Minimum bounded chains and minimum homologous chains in embedded simplicial complexes

03/05/2020 ∙ by Glencora Borradaile, et al. ∙ 0

We study two optimization problems on simplicial complexes with homology over ℤ_2, the minimum bounded chain problem: given a d-dimensional complex 𝒦 embedded in ℝ^d+1 and a null-homologous (d-1)-cycle C in 𝒦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold ℳ and a d-chain D in ℳ, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(√(logβ_d))-approximation algorithm for the minimum bounded chain problem where β_d is the dth Betti number of 𝒦. Finally, we provide an O(√(log n_d+1))-approximation algorithm for the minimum homologous chain problem where n_d+1 is the number of d-simplices in ℳ.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Simplicial complexes are best known as a generalization of graphs, but have more structure than other generalizations such as hypergraphs. Despite the structure, simplicial complexes are sufficiently expressive to make many algorithmic questions computationally intractable. For example, the generalization of shortest path that we examine in this work is NP-hard in 2-dimensional simplicial complexes [14]. Since planar graphs (1-dimensional simplicial complexes embeddable in ) exhibit structure that is algorithmically useful, resulting in more efficient or more accurate algorithms than for general graphs, we ask whether 2-dimensional simplicial complexes that are embeddable in (and more generally, -simplices emdeddable in ) also have sufficient structure that can be exploited algorithmically. To this end, we examine the algebraic generalization of the shortest path problem in graphs to simplicial complexes of higher dimension. This restriction via embedding in Euclidean space would still result in a useful algorithmic tool, given the connection of embedded simplicial complexes to meshes arising from physical systems.

Formally we study the minimum bounded chain problem which is the algebraic generalization of the shortest path problem in graphs [21]. The goal of the minimum bounded chain problem is to find a subcomplex whose boundary is a given input cycle . More precisely: Given a -dimensional simplicial complex and a null-homologous -dimensional cycle , find a minimum-cost -chain whose boundary . The requirement that the cycle be null-homologous is necessary and sufficient for the existence of a solution and we study the problem in the context of -homology.111Formal definitions are presented in Section 2. In -homology, a -chain is a subset of -simplices of the simplicial complex. We see this as a generalization of the shortest path problem in graphs as follows: Let be a one dimensional simplicial complex (i.e. a graph). A pair of vertices in the same connected component, and , is a null-homologous -chain and the minimum 1-chain whose boundary is is the shortest -path. Grady has written on why this generalization is useful in the context of 3D graphics [16].

The minimum bounded chain problem is closely related to the minimum homologous chain problem which asks: given a -chain , find a minimum-cost -chain such that the symmetric difference of and form the boundary of a -chain. Alternatively, is the minimum-cost -chain that is homologous to . Dunfield and Hirani [14] show the minimum bounded and homologous chain problems are equivalent under additional assumptions. We study the minimum homologous chain problem for -chains in -manifolds.

1.1 Our results

We present approximation and fixed-parameter tractable algorithms for the minimum bounded chain and the minimum homologous chain problem. In this paper we consider both problems in the context of simplicial homology over . We denote by the number of -simplices of the -dimensional simplicial complex .

Theorem 1.1.

There exists an -approximation algorithm for the minimum bounded chain problem for a simplicial complex embedded in , with th Betti number .

Theorem 1.2.

There exists an time exact algorithm for the minimum bounded chain problem for simplicial complexes embedded in , where is the number of -simplices in the optimal solution.

Theorem 1.3.

There exists an -approximation algorithm for the minimum homologous chain problem for -chains in -manifolds.

Theorem 1.4.

There exists an time exact algorithm for the minimum homologous chain problem for -chains in -manifolds, where is the size of the optimal solution.

The running times for the first two theorems is computed assuming that the dual graph of the complex in is available. The last two theorems hold, more generally, for weak pseudomanifolds studied by Dey et al. in [12].

On the hardness side, we show that constant factor approximation algorithms for these problems (minimum bounded chain and minimum homologous chain) are unlikely.

Theorem 1.5.

