Given an undirected graph with edge capacities , and some pairs of vertices given as edges of the graph , the maximum-multiflow problem with input , asks for the maximum flow that can be routed in , simultaneously between the endpoints of edges in , respecting the capacities .
This is one of many widely studied variants of the multiflow problem. Other popular variants include demand flows, all or nothing flows, unsplittable flows etc. In this paper, we are mainly interested in the integer version of this problem, and the half-integer or fractional versions also occur as tools. When capacities are , the capacity constraint specialises to edge-disjointness, whence the maximum edge disjoint paths problem (MEDP) between a given set of pairs is a special case; MEDP is NP-Hard to compute for general graphs, even in very restricted settings like when is a tree [garg1997primal].
The edges in are called demand edges (sometimes commodities), those in are called supply edges; accordingly, is the demand graph, and is the supply graph. If is planar we call the problem a plane multiflow problem. Plane multiflows have been studied for the past forty years, starting with Seymour [seymour1981odd].
Let be the set of paths in between the endpoints of , and . For , the edge is said to be the demand-edge of , denoted by . A multiflow, or for simplicity a flow in this paper is a function . The flow is called feasible, if for all . The value of a flow is defined as .
For a path , we refer to as the flow on . If the flow on every path is integer or half-integer, we say that the flow is integer or half-integer, respectively. A circuit is a connected subgraph with all degrees equal to two.
Multiflows (without restrictions on integrality) can be maximised in general, in (strongly) polynomial time [schrijver2003combinatorial, 70.6, page 1225]
using a linear programming algorithm.
A multicut for is a set of edges such that every contains at least one edge in . 111Given a partition of so that all edges in go across different partitions, the edges of going across different classes form a multicut, and inclusionwise minimal multicuts are of this form. A multicut is the simplest possible and most natural dual to the maximum multiflow problem.
It is easy to see that the value of any feasible multiflow is smaller than or equal to the capacity of any multicut. Klein, Mathieu and Zhou [klein2015correlation] prove that computing the minimum multicut is NP-hard if is planar and also provide a PTAS.
There is a rich literature on the maximum ratio of a minimum capacity of a multicut over the maximum multiflow. With an abuse of terminology we will call this the (possibly integer or half-integer) flow-cut gap, sometimes also restricted to subclasses of problems. This is not to be confused with the same term used for demand problems. The integer flow-cut gap is when is a path and when is a tree. For arbitrary , the flow-cut gap is [garg1996approximate]. Building on decomposition theorems from Klein, Plotkin and Rao [klein1993excluded], Tardos and Vazirani [TV] showed a flow-cut gap of for graphs which do not contain a minor; note that for this includes the class of planar graphs. A long line of impressive work, culminated in [seguin2011maximum] proving a constant approximation ratio for maximum half-integer flows, which together with [TV] implies a constant half-integer flow-cut gap for planar supply graphs. A simple topological obstruction proves that the integer flow-cut gap for planar supply graphs, even when all demand edges are on one face of the graph, also called Okamura-Seymour instances, is [garg1997primal].
These results are often bounded by the integrality gap of the multiflow problems for the problem classes, which is the infimum of MAX over all instances of the problem class, and where MAX is the maximum value of an integer multiflow, and is a multiflow for this same input, and the half-integrality gap is defined similarly by replacing “integer” by “half-integer” in the nominator. A -approximation algorithm for a maximisation problem is a polynomial algorithm which outputs a solution of value at least times the optimum; is also called the approximation ratio (or guarantee).
Our first result (Section 3, Theorem 3.1) is an upper bound of for the flow-cut gap (i.e. multicut/multiflow ratio) for plane instances, the missing relation we mentioned. We prove this by relating multicuts to -edge-connectivity-augmentation in the plane-dual, and applying a bound of Williamson, Goemans, Mihail and Vazirani [williamson1995primal] for this problem.
We next show (Section 4, Theorem 4.1) how to obtain a half-integer flow from a given (fractional) flow in plane instances, reducing the problem to a linear program with a particular combinatorial structure, and solving it.
