Supermodularity in Unweighted Graph Optimization III: Highly-connected Digraphs
By generalizing a recent result of Hong, Liu, and Lai [Hong-Liu-Lai] on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple -node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple.
There is an extensive literature of problems concerning degree sequences of graphs or digraphs with some prescribed properties such as simplicity or -connectivity. For example, Edmonds [Edmonds64] characterized the degree-sequences of simple -edge-connected undirected graphs, while Wang and Kleitman [Wang-Kleitman] solved the corresponding problem for simple -node-connected graphs.
In what follows, we consider only directed graphs for which the default understanding will be throughout the paper that loops and parallel arcs are allowed. When neither loops nor parallel arcs from to are allowed we speak of simple digraphs. Oppositely oriented arcs and , however, are allowed in simple digraphs. A typical problem is as follows. Given an -element ground-set , decide for a specified integer-valued function if there is a digraph with some prescribed properties realizing (or fitting) , which means that for every node where denotes the number of arcs of with head . Often we call a function an in-degree specification or sequence or prescription. An out-degree specification is defined analogously, and a pair of functions is simply called a degree-specification.
For any function , the set-function is defined by for , and we shall use this tilde-notation throughout the paper. In order to realize with a digraph, it is necessary to require that since both and enumerate the total number of arcs of a realizing digraph . This common value will be denoted by , that is, our assumption throughout is that
The following result was proved by Ore [Ore56] in a slightly different but equivalent form.
Theorem 1 (Ore).
A digraph has a subgraph fitting if and only if
where denotes the number of arcs with and .
This immediately implies the following characterization ([Ore56], see also [Ford-Fulkerson]).
Theorem 2 (Ore).
Let be a degree-specification meeting (1).
(A) There always exists a digraph realizing .
(B) There is a loopless digraph realizing if and only if
(C) There is a simple digraph realizing if and only if
Moreover, it suffices to require the inequality in (4) only for its special case when consists of the largest values of and consists of the largest values of .
Note that (3) follows from (4) by taking and . We also remark that Part (A) can be proved directly by a simple greedy algorithm: build up a digraph by adding arcs one by one as long as there are (possibly not distinct) nodes and with and . Also, Part (B) immediately follows by the following loop-reducing technique. Let be a digraph fitting and suppose that there is a loop sitting at . Condition (3) implies that there must be an arc with (but allowed). By replacing and with arcs and , we obtain a digraph fitting that has fewer loops than has. For later purposes, we remark that the loop-reducing procedure does not decrease the in-degree of any subset of nodes.
We call a digraph strongly connected or just strong if whenever . More generally, is -edge-connected if whenever . is -node-connected or just -connected if and the removal of any set of less than nodes leaves a strong digraph.
One may be interested in characterizing degree-sequences of -edge-connected and -node-connected digraphs. We will refer to this kind of problems as synthesis problems. The more general augmentation problem consists of making an initial digraph -edge- or -node-connected by adding a degree-specified digraph. Clearly, when is empty we are back at the synthesis problem. The augmentation problem was solved for -edge-connectivity in [FrankJ23] and for -node-connectivity in [FrankJ31], but in both cases the augmenting digraphs were allowed to have loops or parallel arcs. The same approach rather easily extends to the case when is requested to be loopless but treating simplicity is significantly more difficult.
The goal of the present paper is to investigate these degree-specified augmentation and synthesis problems when simplicity is expected. In the augmentation problem, this means actually two possible versions depending on whether the augmenting digraph or else the augmented digraph is requested to be simple. Clearly, when has no arcs (the synthesis problem) the two versions coincide. We will consider both variations.
An early result of this type is due to Beineke and Harary [Beineke-Harary] who characterized degree-sequences of loopless strongly connected digraphs. In a recent work Hong, Liu, and Lai [Hong-Liu-Lai] characterized degree-sequences of simple strongly connected digraphs. In order to generalize conveniently their result, we formulate it in a slightly different but equivalent form.
Theorem 3 (Hong, Liu, and Lai).