The minimum bounded chain problem is

  1. hard to approximate within a factor for some assuming , and

  2. hard to approximate within any constant factor assuming the unique games conjecture,

even if is a -dimensional simplicial complex embedded in with input cycle embedded on the boundary of the unbounded volume in .

Theorem 1.6.

The minimum homologous chain problem is

  1. hard to approximate within a factor for some assuming , and

  2. hard to approximate within any constant factor assuming the unique games conjecture,

even when the input chain is a -cycle on an orientable -manifold.

For the sake of completeness, we also give a more general presentation of the result of Kirsanov and Gortler [21], that minimum bounded chain is polynomial time solvable for a -dimensional simplicial complex embedded in and input chain null-homologous on the boundary of the unbounded region in . This can be found in Appendix 5. This algorithmic result is likely to be the most general possible, given Theorem 1.5.

1.2 Related Work

Chain problems over

Research on the minimum bounded chain problem is limited to the case of

-homology, where linear programming techniques can be employed algorithmically. Sullivan described the problem as the discretization of the minimal spanning surface problem 

[27] with Kirsanov reducing the problem to an instance of minimum cut in the dual graph [21]. Sullivan’s work is on the closely related cellular complexes, but under the same restrictions are we study (embedded in ) and Kirsanov studies the problem in embedded simplicial complexes.

Likewise, research on minimum homologous chain has largely worked in -homology. Dey, Hirani and Krishnamoorthy formulate the minimum homologous chain problem over as an integer linear program and describe topological conditions for the linear program to be totally unimodular (and so, poly-time solvable) [11]. Of course, integer linear programming approaches do not extend to -homology.

This linear programming approach was then applied to the minimum bounded chain problem (over ) by Dunfield and Hirani [14]. Moreover, they show the minimum bounded chain problem is NP-complete via a reduction from 1-in-3 SAT. The gadget they use was originally used by Agol, Hass and Thurston to show that the minimal spanning area problem is NP-complete [2].

Chain problems over

Special cases of the minimum homologous chain problem have been studied in homology. The homology localization problem is the case when the input chain is a cycle. The homology localization problem over in surface-embedded graphs is known to be NP-hard via a reduction from maximum cut by Chambers et al. [7]; our reduction is from the complement problem minimum uncut. On the algorithmic side, Erickson and Nayyeri provide a time algorithm where is the genus of the surface [15]. Using the idea of annotated simplices, Busaryev et al. generalize this algorithm for homology localization of -cycles in simplicial complexes; the algorithm runs in time where is the exponent of matrix multiplication, and is the first homology rank of the complex [6].

Using a reduction from the nearest codeword problem Chen and Freedman showed that homology localization with coefficients over is not only NP-hard, but it cannot be approximated within any constant factor in polynomial time [9]. These hardness results hold for a 2-dimensional simplicial complex, but not necessarily for 2-dimensional complexes embedded in . They also give a polynomial-time algorithm for the special case of -dimensional simplicial complex that is embedded in . (This is different from our setting of a -dimensional simplicial complex that is embedded in ; however the algorithm also reduces to a minimum cut problem in a dual graph, much like that of Kirsanov and Gortler.)

Algebraic formulations

The minimum bounded chain problem over

can be stated as a linear algebra problem, but this has little algorithmic use since the resulting problems are intractable. The algebraic formulation is to find a vector

of minimum Hamming weight that solves an appropriately defined linear system . (It is possible to reduce in the reverse direction, but the resulting complex is not embeddable in general, and so provides no new results.)

In coding theory this algebraic problem is a well studied decoding problem known as maximum likelihood decoding, and it was shown to be NP-hard by Berlekamp, McEliece and van Tilborg [4, 28]. Downey, Fellows, Vardy and Whittle show that maximum likelihood decoding is W[1]-hard [13]. Further, Austrin and Khot show that maximum likelihood decoding is hard to approximate within a factor of under the assumption that  [3]. This work was continued by Bhattacharyya, Gadekar, Ghosal and Saket who showed that maximum likelihood decoding is still W[1]-hard when the problem is restricted to sized matrices for some constant  [5].


Paper organization.