Finally, given any feasible half-integer flow, we show how to extract an integer flow of value at least half of the original, in polynomial time, using an algorithm that -colors planar graphs efficiently [robertson1996efficiently].
These results imply an integrality gap of for maximum half-integer flows, for maximum integer flows, and the same approximation ratios for each. In turn, the flow-cut gap of 2 implies a half-integer flow-cut gap of 4 and an integer flow-cut gap of for plane instances.
A summary of the results completed by lower bounds and open problems are stated in Section 7. The Appendix is a guide to some of the tools.
We detail here some notions, notations, terminology, facts and tools we use, including some preceding results of influence.
Demands, the Cut Condition and Plane Duality
We describe the notion of demand flows, which is closely related to multiflows defined in the previous section. The problem is defined by the quadruple , where are as before, demands are given, and we are looking for a feasible (sometimes in addition integer or half-integer) flow satisfying for all .
A cut in a graph is a set of edges of the form for all . Note that and , are called the shores of the cut. For a subset we use the notation , and we adopt this usual way of extending a function on single elements to subsets. For instance, is the demand of the set .
A necessary condition for the existence of a feasible multiflow is the so called Cut Condition: for every . In other words, across every cut, total capacity of supply should be at least the total demand. The cut condition is not sufficient in general, but Seymour [seymour1981odd] showed that if is planar and Eulerian (all capacities and demands are integer), then the cut condition is sufficient for an efficiently computable integer flow. A half-integer flow follows then for arbitrary integer capacities for such plane instances (see Section 1). The same holds for Okamura-Seymour instances [okamura1981multicommodity].There are more examples, unrelated to planarity, where the cut condition is sufficient to satisfy all demands, and with an integer flow, for instance when all demand edges can be covered by at most two vertices.
Seymour’s theorem [seymour1981odd] on the sufficiency of the cut condition and about the existence of integer flows has promoted plane multiflow problems to become one of the targets of investigations. This paper is devoted to plane multiflow maximisation. Seymour’s proof is based on a nice correspondence to other combinatorial problems through plane duality that we also adopt.
Middendorf and Pfeiffer [middendorf1993complexity] showed that the plane multiflow problem (edge disjoint paths already) is NP-Hard. The cut condition can be checked in polynomial time222Seymour’s correspondence through dualisation reduces this problem to checking whether is a minimum weight “-join” (see eg. [schrijver2003combinatorial]) in with weights defined by and , where is the set of odd degree vertices of
is the set of odd degree vertices of. This also provides a polynomial separation algorithm for maximising the sum of (not necessarily integer) demands satisfying the cut condition. so the difficult question to decide is the existence or not of an integer flow when the cut condition is satisfied. As far as multicuts are concerned, we show that their minimisation in plane instances is equivalent to a 2-edge-connectivity augmentation problem in planar graphs.
Following Schrijver [schrijver2003combinatorial, p. 27] we denote the dual of the planar graph by , where is the set of faces of and each edge corresponds to an edge joining the two faces that share . For denote . An important fact we need and use about dualisation: is a minimal cut in if and only if is a circuit in .
Two Lemmas on Laminar Families
This correspondence by plane duality allows to transform any fact on cuts to circuits in the dual and vice versa. For example fractional, half-integer or integer packings of cuts in , where each cut contains exactly one edge of correspond to a fractional, half-integer or integer multiflow in . We provide now some related definitions, notations and facts. We do this directly on the graph where they will be used; we denote by the vertices of this graph, that is the faces of . So . Let , be two crossing cuts, i.e. they are neither disjoint nor contain one another, each of which contains exactly one edge of . Then they can be replaced by and or and (in the plane dual). It is easy to check that every edge is contained in at most as many of these two cuts after the replacement as before, and if both cuts contain exactly one edge of then this also holds for the replacing cuts. Doing this iteratively and using plane duality, we can convert any feasible flow into another one (without changing the total value of flow) in which no two flow paths cross. We formalise this below.