Suppose that there is a simple digraph fitting a degree-specification (that is, (4) holds). There is a strongly connected simple digraph fitting if and only if
holds for every pair of disjoint subsets with . Moreover, it suffices to require (5) only in the special case when consists of the largest values of and consists of the largest values of .
We are going to extend this result in two directions. In the first one, degree-specifications are characterized for which there is a simple realizing digraph whose addition to an initial -edge-connected digraph results in a -edge-connected digraph . The general problem of augmenting an arbitrary initial digraph with a degree-specified simple digraph to obtain a -edge-connected digraph remains open even in the special case when has no arcs at all. That is, the synthesis problem of characterizing degree-sequences of simple -edge-connected digraphs remains open for .
Our second generalization of Theorem 3 provides a characterization of degree-sequences of simple -node-connected digraphs. We also solve the more general degree-specified node-connectivity augmentation problem when the augmented digraph is requested to be simple. It is a bit surprising that node-connectivity augmentation problems are typically more complex than their edge-connectivity counterparts and yet an analogous characterization for the general -edge-connected case, as indicated above, remains open.
In the proof of both extensions, we rely on the following general result of Frank and Jordán [FrankJ31].
Theorem 4 (Supermodular arc-covering, bi-set function version).
Let be a positively crossing supermodular bi-set function which is zero on trivial bi-sets. The minimum number of arcs of a loopless digraph covering is equal to where the maximum is taken over all independent families of bi-sets. There is an algorithm for crossing supermodular , which is polynomial in and in the maximum value of , to compute the optima.
One way to obtain a non-negative positively crossing supermodular function is taking a crossing supermodular function and increase its negative values to zero, but not every non-negative positively crossing supermodular function arises in this way. In deriving applications, it is simpler to work with the more general notion but it should be emphasised that no polynomial algorithm is known when is positively crossing supermodular, and the existing algorithms work only for crossing supermodular functions.
The algorithm described in [FrankJ31] relies on the ellipsoid method and on a subroutine to minimize a submodular function given on a ring-family (see, Schrijver’s algorithm in [Schrijver2000]). Végh and Benczúr [Benczur-Vegh] developed a purely combinatorial algorithm for optimal directed node-connectivity augmentation problem, an important special case of Theorem 4. Though not mentioned explicitly in the paper of Végh and Benczúr, their algorithm can be extended without much effort to the general case described in Theorem 4 when is crossing supermodular. This extended algorithm relies on an oracle for minimizing submodular functions. We should emphasize that the algorithm of Végh and Benczúr is particularly intricate and it is a natural goal to develop simpler algorithms for the special cases considered in the present work.
This theorem was earlier used to solve several connectivity augmentation problems. It should be, however, emphasized that even this general framework did not allow to handle simplicity. Even worse, there is no hope to extend Theorem 4 so as to characterize minimal simple digraphs covering since this problem, by relying on a result by Dürr et al. [DGM], was shown in [Berczi-Frank16a] (Theorem 12) to include NP-complete special cases. In [Berczi-Frank16a] and [Berczi-Frank16b], we developed other applications of the supermodular arc-covering theorem when simplicity could be guaranteed.
A main feature of the present approach to manage the above-mentioned special cases (when simplicity of the augmenting digraph is an expectation) is that, though Theorem 4 remains a fundamental starting point, relatively tedious additional work is needed. (The complications may be explained by the fact that some special cases are NP-complete while others are in NPco-NP.)
There are actually two issues here to be considered. The first one is to develop techniques for embedding special simplicity-requesting connectivity augmentation problems into the framework of Theorem 4. When this is successful, one has to resolve a second difficulty stemming from the somewhat complicated nature of an independent family of bi-sets in Theorem 4.
To demonstrate this second obstacle, consider the following digraph with a particularly simple structure. Let be an arc of if or , where and are two specified disjoint subsets of with . An earlier direct consequence of Theorem 4 (formulated in Section 3 as Theorem 20) does provide a formula for the minimum number of new arcs whose addition to any initial digraph results in a -connected digraph. But to prove that this minimum for our special digraph is actually equal to the total out-deficiency of the nodes of is rather tricky or tedious.