In Section 2, we give formal definitions for the paper. In Section 3, we present our approximation algorithms and fixed parameter tractable algorithms. In Section 4, we present our hardness results.

2 Preliminaries

Simplicial complexes

Given a set of vertices we define an abstract simplicial complex to be a subset of the power set of such that the following property holds: if and then . We call any a simplex and define the dimension of to be if we call a -simplex. Further, we call 0-simplices, 1-simplices, and 2-simplices vertices, edges, and triangles. We define the dimension of to be equal to the largest dimension of any simplex in . If has dimension we refer to as a -simplicial complex or -complex. We refer to any subset of a -simplex as a face of . If is a face of with dimension we refer to as a facet of .

Homology

In this paper we work in simplicial homology with coefficients over the finite field . Here we briefly define the concepts from homology that will be used throughout this paper. We assume familiarity with the basics of algebraic topology, and refer the reader to standard references [17, 23] for the details.

Given a simplicial complex we define the th chain group of to be the free abelian group, with coefficients over , generated by the -simplices in . We denote the chain group as and note that its elements are expressed as formal sums where and is a -simplex. We call the elements of the chain group chains or more specifically -chains. When working over there is a one-to-one correspondence between -chains and sets of -simplices in . It follows that adding two -chains over is the same thing as taking the symmetric difference of their corresponding sets. Hence, we use the notation to denote the sum of two -chains. By abuse of notation we will also use to denote the symmetric difference of sets, but the context should always be clear.

For a -simplex we define its boundary to be the sum of the -simplices contained in . We extend this operation linearly to obtain the boundary operator on chain groups, . We will often drop the subscript when the context is clear. Note that the composition is always equal to the zero map. If we say that is bounded by . We call a chain a cycle if .

By we denote the th cycle group of . This is subgroup of generated by the -simplices in . Similarly, by we denote the th boundary group of , which is the subgroup of generated by the -simplices in . Since we have that is a subgroup of . We define the th homology group of , denoted , to be the quotient group . The th Betti number of , denoted , is defined to be the dimension of . We call a -chain null-homologous if it is a boundary, that is . Further, we call two -chains and homologous if their difference is a boundary, that is .

Embeddings and duality

Given a -complex an embedding of is a function such that restricted to any simplex in is an injection. Further, for any two simplices we require that . That is, the images of two simplices only intersect at their common faces. The function is an embedding of the abstract simplicial complex . In this paper we make no distinction between and an embedding of . Hence, we use the notation to refer to both and refer to as an embedded simplicial complex.

The Alexander duality theorem, a higher dimensional analog of the Jordan curve theorem, states that is partitioned into connected components. Exactly one of these connected components is unbounded, and we refer to the unbounded component as . Using this partition we define the dual graph of . has one vertex for each connected component of with the vertex corresponding to denoted by . Further, has one edge for each -simplex in . There is an edge between two vertices representing connected components and in if there is a -simplex contained in the intersection of the topological closures of and . Note that can have parallel edges and self-loops. Since each -simplex can be in the closure of at most two connected components we have a one-to-one correspondence between -simplices in and edges in . If is a set of -simplices in we denote their corresponding edges in by . Similar to planar graphs, there is a duality between -cycles in and edge cuts in . There exists a one-to-one correspondence between -cycles in and minimal edge cuts in . We refer to this correspondence as cycle/cut duality, and it will play a central role in many of our proofs.

By we denote the outer shell of . This is defined to be the subcomplex of consisting of all -simplices whose corresponding edges in are incident to . Equivalently, it is also the subcomplex of consisting of all -simplices contained in the boundary of .

We endow the geometric realization of a simplicial complex with the subspace topology inherited from . We call a -dimensional manifold if every point in its geometric realization is contained in a neighborhood homeomorphic to . If every point in the geometric realization of is contained in a neighborhood homeomorphic to either or the -dimensional half-space we call a manifold with boundary.

Graph cuts

Let be a graph. For any two subsets a -cut is a set of edges such that the graph contains no path from to . Often we will consider -cuts for some where denotes the complement of in . By we refer to the edge set corresponding to all edges that have one endpoint in and the other in , which is the minimum -cut. We extend this notation to vertices. For any two vertices an -cut refers to a set of edges whose removal disconnects from .

