The multi-commodity flow-cut gap is a fundamental parameter that has been proven instrumental in the design of routing and divide & conquer algorithms in graphs. Bounds on this parameter generalize the max-flow/min-cut theorem, and lead to deep connections between algorithm design, graph theory, and geometry [15, 3, 2]. While the flow-cut gap for several classes of undirected graphs has been studied extensively, the case of directed graphs is poorly understood despite significant efforts. In this work we make progress towards overcoming this limitation by showing constant flow-cut gaps for some directed graph families. Consequently we also develop constant-factor approximation algorithms for certain directed cut problems on these graphs.
1.1 Multi-commodity flow-cut gaps
A multi-commodity flow instance in an undirected graph is defined by two non-negative functions: and . We refer to and as the capacity and demand functions respectively. The maximum concurrent flow is the maximal value such that for every , can be simultaneously routed between and , without violating the edge capacities. We refer to this value as .
For every , the sparsity of is defined as follows:
where is the indicator variable for membership in . The sparsity of a cut is a natural upper bound for . The multi-commodity max-flow min-cut gap for , denoted by , is the maximum gap between the value of the flow and the upper bounds given by the sparsity formula, over all multi-commodity flow instances on . The flow-cut gap on undirected graphs has been studied extensively, and several upper and lower bounds have been obtained for various graph classes. The gap is referred to as the uniform multi-commodity flow-cut gap for the special case where there is a unit demand between every pair of vertices. Leighton and Rao  showed that the uniform flow-cut gap is in undirected graphs. Subsequently Lineal, London and Rabinovich  showed that the non-uniform multi-commodity flow-cut gap for the Sparsest Cut problem with demand pairs is upper bounded by . Besides these there are various studies of the flow-cut gap for specific graph families. A central conjecture posed by Gupta, Newman, Rabinovich, and Sinclair in  asserts the following.
Conjecture 1 (GNRS Conjecture ).
For every family of finite graphs , we have iff forbids some minor.
Conjecture 1 has been verified for the case of series-parallel graphs , -outerplanar graphs , -pathwidth graphs , and for some special classes of planar metrics . For graphs excluding any fixed minor the flow-cut gap with terminal pairs is known to be for uniform demands and for arbitrary demands .
For the case of directed graphs, the flow-cut gap is defined in terms of the Directed Non-Bipartite Sparsest Cut problem which is an asymmetric variant of the Sparsest Cut problem, and is defined as follows. Let be a directed graph and let be a capacity function. Let be a set of terminal pairs, where each terminal pair has a non-negative demand . A cut in is a subset of directed edges of . For a cut in , let be the set of all indices such that all paths from to have at least one edge in . Let be the demand separated by . Let be the sparsity of . The goal is to find the cut with minimum sparsity. The LP relaxation of this problem corresponds to the dual of the LP formulation of the directed maximum concurrent flow problem, and the integrality gap of this LP relaxation is the directed multi-commodity flow-cut gap. Hajiyaghayi and Räcke  showed an upper bound of for the flow-cut gap. This upper bound on the gap has been further improved by Agarwal, Alon and Charikar to in . For directed graphs of treewidth , it has been shown that the gap is at most by Mémoli, Sidiropoulos and Sridhar . On the lower bound side Saks et al.  showed that for general directed graphs the flow-cut gap is at least , for any constant , and for any . Chuzhoy and Khanna showed a lower bound for the flow-cut gap in .
A natural generalization of the GNRS Conjecture for directed graphs poses the question of whether the multi-commodity flow-cut gap is for any family of minor free directed graphs. In this paper, we provide the first constant gaps for some non-trivial family of graphs. Throughout this paper, when we refer to a directed family of graphs we mean that it is obtained from an undirected family of graphs by assigning arbitrary directions to the edge sets. We state below our two main results pertaining to the flow-cut gap.
The uniform multi-commodity flow-cut gap on directed series-parallel graphs and directed bounded pathwidth graphs is .