From an algorithmic point of view, it should be noted that the original proof of Hong et al. [Hong-Liu-Lai] gives rise to a polynomial algorithm which is purely combinatorial. The present approach makes use of the supermodular arc-covering theorem. Since [FrankJ31] and [Benczur-Vegh] describe polynomial algorithm to compute the optima in question, there are polynomial algorithms for finding the simple degree-specified digraphs with the prescribed connectivity properties. However, the algorithm in [FrankJ31] relies on the ellipsoid method while the general algorithm of Végh and Benczúr is particularly complex. Therefore developing a simple combinatorial algorithm for our cases requires further investigations.
Also, when, instead of exact degree-specifications, upper and lower bounds are prescribed for the in-degrees and out-degrees, the problem of characterizing the existence of simple -node-connected and degree-constrained digraphs, even in the special case , remains a challenging research task for the future.
1.1 Notions and notation
For a number , let . For a function and for , let . For a set-function and a family of sets, let For a family of sets, let denote the union of the members of .
Two subsets of a ground-set are said to be co-disjoint if their complements are disjoint. By a partition of a ground-set , we mean a family of disjoint subsets of whose union is . A subpartition of is a partition of a subset of . A co-partition (resp., co-subpartition) of is a family of subsets arising from a partition (subpartition) of by complementing each of its member. For a subpartition , we always assume that its members are non-empty but is allowed to be empty (that is, ).
Two subsets are crossing if none of is empty. A family of subsets is cross-free if it contains no two crossing members. A family of subsets is crossing if both and belong to whenever and are crossing members of .
When is a subset of , we write , while means that is a proper subset. The standard notation for set difference will be replaced by . When it does not cause any confusion, we do not distinguish a one-element set (often called a singleton) from its only element , and we use the notation for the singleton as well. For example, we write rather than , and stands for . In some situations, however, the formally precise notation must be used. For example, an arc in a digraph is said to enter a node even if is a loop (that is, ) while enters the singleton only if . That is a loop sitting at enters but does not enter . Therefore the in-degree (the number of arcs entering ) is equal to the in-degree plus the number of loops sitting at .
In a digraph , an arc enters a subset or leaves if . The in-degree of a subset is the number of arcs entering while the out-degree is the number of arcs leaving . Two arcs of a digraph are parallel if their heads coincide and their tails coincide. The oppositely directed arcs and are not parallel. We call a digraph simple if it has neither loops nor parallel arcs.
By the complete digraph , we mean the simple digraph on in which there is one arc from to for each ordered pair of distinct nodes, that is, has arcs. For two subsets , let denotes the subgraph of consisting of those arcs for which or . Then has arcs.
For two digraphs and on the same node-set, denotes the digraph consisting of the arcs of and . That is, has arcs.
A digraph covers a family of subsets if for every . A digraph covers a set-function on if for every .
By a bi-set we mean a pair of subsets for which . Here and are the outer set and the inner set of , respectively. A bi-set is trivial if or . The set is the wall of , while is its wall-size. Two bi-sets and are comparable if and or else and . The meet of two bi-sets and is defined by while their join is . Note that is a modular function in the sense that . Two bi-sets and are crossing if , , and they are not comparable. A family of bi-sets is crossing if both and belong to whenever and are crossing members of . A bi-set function is positively crossing supermodular if
whenever , and are crossing bi-sets.
An arc enters (or covers) a bi-set if enters both and . The in-degree of a bi-set is the number of arcs entering . Two bi-sets are independent if no arc can cover both, which is equivalent to requiring that their inner sets are disjoint or their outer sets are co-disjoint. A family of bi-sets is independent if their members are pairwise independent. Given a digraph , a bi-set is -one-way or just one-way if no arc of covers .
The degree-specified augmentation problem for -edge-connectivity was shown by the second author [FrankJ23] to be equivalent to Mader’s directed splitting off theorem [Mader82].
Theorem 5 ([FrankJ23]).
An initial digraph can be made -edge-connected by adding a digraph fitting if and only if and hold for every subset . If, in addition, (3) holds, then can be chosen loopless.