The minimum bounded/homologous chain problems

Now we give the formal statement of the minimum bounded chain problem. Given a -dimensional simplicial complex and a -cycle contained in the minimum bounded chain problem asks to find a -chain with such that the cost of is minimized. The cost of is given by its norm . Here we are treating as an -dimensional indicator vector where is the number of -simplices in . The simplicial complex may be weighted by assigning a real number to each -simplex in . In this case the cost of is given by , where is a vector assigning weights to the -simplices of .

Now let be a -chain, which may or may not be a cycle. The minimum homologous chain problem asks to find a minimum -chain such that for some -chain , equivalently, the minimum -chain such that is null-homologous. The cost of as well as the weighted problem are defined the same as in the previous paragraph.

In this paper, we study the minimum bounded chain problem for complexes embedded in , and the minimum homologous chain problem for -chains in -manifolds.

3 Approximation algorithm and fixed parameter tractability

In this section, we describe approximation algorithms and parameterized algorithms for both minimum bounded chain and minimum homologous chain problems. Our algorithms work with the dual graph of the input space. In order to simplify our presentation we assume that the dual graph of the input complex contains no loops. The following lemma shows that we can make this assumption without any loss of generality. The proof is in the appendix.

Lemma .

In polynomial time we can preprocess an instance of the minimum bounded chain problem into a new instance such that (i) contains no loops and (ii) an -approximation algorithm for implies an -approximation algorithm for .

Proof.

Let be the set of -simplices corresponding to the loops in . The cycle/cut duality implies that no -simplex can be on a -cycle; as a loop cannot be on any cut. Therefore, for any with boundary , is either on both of them, or on none of them. Thus, each is either on all -chains with boundary , or none of such chains. Let be the -simplices that are on all with boundary , and let be the -simplices that are on no with boundary , we have .

Now, we compute a feasible solution with by solving the linear system using standard methods [18]. Using we can partition into and : a -simplex is in if it is in , and in otherwise. We can remove from without changing the optimal solution. Further, any chain that bounds contains all . That is, we can write . It follows that

Hence, we can find the minimum chain in that bounds . Then, is the minimum bounded chain for in . Furthermore, any approximation algorithm for implies an approximation algorithm with the same ratio for . To see that, let be an approximation to in , and let . So, we have:

The second equality holds as and are disjoint from ; as they are solutions in that does not contain . The last equality holds as , , and are non-negative and . ∎

3.1 Reductions to the minimum cut completion problem

Given and , the minimum cut completion problem asks for a cut with edge set that minimizes . First, we show that the minimum cut completion problem generalizes the minimum bounded chain problem.

Lemma .

For any -dimensional instance of the minimum bounded chain problem, , there exists an instance of the minimum cut completion problem that can be computed in polynomial time, and a one-to-one correspondence between cuts in and -chains with boundary in . Moreover, if the cut with edge set in corresponds to the -chain in then .

Proof.

Let be any -chain such that , such an can be computed in polynomial time, by solving the linear system. In turn, let , and .

Now, let be any -chain such that . So, . Thus, by cycle/cut duality partitions , let be the corresponding dual cut in , and let be the edge set of this cut. We have .

On the other hand, let be a cut in , with edge set . By cycle/cut duality . Now, let . It follows that . Moreover, we have . ∎

Next, we show via a similar argument that the cut completion problem also generalizes the minimum homologous chain problem when the input complex is a weak pseudomanifold (see the appendix for the proof). A weak pseudomanifold is a pure -complex such that every -simplex is a face of at most two -simplices. Weak pseudomanifolds generalize manifolds and the definition was first introduced by Dey et al. in [12]. Although recognizing -manifolds is undecidable [10], weak pseudomanifolds can be recognized in polynomial time.

Lemma .

For any -dimensional instance of the minimum homologous chain problem , where is a weak pseudomanifold, there exists an instance of the minimum cut completion problem that can be computed in polynomial time, and a one-to-one correspondence between cuts in and -chains in that are homologous to . Moreover, if the cut with edge set in corresponds to the -chain in then .