The non-uniform multi-commodity flow-cut gap on directed cycles and directed trees is .
1.2 Cut problems of directed graphs
Better bounds on the flow-cut gap typically also imply better approximation ratios for solving cut problems. For the Directed Non-Bipartite Sparsest Cut problem the flow-cut gap upper bounds of  and  are also accompanied by and polynomial time approximation algorithms respectively. Similarly for graphs of treewidth , a polynomial time approximation algorithm is also provided in .
Another closely related cut problem is the Directed Multicut problem which is defined as follows. Let be a directed graph and let be a capacity function. Let be a set of terminal pairs. A cut in is a subset of . The capacity of a cut is . The goal is to find a cut separating all terminal pairs, minimizing the capacity of the cut. This problem is NP-hard. An approximation algorithm for Directed Multicut was presented by Cheriyan, Karloff and Rabani . Subsequently an -approximation was given due to Kortsarts, Kortsarz and Nutov . Finally  also gives an improved -approximation algorithm for this problem. Again for graphs of treewidth a approximation algorithm was also shown in .
On the hardness side  demonstrated an -hardness for the Directed Non-Bipartite Sparsest Cut problem and the Directed Multicut problem under the assumption that NP DTIME . This was further improved by them in a subsequent work  to obtain an -hardness result for both problems for any constant assuming that NP ZPP .
Our main results for these problems are the following theorems.
There exists a polynomial time -approximation algorithm for the Uniform Directed Sparsest Cut problem on series parallel graphs and graphs of bounded pathwidth.
There exists a polynomial time -approximation algorithm for the Directed Multicut problem on series parallel graphs and graphs of bounded pathwidth.
We remark that in the above results the running time in the case of graphs of pathwidth is . That is, the running time does not depend on . Typically, algorithms for graphs of pathwidth have running time of the form either , or , for some function , due to the use of dynamic programming. Our algorithms are based on LP relaxations, and thus avoid this overhead.
1.3 Quasimetric spaces and embeddings
A Quasimetric space is a pair where is a set of points and , that satisfies the following two conditions:
(1) For all , iff .
(2) For all , .
The notion of random quasipartitions was introduced in . A quasipartition is a transitive reflexive relation. Let be a quasimetric space. For a fixed , we say that a quasipartition of is -bounded if for every with , we have . Let be a distribution over -bounded quasipartitions of . We say that is -bounded. Let . We say that is -Lipschitz if for any , we have that
Given a distribution over quasipartitions we sometimes use the term random quasipartition (with distribution ) to refer to any quasipartition sampled from . We consider the quasimetric space obtained from the shortest path distance of a directed graph. Mémoli, Sidiropoulos and Sridhar in  find an -Lipschitz distribution over -bounded quasipartitions of tree quasimetric spaces. They also prove the existence of a -Lipschitz distribution over -bounded quasipartitions for any quasimetric that is obtained from a directed graph of treewidth .
Our main results for finding Lipschitz quasipartitions are the following theorems.
Let be a directed graph of treewidth 2. Let denote the shortest-path quasimetric space of . Then for all , there exists some -Lipschitz distribution over -bounded quasipartitions of .
Let be a directed graph of of pathwidth . Let denote the shortest-path quasimetric space of . Then for all , there exists some -Lipschitz distribution over -bounded quasipartitions of .
Before stating our embedding results, we first need to introduce some notations and definitions. Let and be quasimetric spaces. A mapping is called an embedding of distortion if there exists some , such that for all , we have . We say that is isometric when . Let be a distribution over pairs , where . We say that is a random embedding of distortion if for all , the following conditions are satisfied:
Directed (Charikar et al. )
The directed distance between two points and is given by .
The following theorems are our main results for random embeddings.
Let be a directed cycle and let be the shortest-path quasimetric space of . Then admits a constant-distortion embedding into directed . Moreover the embedding is computable in polynomial time.