A family of subsets of is called laminar if any two of its members are disjoint or one of them contains the other. If for any two members one of them contains the other we say that the family is a chain. Given a laminar family , a chain is full (in ) if and implies . We call a multiflow laminar, if , where is laminar. We state the following Lemma without proof (see the Appendix for references and explanations):
For every feasible multiflow there exists a laminar feasible multiflow so that , and can be found in polynomial time.
The following useful properties are easy to check:
For a family of subsets of and , denote ; .
Let be a laminar family of subsets of . Then
b. both form full chains of subsets of . ∎
Integrality in demand flows and stable sets
We first compute a half integer flow of value at least half the fractional flow and then convert this into an integer flow. We now describe an instance which illustrates the difficulty in finding an integer flow. Consider a planar graph with a perfect matching without any nontrivial tight cut333i.e. a cut with both shores containing more than one vertex, and meeting every perfect matching in exactly one edge. Lovász characterised graphs without nontrivial tight cuts as “bicritical -connected graphs” [LovaszPlummer1986MatchingTheory]. Such graphs may have arbitrarily many vertices, even under the planarity constraint. to be , and to be any perfect matching in it. Let all capacities be . Upper bound the demands by , by replacing each demand edge by two edges in series, a demand and a supply edge, the latter of capacity . Then multiflows can use only the dual edge-sets of stars of vertices in , so an integer multiflow of value corresponds exactly to a stable set, that is, a set of vertices not inducing any edge, of size in . This indicates that in order to find a large integer flow, we need to find large stable sets in planar graphs. Despite these restricted multiflows, the gap between integer and half-integer flow is at least for these graphs: it follows from the -color theorem [AppelHaken1976proof] that a stable set of size at least exists while the maximum half integer flow cannot exceed . We will be able to reach this ratio in general (see Section 5, Theorem 5.1), and with a matching shows that this cannot be improved.
In order to reach this integrality gap of in general, we will actually need to find a stable-set of size in . The maximum stable set problem is NP-hard, but there is a PTAS for it in planar graphs [Baker1994], which, combined with the -color theorem [AppelHaken1976proof] provides a stable-set of size ; the -coloring algorithm of Robertson, Sanders, Seymour and Thomas [robertson1996efficiently] directly provides a -coloring of a planar graph in polynomial time, and the largest color class is clearly of size at least . Either of these will suffice, and be used as a black-box-tool for rounding half-integer flows, so we state the result:
In a planar graph on vertices, a stable set of size can be found in polynomial time.
3 Multicuts versus Multiflows via 2-Connectors
We show in this section that the flow-cut gap is at most two for plane instances, via a reduction to the 2-edge-connectivity augmentation problem.
Given , , a -connector for in is a set of edges such that none of the edges is a cut edge of ; equivalently, is a -connector if and only if each is contained in a circuit of . The 2-edge-connectivity Augmentation Problem (2ECAP) is to find, for given edge costs on , a minimum cost -connector.
Let , , be the input of a plane multiflow maximisation problem, and , where is the set of faces of . Define .
The edge-set is a multicut for if and only if is a -connector for in .
The edge-set forms a multicut in if and only if the endpoints , of any edge are in different components of , that is, if and only if for all there exists an inclusionwise minimal cut of such that . But we saw among the preliminaries concerning duality that is an inclusionwise minimal cut in if and only if is a circuit in . Summarizing, forms a multicut in , if and only if for all there exists a circuit in such that . This means exactly that is a -connector for , in , finishing the proof. ∎∎
Let be an instance of multiflow problem with planar. Let with if and only if , otherwise . The following linear program specialises the one investigated in [williamson1995primal]:
Since is -valued so are the coordinatewise minimal integer solutions including the integer optima of (2ECAP).