The second part immediately follows from the first one by applying the loop-reducing technique mentioned in Section 1 since loop-reduction never decreases the in-degree of a subset. Though the problem when simplicity of is requested remains open even in the special case when has no arcs, we are able to prove the following straight extension of Theorem 3.
Suppose that there is a simple digraph fitting a degree-specification (that is, (4) holds). A digraph can be made strongly connected by adding a simple digraph fitting a degree-specification if and only if inequality (5) holds for every pair of disjoint subsets for which there is a non-empty, proper subset of so that and .
This theorem is just a special case of the following.
Suppose that there is a simple digraph fitting a degree-specification . A -edge-connected digraph can be made -edge-connected by adding a simple digraph fitting a degree-specification if and only if inequality (5) holds for every pair of disjoint subsets for which there is a non-empty, proper subset of so that and .
Since the family of subsets of in-degree in a -edge-connected digraph is a crossing family, the following result immediately implies Theorem 7.
Let be a crossing family of non-empty proper subsets of . Suppose that there is a simple digraph fitting a degree specification , that is, (4) holds. There is a simple digraph fitting which covers if and only if
holds for every pair of disjoint subsets for which there is a member with .
Proof. Suppose that there is a requested digraph . By the simplicity of , there are at most arcs from to . Therefore the total number of arcs with tail in or with head in is at least . Moreover at least one arc enters and such an arc neither leaves an element of nor enters an element of , from which we obtain that , that is, (6) is necessary.
To prove sufficiency, observe first that the theorem is trivial if so we assume that . We need some further observations.
In particular, if for some , then , and if for some , then .
Proof. follows by applying (6) to and , while follows with the choice and . ∎
This claim immediately implies the following.
has at most pairwise disjoint and at most pairwise co-disjoint members.
and for every
Proof. By applying (4) to and , one gets , that is, , and is obtained analogously by choosing and . ∎
Define a bi-set function as follows. Let be zero everywhere apart from the next three types of bi-sets.
Type 1: For with , let .
Type 2: For , let .
Type 3: For , let
Note that the role of and is not symmetric in the definition of . Since , each bi-set with positive belongs to exactly one of the three types.
The bi-set function defined above is positively crossing supermodular.
Proof.Let and be two crossing bi-sets with and . Then neither of and is of Type 2. Suppose first that both and are of Type 1. Observe that if for some , then is of Type 3 and hence by Claim 9. Similarly, if for some , then is of Type 2 and hence . Therefore the supermodular inequality in this case follows from the assumption that the set-system is crossing.
If both and are of Type 3, then for some , and in this case the supermodular inequality holds actually with equality.
Finally, let be of Type 1 and let be of Type 3. Then for some with and for some . Observe that does not belong to any of the three types and hence . Furthermore, since is of Type 3, we have .
Since and are not comparable, and therefore ∎
It follows from the definition of that every digraph covering must have at least arcs.
Case 1. There is a loopless digraph with arcs covering .
Now from which follows for every . Analogously, we get for every . By the definition of , it also follows that covers .
Proof.Suppose, indirectly, that has two parallel arcs and from to . Consider the bi-set for . We have , a contradiction. ∎
We can conclude that in Case 1 the digraph requested by the theorem is indeed available.
Case 2. The minimum number of arcs of a loopless digraph covering is larger than .
Theorem 4 implies that in Case 2 there is an independent family of bi-sets for which . Then partitions into three parts according to the three possible types its members belong to. Therefore we have a subset , a subset , and a family of bi-sets so that
We may assume that is as small as possible.
There are no two members and of with the same inner set .
Proof.If indirectly there are two such members, then by the independence of . If we replace the two members and of by the single bi-set , then the resulting family is also independent since any arc covering covers at least one of and . Furthermore,
We claim that . Indeed, this is equivalent to , that is, , but this holds by Claim 11. Consequently, , contradicting the minimality of . ∎
We may assume that is as large as possible, and modulo this, is minimal.
For , one has , and for , one has .