Proof.

Let be the instance of the cut completion problem constructed as follows. For each -simplex in , we have a vertex in . Moreover, contains one extra vertex, . For each -simplex in , if is a face of two -simplices and , we add the edge to ; we call such an edge a regular edge. Otherwise, when is only adjacent to one -simplex , we add the edge to ; we call such an edge a boundary edge. Since is a -weak pseudomanifold, each -simplex is a face of either one or two -simplices. Finally, let be the edges dual to .

Now, let be a -chain in that is homologous to . Therefore, is null-homologous, that is there exists a -chain , such that . It follows that is a cut in with cost , as is dual to and is dual to .

On the other hand, let be a cut in with edge set . Thus, is null-homologous. So, is homologous to , and its cost is . ∎

3.2 Algorithms for the minimum cut completion problem

We show an -approximation algorithm and a fixed parameter tractable algorithm for the cut completion problem. We obtain both of these results via reduction to 2CNF Deletion: given an instance of 2SAT, find the minimum number of clauses to delete to make the instance satisfiable. Agarwal et al. [1] show an -approximation algorithm for 2CNF Deletion, where is the number of clauses, and Razgon and O’Sullivan show that the problem is fixed parameter tractable.

Lemma (Agarwal et al.[1], Theorem 3.1).

There is a randomized polynomial-time algorithm for finding an -approximation for the minimum disagreement 2CNF Deletion problem.

Lemma (Razgon and O’Sullivan [26], Theorem 7).

Let be an instance of 2CNF Deletion problem with clauses that admits a solution of size . There is an time exact algorithm for solving .

The next lemma shows similar results for the cut completion problem.

Lemma .

For the cut completion problem ,

  1. there is a randomized polynomial-time -approximation algorithm, and

  2. there is an time exact algorithm, where is the size of the optimal solution.

Proof.

Let , and . We show a 2CNF Deletion instance such that for any cut with edge set , the number of unsatisfied clauses in is exactly . The statement of the lemma will follow from Lemma 3.2 and 3.2.

Let be the instance of the 2CNF Deletion problem defined on as follows:

  • For each vertex , we have variable .

  • For each edge :

    • if , we add and to , and

    • if , we add and to .

    (Note that in both cases, any assignment of and satisfies at least one of the clauses. Again in both cases, assignments exist that satisfy both clauses.)

Let be a cut with edge set . Let be the natural boolean vector that corresponds to the cut: for all . We show that is equal to the number of clauses that are not satisfied in . Specifically, we show (I) for each edge , exactly one of its corresponding clauses is satisfied, and (II) for each edge both of its corresponding clauses are satisfied.

If there are two cases to consider: (I.1) and , that is and the corresponding clauses are and . Exactly one of the clauses is satisfied. (I.2) and , that is , and the corresponding clauses are and ; exactly one of the clauses is satisfied.

If there are two cases to consider: (II.1) and , that is and the corresponding clauses are and . Both of the clauses are satisfied. (II.2) and , that is , and the corresponding clauses are and . Both of the clauses are satisfied. ∎

3.3 Wrap up (Proofs of Theorems 1.1, 1.2, 1.3, and 1.4)

Lemma 3.1 and Lemma 3.1 show that the bounded chain problem and the minimum homologous chain problem are special cases of the cut completion problem, and Lemma 3.2 shows that we obtain -approximation algorithm and time exact algorithm for the cut completion problem. The number of vertices translates to for simplicial complexes embedded in (Theorem  1.1), and , the number of -dimensional simplices for -manifolds (Theorem 1.3). The number of edges translates to to in both simplicial complexes embedded in (Theorem 1.2) -manifolds (Theorem 1.4).

4 Hardness of approximation

In this section, we show it is unlikely that either of the minimum bounded chain or minimum homologous chain problems admit constant factor approximation algorithms, even for their low dimensional instances. Our hardness results follow from reductions from the minimum cut completion problem, defined in the previous section.

