the four blown-up half-edges around a grid replacing a vertex corresponds to the cyclic ordering of the four half-edges around that is prescribed by the cellular embedding. This new graph is also cellularly embedded on , since the embedding of can be “thickened” to an embedding of . Moreover, clearly has vertices.
The properties of that we are interested in are summarized in the following lemma.
Let be a four-regular graph cellularly embedded on a surface , such that for some universal constant , and let be a positive integer. The graph can also be cellularly embedded on , has vertices and treewidth .
For , an -separator of a graph is a subset of vertices of such that each connected component of has a fraction at most of the vertices of . To prove the previous lemma, we will use the following auxiliary result.
Let be the -grid where , and let be given. There exists , depending only on and , such that any -separator of has more than vertices.
Let . Let be a subset of vertices of of size at most . Thus, lies in at most rows and columns, which implies that there is a connected component of of size at least . Thus, is not an -separator. ∎
Proof of Lemma 6.2.
Following the construction, there just remains to prove that . The approach mimicks the lower bound proof of Gilbert et al. [ght-stgbg-84] and adapts it to the case of four-regular graphs. Their initial proof is for three-regular graphs. Let . For each vertex of the original graph , we denote by the associated -grid in .
Most of the proof consists in proving that if