The -solutions are exactly the (incidence vectors of)
-solutions are exactly the (incidence vectors of)-connectors of in . Williamson, Goemans, Mihail and Vazirani [williamson1995primal] developed a primal-dual algorithm finding for given input and , an integer primal solution to a class of linear programs including (2ECAP), together with a (not necessarily integer) dual solution in polynomial time, proving the following WGMV inequality [williamson1995primal, Lemma 2.1], see also [kortevygen2018combinatorial, Section 20.4]:
where OPT is the minimum cost of a -connector, and LIN is the optimum of (2ECAP). We will refer to this algorithm as the WGMV algorithm.
Note that the algorithm works for the class of weakly supermodular functions. A function is called weakly supermodular if and for any , . It can be verified that defined above is weakly supermodular.
Let be a plane multiflow problem. Then there exists a feasible multiflow and a multicut , such that , where both and can be computed in polynomial time.
Recall that the WGMV algorithm finds and satisfying the WGMV inequality, where is the incidence vector of a -connector of in , let us denote its plane dual set in by . By Lemma 4 is a multicut.
Note that if and only . This implies can be supposed for all with ; furthermore, is a circuit in , so is a path in , denote it by . Define a multiflow in by . The dual feasibility of means exactly that the multiflow is feasible. Finally, by our construction and the WGMV inequality we have
So the multicut and the multiflow satisfy the claimed inequality; all operations, including the WGMV algorithm run in polynomial time. ∎∎
Note that if is half-integer (assuming integer edge-costs), the obtained multiflow is half-integer and a half-integer flow-cut gap of directly follows. There are examples where the WGMV algorithm does not produce a half-integer dual solution, but we do not know of an instance where half-integer flow-cut gap is more than 2.
4 From Fractional to Half-Integer
We show here how to convert a flow to a half-integer one, loosing at most half of the flow value, provided the capacities are integers.
Let be a plane multiflow problem, where is an integer capacity function. Given a feasible multiflow , there exists a feasible half-integer multiflow , such that , and it can be computed in polynomial time.
By Lemma 1 we can suppose that the given feasible multiflow is laminar and can be found in polynomial time. Let be the laminar family of cuts in , with (see Section 2, just above Lemma 1). Denote , . The feasibility of the multiflow means , that is, satisfies
Clearly, the edge is contained in exactly those sets for which , where , and both and form full chains (Lemma 2). So the linear program
is a relaxation of (1): for each the coefficient vector of (1) associated with is the sum of the coefficient vectors, one for each of and , of the two inequalities associeated with in (2). Both of these ( and ) correspond to full chains in the laminar family , and the right hand side is repeated for both.
Denote the coefficient matrix of (2) (without the non-negativity constraints).
According to Edmonds and Giles [edmondsgiles1977], has a rooted tree (arborescence) representation in which the full chains correspond to subpaths of paths from the root, so is a network matrix. As such, it is totally unimodular by Tutte [tutte1965networkunimodular] and (2) has an integer optimum , computable in polynomial time by [hoffmankruskal1956integral], [schrijver1986LPandIP, Theorem 19.3 (ii), p. 269].
To finish the proof now, note that on the one hand, is a solution to (1), and therefore it is also a feasible solution of the relaxation (2). Since is the maximum of (2), . According to the two inequalities of (2) associated to , the sum of coefficients of the paths containing any given edge is at most , so defines a half-integer feasible flow in (by assigning the flow value to the path ), so , finishing the proof. ∎∎
For more explanations and references, various ways of showing that is a network matrix and an alternative direct combinatorial solution of the integer linear program with a simple greedy type algorithm, see the Appendix. The optimum solution to this linear program is integer and can be computed efficiently [schrijver2003combinatorial].
This proof does not fully exploit the possibilities of totally unimodular matrices: instead of putting as right hand side for both inequalities of (2) associated with we can put everywhere the smallest integer capacities satisfied by the fractional flow. Since the fractional values on the mentioned two inequalities sum to at most and not we get a sharper result this way. The proof works if we replace the capacities by the rounded up fractional flow, but only with an error of , because of the round-up. Let us denote the all function on , and check this precisely:
Let be a plane multiflow problem, where . Given a feasible multiflow , there exists a feasible integer multiflow , computable in polynomial time, feasible for the capacity function , and .