Let be a directed tree, and let be the quasimetric induced by . Then embeds into directed .
We further discuss some limitations of methods that were developed for undirected graphs. Klein, Plotkin, and Rao in  introduced the notion of random partitions for undirected graphs. In Section 8, we show that this method can not be used or generalized for the case of directed graphs. Furthermore, we complete our paper with a lower bound result that is stated in the following theorem.
There exists a directed cycle such that any non-contracting random embedding of into directed trees has distortion .
In Sections 3 and 4, we provide efficient algorithms for computing random quasipartitions for directed graphs of treewidth 2 and bounded pathwidth graphs respectively. In Section 5, we describe an algorithm for computing an -distortion embedding of the directed cycles into directed . In Section 6, we provide an algorithm for embedding directed trees into directed with distortion one. In Section 7 we discuss the applications to directed cut problems. In Section 8 we discuss the limitations of random partitions for the directed case, and finally in Section 9 we provide a lower for non-contracting embeddings of directed cycle into directed trees.
2 Notation and preliminaries
We now introduce some notation that will be used throughout the paper.
Directed graphs and treewidth.
From any undirected graph we can obtain a directed graph as follows. We set and ; i.e. for every undirected edge in we add directed edges and to . We refer to as the underlying undirected graph of . We say that is a directed graph of treewidth if its underlying undirected graph has treewidth , for some . Similarly, we say that is a directed tree (resp. directed cycle) if its underlying undirected graph is a tree (resp. cycle).
Directed cut metrics and 0-1 quasimetrics (Charikar et al. )
Given a set and a subset , the corresponding directed cut metric distance for any pair of elements is given by,
A 0-1 Quasimetric space is a pair where for all we have that or and for all we have that .
3 Lipschitz quasipartitions of treewidth-2 directed graphs
In this Section we provide a proof for Theorem 1.5. Note that since all series-parallel graphs have treewidth at most this result automatically holds for any series-parallel graph. We present an efficient algorithm for computing a random quasipartition of a directed graph of treewidth 2. We begin by describing some special type of graphs of treewidth 2, which we refer to as trees of hexagons. We show that any graph of treewidth 2 admits an isometric embedding into some tree of hexagons. We then further show how to preprocess a tree of hexagons such that it can be inductively constructed via a sequence of either slack or tight paths similar to . Finally, we present the algorithm for computing the random quasipartition, and we analyze the correctness of the algorithm.
3.1 Trees of hexagons
Let be a directed graph of treewidth . We can construct as follows. Start with a single edge and sequentially perform the following operation. Pick an arbitrary existing edge . Add a new vertex and edges and . Let be the added path. We say that is the parent of . Finally, remove an arbitrary subset of edges. Now we may assume w.l.o.g. that no edges are removed in the last step while constructing . Suppose this is not the case then we can replace the removed edges with edges that have weight equal to the shortest path distance between the two end points. This will ensure that the induced shortest path quasimetric of remains the same. The parent relation induces a rooted tree decomposition of of width , where each bubble induces a triangle in .
For any path of whose parent edge is we say that the directed edge is the parent of the directed path in and the directed edge is the parent of the directed path in .
We construct a new graph as follows. We start with and modify it in the following fashion. For all we consider the sub-graph and proceed as follows. Let be the vertices of . We duplicate each vertex of and add edges of weight in both directions between the two copies of a vertex (See Figure 1). By doing so, every directed triangle in corresponds to a directed hexagon in , where are the vertices of the hexagon. The edges , , correspond to , , and respectively in with the same weight. Similarly, the edges , , and correspond to , , and respectively in with the same weight (See Figure 1). Therefore, we get a new graph .
For any triangle in where the edge is the parent of the path , let the corresponding hexagon in be . We call the directed edge the parent edge of the directed path and of every edge in it. Similarly we call the parent edge of the directed path and every edge in it. This parent relation induces a rooted tree decomposition of , of width , where each bubble induces a hexagon in . We say that is a tree of hexagons.