4.1 Minimum bounded chain to minimum cut completion

We show that the minimum cut completion problem reduces to a -dimensional instance of the minimum bounded chain problem , where is in fact a manifold and is a (possibly not connected) cycle on . Our hardness of approximation result for the minimum bounded chain problem is based on this reduction.

Lemma .

Let be any instance of the minimum cut completion problem. There exists an instance of the -dimensional minimum bounded chain problem with on the outer shell of that can be computed in polynomial time, and a one-to-one correspondence between cuts in and -chains with boundary in . Moreover, if the cut with edge set in corresponds to the -chain in then

where and is the number of edges in .

Proof.

Our construction is simple in high-level. We start from any embedding of in , and we thicken it to obtain a space, in which each edge corresponds to a tube. We insert a disk in the middle of each tube; we call these disks edge disks. Then we triangulate all of the -dimensional pieces. The dual of the complex that we build is almost , except for one extra vertex corresponding to its outer volume, and a set of extra edges, all incident to the extra vertex. We give our detailed construction below.

We consider the following piecewise linear embedding of in ; let and be the number of vertices and edges of , respectively. First, map the vertices of into on the -axis. Now, consider planes all containing the -axis with normals being evenly spaced vectors ranging from to . We use for drawing the edges . We arbitrarily assign edges of to these plane, so each plane will contain exactly one edge. Each edge is drawn on its plane as a three-segment curve; the first and the last segment are orthogonal to -axis and the middle one is parallel. All edges are drawn in the upper half-space of . See Figure 4, left.

Next, we place an axis parallel cube around each vertex. The size of the cubes must be so that they do not intersect, fix the width of each cube to be . We refer to these cubes as vertex cubes Then, we replace the part of each edge outside the cubes with a cubical tube, called edge tube. We choose the thickness of these tubes sufficiently small so that they are disjoint. We also puncture the cubes so that the union of all vertex cubes and edges tubes form a surface; see Figure 4, left. (This surface will have genus by Euler’s formula, which is the dimension of the cycle space of )

Figure 1: Left: an embedding of in , and the thickened surface composed of blue vertex cubes and pink edge tubes, right: an edge tube subdivided by an edge square.

Next, we subdivide each tube by placing a square in its middle; see Figure 4, right. We refer to these squares as edge squares. Edge squares partition the inside of the surface into volumes. We observe that each of these volumes contains exactly one vertex of the drawing of , thus, we call them vertex volumes.

For our reduction to work, we need that the weight of each -cycle to be dominated by the weight of its edge squares. To achieve that we finely triangulate each edge square. For an edge tube, we first subdivide its surface to quadrangles as shown in Figure 2, left. Then, we obtain a triangulation with triangles by splitting each quadrangle into two triangles. For a vertex cube, note that all the punctures are on the top face by our construction. We split all the other faces by dividing each of them into two triangles. For the top face, we can obtain a triangulation in polynomial time; this triangulation will have triangles by Euler’s formula, where is the degree of the vertex corresponding to the cube. Therefore, the triangulation of each vertex cube will have triangles, see Figure 2, right. Therefore, there are triangles that are not part of edge squares. Finally, we triangulate each edge square into triangles so that the cost of one edge square is greater than the sum of all triangles not contained in edge squares. This triangulation can be done efficiently by subdividing triangles. The subdivision is performed by inserting a vertex into the interior of the triangle and connecting it with an edge to each vertex on the boundary of the triangle. The result is a new complex, homeomorphic to the original, with two additional triangles. Overall, our complex has triangles.

Figure 2: Left: subdividing the surface of an edge-tube to quadrangles, right: triangulating the surface of a vertex cube.

We are now done with the construction of . Let be the set of all triangles in edge squares that correspond to edges in . Then, let . We show an almost cost preserving one-to-one correspondence between cuts in the cut completion problem in and chains with boundary in .

Let be a cut with edge set , note that the cost of this cut is in the cut completion problem . In , let be the symmetric difference of the vertex volumes that correspond to vertices of . The total weight of is between and . Similarly, the total weight of is between and . Since we cannot get an exact count on the number of edges in the subgraph induced by we have a range of values for the weight of instead of an exact weight. However, if and are two cuts with then the weight of is strictly less than the weight of by the construction of the edge squares.