Let be a feasible multiflow, and the coefficient matrix defined in the proof of Theorem 4.1. Define , and consider the linear program
where . In words, (3) has exactly the same coefficients as (2), but the right hand sides and of the two inequalities associated with , are defined with the sum of flow values on , for and respectively.
Since is a feasible flow for , , so , and is feasible for (3) since the capacities have been defined by rounding up the flow values. On the other hand, the coefficient matrix is totally unimodular, so by [hoffmankruskal1956integral], [schrijver1986LPandIP, Theorem 19.3 (ii), p. 269] the linear program (3) has an integer maximum solution , again computable in polynomial time, and , . ∎∎
This result extends a natural consequence for maximisation of the tight additive integrality gap known for plane demand flow problems with integer capacities and demands satisfying the cut condition: according to a result of Korach and Penn [korachpenn1992gap], if all demand edges lie in two faces of the supply graph, there exists an integer multiflow satisfying all demands but at most . This readily implies that increasing each capacity by , an integer flow of the same value as the maximum flow for the original capacities, can be reached.
From Frank and Szigeti [frankszigeti1995surficit] the same can be deduced for demand-edges lying on an arbitrary number of faces. Indeed, increasing all capacities by , the surplus of the cut condition will be at least , which is the Frank-Szigeti condition for integer multiflows. Theorem 4.2 states that this consequence is actually true in general, without requiring the integrality of the demands, and also for the maximisation problem; the same holds for maximum “packings of -cuts”.
5 From Half-Integer to Integer
In this section, we show how to round a half-integer flow to an integer one, losing at most one half of the flow value.
Let be a plane multiflow problem, where , and . Given a feasible half-integer multiflow , there exists a feasible integer multiflow , computable in polynomial time, and .
Let and be as assumed in the condition, moreover that is laminar (Lemma 1) We proceed by induction on the integer . We suppose that all nonzero values are actually : if , we can decrease by , as well as all capacities of edges of , and the statement follows from the induction hypothesis. (Such a step can be repeated only a polynomial number of times, since by Lemma 2.)
To choose the values to round we replace , by parallel edges of capacity each.444This is not an allowed step for a polynomial algorithm, but it will not really be necessary to do it. The choice of the cuts to be rounded down or up will be clear from the proof without actually executing this subdivision. The choices for rounding concern a family of size . Then take the plane dual of the resulting graph, which is with each edge replaced by a path of size . We consider the laminar system defining the paths , 555, as before. in this subdivided graph so that every edge is contained in at most two sets , and remains laminar (this is clearly possible, since all positive values are ). For simplicity, we keep the notations of the original graph - as if what we get in this way were the given graph with all capacities equal to .
Let be the intersection graph of the cuts defined by , that is, . We have Claim a. and b. so far, and now we check Claim c.:
Claim: a. Each is contained in at most two sets in .
c. is planar.
To check Claim c., note first its validity if consists of disjoint sets. If this does not hold, not even by complementing some (ie. replacing it by ), then it is easy to find (possibly after coplementation) three sets in . We claim that is a cut-vertex in . By laminarity, every cut has either a shore contained in (like ), or a shore disjoint from (like ). If there exists an , this would mean that has an endpoint in and the other endpoint in , so , contradicting Claim a. This implies that is a cut vertex.
Hence, , , with no edge between and in . Since the graphs induced by () are both defined by flows of smaller value, we can apply the induction hypothesis to them: they are planar, so is also planar, and the Claim follows.
To finish the proof of the theorem using Claim c., find a stable set of size in , by Lemma 3, and increase the flow on the corresponding paths to , while decreasing the flow on the other paths to . This results in an integer flow , finishing the proof of the bound.