Let be the parent edge of a path . For any edge in we define the tail of , , to be the subpath of from to .
The above discussion immediately implies the following.
Lemma 3.1 (Embedding into a tree of hexagons).
There exists a polynomial-time algorithm which given a directed graph of treewidth 2, computes an isometric embedding of into some tree of hexagons .
3.2 Slack and tight paths
Let be the input graph, and let be a non-negative weight function on the edges of . By Lemma 3.1 we may assume w.l.o.g. that is a tree of hexagons, and we have a tree decomposition of , rooted at some . For any two vertices , we pick a unique shortest path from to denoted by to use in our algorithm. We always pick to be a shortest path with the fewest number of edges. If there are multiple such paths we pick one maintaining the condition that the intersection of any two shortest paths is a (possiply empty) path. For a path , let denote the length of . Let . For any child path of , we say that is slack if , and we say that is tight if .
Let be an arbitrary vertex. Let be a level function where , and for any other , denotes the length of the shortest path from to in . Let be the leaf vertices of . For every , let be an arbitrary vertex in . For every , let be the unique shortest paths in from to . Let and let . For every , we define the complementary path as follows. Let be the path of hexagons corresponding to . Let be the subgraph of obtained by deleting all the edges of , i.e. . We set to be the unique path from to in (See Figure 2). For every , we define the flattened complementary graph as follows. Start with , and for every tight path with a parent edge add to and repeat until we don’t add any new paths. We can similarly define the complementary path and the flattened complementary graph for every .
Let be a path in . We say that is down-monotone if when traversing we visit the bubbles of in non-decreasing distance from the root of . Similarly, we say that is up-monotone if when traversing we visit the bubbles of in non-increasing distance from the root of .
We say that some tree of hexagons is canonical if for all , every child of is either tight or slack. We first show that any directed graph of treewidth 2 admits a constant-distortion embedding into a canonical directed tree of hexagons. This allows us to focus on canonical graphs.
Lemma 3.2 (Embedding into a canonical graph).
Given a directed tree of hexagons , we can compute in polynomial time some embedding of into some canonical tree of hexagons , with distortion at most .
The algorithm proceeds by inductively modifying the graph . We intially mark all edges as unresolved. We mark all edges with no parent as resolved. While there are unresolved edges, we pick some unresolved edge , whose parent is resolved, and let be the child path of that contains . If is neither slack nor tight then for all , we set . We all mark edges in as resolved. We set be the graph obtained at the end of this inductive process. It is immediate the is canonical. At each iteration the number of unresolved edges decreases by at least one, so the algorithm terminates in polynomial time. By the definition of tight and slack paths, it follows that the length of each edge changes by at most a factor of 2. Thus the distortion of the induced embedding is at most 2, which concludes proof. ∎
3.3 Computing a random quasipartition
The algorithm for computing a random quasipartition is as follows. The input consists of some directed tree of hexagons , a non-negative weight function on the edges of , and some . The output is a random -bounded quasipartition of the shortest-path quasimetric space of .
Input: A directed canonical tree of hexagons , and a tree decomposition of , rooted at some . and .
Output: Random quasipartition of the shortest-path quasimetric space of , .
Initialization. Set and .
Step 1. Let be an arbitrary vertex. Pick uniformly at random.
Step 2. For all remove from if and for some integer .
Step 3. For all remove from if and for some integer and some integer .
Step 4. For all that are removed from in Step 3 do the following:
Step 4.1. For each uncut child path of remove one of the edges , , , or from
, chosen randomly with probability, , , and respectively.
Step 4.2. Recursively perform Step 4.1 on the removed edge.
Step 5. For all remove from if and for some integer .
Step 6. For all remove from if and for some integer and some .
Step 7. For all that are removed from in step 6 do the following:
Step 7.1. For each uncut child path of remove one of the edges , , , or from , chosen randomly with probability , , , and respectively.