On the other hand, let be a -chain with boundary in . As does not intersect the interior of any edge square, for each edge square either contains all of its triangles or none of them. Also, has no boundary, thus its complement is disconnected. The interior of each vertex volume is completely inside one of the connected components of , as by the construction must either contain the entire vertex volume or none of it. Now, let be the set of all vertices whose corresponding vertex volumes are in the unbounded connected component of . The edges of the cut correspond to edge squares in , where is the set of edge square triangles of . As is in one-to-one correspondence to , it follows that the cut completion cost of is . We have where is the set of triangles in not contained in edge squares. The size of is per edge square, and by construction. It follows that we have our desired inequality,

The next lemma shows that an approximation algorithm for the minimum bounded chain problem implies an approximation algorithm with almost the same quality for the minimum cut completion problem.

Lemma .

Let be any instance of the minimum cut completion problem. For any and any , there exists an instance of the -dimensional minimum bounded chain problem that can be computed in polynomial time, such that an -approximation algorithm for implies a -approximation algorithm for , and is on the outer shell of .

Proof.

Let . Given an -approximation algorithm for the minimum bounded chain problem, we describe an -approximation algorithm for the cut completion problem. Let , and be any instance of the cut completion problem, and let with edge set be an optimal solution for this instance. Our algorithm considers two cases, based on whether or not. It solves the problem under each assumption and outputs the best solution it obtains in the end.

If , then our algorithm finds the optimal solution in time by considering all subsets of edges of size at most as candidates for . From all candidates, we return the minimum such that is a cut. Note this is an exact algorithm, so in this case we find the optimal solution.

Otherwise, if , we use the given -approximation algorithm for the minimum bounded chain problem for a simplicial complex , and chain that corresponds to by Lemma 4.1. Note that is an unweighted simplicial complex piecewise linearly embedded in and is a cycle in its outer shell.

Let be the corresponding -chain to in . Thus, . In addition, let be the surface with boundary that the -approximation algorithm finds, so . Finally, let be the cut corresponding to in via the one-to-one correspondence of Lemma 4.1. Therefore, . Putting everything together,

(1)

Since , we have: . Therefore, together with (1), we have a -approximation algorithm, as desired. ∎

4.2 Minimum homologous cycle to minimum cut completion

We show a similar reduction from the cut completion problem to the minimum homologous cycle problem for -dimensional cycles on orientable -manifolds. The minimum homologous cycle problem is the special case of the minimum homologous chain problem when the input chain is required to be a cycle, so showing hardness of approximation for it implies hardness of approximation for the more general minimum homologous chain problem.

Lemma .

Let be any instance of the minimum cut completion problem. For any , there exists an instance of the -dimensional minimum homologous cycle problem that can be computed in polynomial time such that an -approximation for implies an -approximation for .

Proof.

We construct a -manifold as in the proof of Lemma 4.1, but we omit the edge squares. Hence, the genus of is equal to the dimension of the cycle space of . Note that each edge of corresponds to a cycle with edges in , we refer to these cycles as edge rings. Each connected component of the complement of the edge rings on corresponds to a vertex, referred as a vertex region. We set to be equal to the set of edge rings corresponding to . Intuitively, if is the minimum cycle homologous to we do not want to intersect the interior of any vertex region, so that it corresponds to a cut in . To achieve this we can heavily weigh edges that are not on edge-cycles, or equivalently replace them with long paths, via a sequence of subdivisions. However, as a result of these subdivisions, we obtain an embedded graph with non-triangular faces, which is not a simplicial complex. To obtain a simplicial complex again, we triangulate the inside of each non-triangular face, but, so that for any pair of vertices on the face, their shortest path remains on the face. That is the triangulation of a face should not introduce new shortest path between the vertices of the face. Then, we can shortcut any possible -approximation to obtain a solution that only uses edge rings, thus, it corresponds to a cut. Our formal construction follows.