For the computational complexity results, first recall that the support of the half integer laminar flow obeys Lemma 2a. Then note, that among this linear number of sets, the proof finds in polynomial time at least one fourth of the cuts to round up, while the other cuts are rounded down, so that the capacity constraints are not violated. It is straightforward to mimick the subdivision of edges without doing it, and to compute an input to Lemma 3, in strongly polynomial time. The choice of the paths to round up among the polynomial number of different paths is provided by the output stable set. ∎∎
6 Lower bounds on the flow-cut gap
We show a class of plane multiflow instances on which the half-integer flow-cut gap is tending to as . Cheriyan et.al. [cheriyan2008integrality] used these instances to show integrality gap results for the Tree Augmentation Problem.
Let be an instance of the multiflow problem defined as follows: and . The capacity of all edges in is (see Fig. 1).
The graph is planar for all , and the following hold:
– The minimum multicut capacity is .
– The maximum multiflow value is .
– The maximum half-integer multiflow value is .
– The maximum integer multiflow value is .
The minimum multicut capacity is clearly at most , since deleting each edge of the -path, the endpoints of all demand-edges are separated. We check now for an arbitrary multicut , by induction. For the statement is obvious. Deleting and , clearly, the remaining is (isomorphic to) and is a multicut in it. By the induction hypothesis, the minimum multicut of is of size .
Now either both and in which case is not an inclusonwiwe minimal multicut for , since deleting one of we already disconnect the same terminal pairs. So in this case , and we are done; or one of is not in , but then one of must be in it, since otherwise is not disconnected by from or from . So in this case finishing the proof of the first assertion.
For the maximum multiflow value , or maximum half-integer and integer multiflow values , , note that the supply edges form a tree, the set of paths between the endpoints of demand edges contains exactly one path for each demand edge , so . Defining and for each other path , we have . To prove that this is a maximum flow, note that is contained in if and only if is contained in it, so for any multiflow ,
Claim: , and if , then
Indeed, on the one hand, causes in isomorphic to with max flow value , after the deletion of , and then by induction, . On the other hand, if we have by induction , and the claim is proved, finishing the proof of the second statement.
The proof of the third assertion, concerning works similarly. Defining , and , otherwise , we is a feasible half-integer flow, . To prove a similar induction works as before: the stronger statement follows by induction.
Finally, the fourth assertion is immediate from the third one: is integer, and an integer flow , is also easy to construct. ∎∎
This paper established bounds on the integrality gap of multiflows, developed approximation algorithms for them and bounded the flow-cut gap. Applying Theorem 4.1 to a maximum multiflow, and then applying Theorem 5.1 to the result, we arrive at the following summary:
There exists a -approximation algorithm for integer plane multiflow maximisation, with an integer flow-cut gap of ; there exists a -approximation algorithm for half-integer plane multiflows with a half-integer flow-cut gap of ; the flow-cut gap is at most ; the approximation algorithms provide lower bounds to the integrality or half-integrality gap equal to the approximation ratio.
a. Multicuts and flow-cut gap
The minimum multicut is NP-hard to find, but has a PTAS by Lemma 4 and [klein2015correlation]. The half-integer flow-cut gap is at least by the example illustrated in Figure 1 (and Theorem 6.1) and it is at most by Theorem 7.1. The (fractional) flow-cut gap is in the interval , again by the same example and the theorem. For the integer flow-cut gap, a lower bound of is shown by and consisting of two demand edges forming a matching. The true value is thus wide open in the interval .
b. Half-integrality Gap
We do not know the complexity of finding a half-integer multiflow of maximum value. The worst case ratio between half-integer and fractional flow is in the interval , again by the Theorem 4.1 and Theorem 6.1. It remains an interesting open problem to pin down the exact half-integrality gap in this interval. This problem does not look easy though, since there is a simple example in which there is a unique maximum multiflow which is integer (Figure 2).
c. Integrality Gap The worst case ratio between integer and fractional flow lies in the interval , with lower bound given by Theorem 7.1 and upper bound by and consisting of two demand edges forming a matching. Finally, the worst integer flow/half-integer flow ratio is closed: it is exactly , as the same example of and Theorem 5.1 together show.