Step 7.2. Recursively perform Step 7.1 on the removed edge.
Step 8. For any , if , remove from .
Step 9. Enforce transitivity on ; that is, for all if and then add to .
This concludes the description of the algorithm for computing a random quasipartition.
We now analyze the performance of the above algorithm. We begin by showing that the probability that an edge is removed from the quasipartition is small. This statement is shown by considering separately all possible steps of the algorithm where an edge can be removed.
For all , we have .
The edge is removed from in Step 2 when and for some integer . By the triangle inequality this implies that , as required. ∎
For all , we have .
The proof of this case is similar to the proof of Lemma 3.3. ∎
Let . Suppose that and for some , then and .
Let be the hexagon containing and . Let be the parent edge of . Note that it is possible that or . Since any complement subpath from to must end with the unique subpath , it follows that and . The fact that the complement contains also means that is contained in and . Therefore we have that and share the same sub-path from to . Combined with the fact that and this implies that and share the same sub-path from to ending with the edge .
Since and share the same sub-path from to ending with the edge we have that and . ∎
Let . Suppose that traverses the parent edge of a tight path . Then does not visit any vertex in other than and .
Let . Since is the parent edge of a tight path we have that . Suppose visits some vertex in other than and we have that intersects the shortest path more than once which is a contradiction. ∎
Let . Suppose that , and for some , then and .
Let be the unique ancestor edge of that is contained in . We have that is the ancestor edge of a tight path that contains . This implies that . Now let be the parent edge of . This implies that .
Let us suppose that the parent edge of is . This implies that contains . Since is the parent edge of a tight path we also have that . This implies that . From Lemma 3.6 we have that does not contain . Recursively applying Lemma 3.6 we have that does not intersect with . Therefore we have that and that . Since is common to both and , we have that and share the same subpath from to . Moreover they both also contain which concludes the proof. ∎
Let . Suppose that and for some , then and .
Let be the unique ancestor edge of that is contained in . Let be the unique ancestor edge of that is contained in . W.l.o.g. let be the ancestor edge of a tight path that contains . From Lemma 3.7 we have that . Since is an ancestor edge of a tight path that contains it follows that and both contain . This implies that and concluding the proof. ∎
For all , we have .
For all , we have .
The proof of this case is similar to the proof of Lemma 3.9. ∎
For all , we have .
We prove this by induction. For the base case suppose that is an edge in then it has no parent edge and therefore the assertion is immediate. Otherwise let be the parent edge of a child path containing and assume, by the inductive hypothesis, that . There are two cases:
Case 1: Suppose that is a tight child path of . Then we have
Case 2: Suppose that is a slack child path of . Then we have
Thus in either case the assertion is satisfied, concluding the proof. ∎
For all , we have .
The proof for this is similar to the proof of Lemma 3.11. ∎
For all , we have .
Since only edges of length greater than are removed in Step 8 we have that
For all , we have .
Finally we show that is -bounded.
Let where and . Suppose that is in the path from to in . If then there exists a monotone-down path in , where for all we have .
Since , we have that at the beginning of step there must have been a path such that for all we have . If is monotone-down, we are done. Otherwise, we start with and modify it to obtain the desired . Suppose that is not monotone-down. Let be the shortest path from to in . Let . Since is not monotone-down, there exists such that . Let be the smallest number such that has such property. Let be the hexagon containing and (See Figure 3). Let be the other neighbor of in , and let . Let be the next hexagon traversed by after , and let be the other edge in which is not traversed by . We similarly define , and .
The main idea is that we are able to replace the subpath of from to with . Suppose that we are not able to do such replacement, and thus is removed in the algorithm. Let . is a child of . First suppose that is tight. In this case, since is removed, at least one of the edges of should be removed after step , and thus should be removed by the algorithm. Second case is where is a slack. In this case, since is removed, at least one of the edges of should be removed after step