Let , we subdivide each edge not contained in an edge ring times. The resulting complex is not simplicial. To make it simplicial we triangulate the faces of length by adding concentric cycles, each with vertices, labeled , where is the original face from . By we denote the th vertex in . We add the edges and , where the addition in is modulo . To complete the triangulation we add one additional vertex at the center of and add an edge between it and each vertex on . We call the new simplicial complex . See Figure 3 for an example.

Figure 3: Subdividing a face of length five; the outer face with white vertices is the original face.

We have that the shortest path between any two vertices and is entirely contained within . First note that such a shortest path can never include as a path contained entirely in is always shorter. For at any vertex we can move to the following vertices: and . Any time a path contains an edge from to the path must also include an edge from to , hence the shortest path in .

Now, suppose we can compute an -approximation of the minimum homologous cycle for the instance , hence . By our construction an optimal solution to has the same size as an optimal solution to . As is a cycle, if intersects a face with depth 0, it must intersect it an even number of times. For any two consecutive depth 0 vertices and in we replace the path between them with the shortest path contained in . We call the new cycle , since we have that is also an -approximation for . It follows that is a union of edge rings, otherwise we have . Thus, corresponds to a cut and so an -approximation algorithm for the minimum homologous cycle problem implies an -approximation for the minimum cut completion problem. ∎

4.3 Wrap up

It remains to show that the cut completion problem is hard to approximate. We show this via a straightforward reduction from the minimum uncut problem: given a graph , find a cut with minimum number of uncut edges. Note that the optimal cuts for the minimum uncut problem and the maximum cut problem coincide, yet, approximation algorithms for one problem do not necessarily imply approximation algorithm for the other one.

Lemma .

The minimum uncut problem is a special case of the minimum cut completion problem.

Proof.

Consider the cut completion problem for , and let . Let be any cut with edge set . The cut completion cost of this cut is

which is the number of uncut edges by . ∎

Now, we are ready to prove our hardness results.

Proof of Theorem 1.5 and 1.6.

The minimum uncut problem is hard to approximate within for some  [25]. In addition, it is hard to approximate within any constant factor assuming the unique games conjecture [22, 20, 8, 19]. By Lemma 4.3, the cut completion problem generalizes the minimum uncut problem. Finally, by Lemma 4.2 and 4.1, for any and , an -approximation algorithm for the minimum bounded chain problem or the minimum homologous cycle problem implies a -approximation algorithm for the cut completion problem. ∎

5 A polynomial time special case

We have shown hardness results, approximation algorithms and parameterized algorithms for the minimum bounded chain problem. We showed that the problem is hard to approximate even when the input cycle is on the outer shell of an unweighted -complex embedded in . If is null-homologous on the outer shell of there is an exact polynomial time algorithm to find the minimum chain bound by . The assumption that is null-homologous on the outer shell of allows us to treat the problem as a generalization of the shortest -path problem in planar graphs when and are contained on the boundary of the unbounded face. Hence, we can generalize the duality between shortest paths and minimum cuts in planar graphs to -complexes embedded in . The algorithm was first found by Kirsanov and Gortler under the assumption that is trivial [21]. For the sake of completeness we include the same algorithm but described for any -complex embedded in . Before describing the algorithm we prove the following lemma about graph cuts, which will be useful in the proof of correctness of the algorithm.

Lemma .

Let and be two -cuts of a graph with edge sets and , respectively. The symmetric difference is the set of edges of a cut that has and on the same side.

Proof.

We show the edge set of the cut is . The statement follows as .

Let . Either, and or and .

In the first case, there are two possibilities up to symmetry of . Either and , which implies and , or and , which implies and .

In the second case, there are again two possibilities up to symmetry of . Either and , which implies and , or and , which implies and . ∎

Let be a -chain such that . Such an exists by the assumption of this section. We define a cut problem based on . Let be the dual graph of the complex . By the definition of , each edge of is adjacent to . We build the graph from by splitting as follows. We replace with two vertices and . We replace the incident edges to as follows:

  1. A loop that is dual to a face in is replaced by a edge.

  2. A loop that is dual to a face not in is replaced by -loops.

  3. A non-loop edge that is dual to a face in is replaced by a -edge.

  4. A non-loop edge that is dual to a face not in <