Proof.The independence of implies . Suppose, indirectly, that there is an element . Replace the member of by . Since the resulting system is not independent, therefore there exists a member of that is covered by arc . Since is independent, this member is unique. As , Claim 14 implies that must be in , that is, for some . By leaving out from , we obtain an independent for which , contradicting the minimal choice of . ∎
Due to these claims, Condition (8) reduces to , which is equivalent to
Proof.If indirectly , then in which case (12) reduces to
Proof. Suppose, indirectly, that Then (12) reduces to
The independence of implies that all the members of include and these members are pairwise co-disjoint. Let consist of the complements of the members of (that is, ). Then is a subpartition of .
We have concluded that and .
Case A .
Therefore we must have . For , the independence of implies that and . As and , the independence of also implies that cannot have more than one member, that is, , and hence violate (6).
Case B .
Let be a simple digraph on nodes. Recall that a bi-set was called -one-way or just one-way if , that is, if no arc of enters both and . Recall the notation . The following lemma occured in [FrankJ31]. For completeness, we include its proof.
The following are equivalent.
(A1) is -connected.
(A2) for every non-trivial bi-set .
(A3) for every non-trivial one-way bi-set .
(B) There are openly disjoint -paths in for every ordered pair of nodes .
Proof.(A1) implies (B) by the directed node-version of Menger’s theorem.
(B)(A2). Let be a non-trivial bi-set, let and By (B), there are openly disjoint -paths. Each of them uses an arc entering or an element of the wall of , from which , that is, (A2) holds.
(A2)(A3). Indeed, (A3) is just a special case of (A2).
(A3)(A1). If (A1) fails to hold, then there is a subset of less than nodes so that is not strongly connected. Let be a non-empty proper subset of with no entering arc of and let . Then is a non-trivial bi-set with for which and , that is, (A3) also fails to hold. ∎
3.1 Connectivity augmentation: known results
Let be a starting digraph on nodes. The in-degree and out-degree functions of will be abbreviated by and , respectively. In the connectivity augmentation problem we want to make -connected by adding new arcs. Since parallel arcs and loops do not play any role in node-connectivity, we may and shall assume that is simple.
In one version of the connectivity augmentation problem, one strives for minimizing the number of arcs to be added. In this case, the optimal augmenting digraph is automatically simple. The following result is a direct consequence of Theorem 4.
Theorem 20 (Frank and Jordán, [FrankJ31]).
A digraph can be made -connected by adding a simple digraph with at most arcs if and only if
holds for every independent family of non-trivial -one-way bi-sets.
In what follows, denotes the family of non-trivial -one-way bi-sets. For any bi-set , let . With these terms, (16) requires that for every independent .
In a related version of the connectivity augmentation problem, the goal is to find an augmenting digraph fitting a degree-specification (meeting (1)) so that the augmented digraph is -connected. The paper [FrankJ31] described a characterization for the existence of such a , but in this case the augmenting digraph is not necessarily simple. This characterization can also be derived from Theorem 4.
There exists a digraph fitting such that is -connected if and only if
holds for every and for every independent family of non-trivial -one-way bi-sets with , and
holds for every and for every independent family of non-trivial one-way bi-sets with .
Note that in this theorem both parallel arcs and loops are allowed in the augmenting digraph . By using Theorem 4 and some standard steps, one can derive the following variation when loops are excluded.
If there is a loopless digraph fitting and if there is a digraph fitting for which is -connected, then can be chosen loopless.
Note that an analogous statement for -edge-connectivity in Theorem 5 follows immediately by applying the loop-reducing technique, but this approach does not seem to work here since a loop-reducing step may destroy -node-connectivity. To see this, let be the digraph on node-set with no arc. Let and let , , , and . Consider the digraph with arc set . This digraph is 2-node-connected and fits the degree-specification, but the loop-reducing technique replaces, say, the two arcs and by and , and the digraph arising in this way is not 2-connected since is not strongly connected. Analogously, the same happens when the two arcs and are replaced by and . Note that the digraph on with arc set is 2-connected, loopless, and fits the degree-specification.
3.2 Degree-specified connectivity augmentation preserving simplicity
Our present goal is to solve the degree-specified node-connectivity augmentation problem when simplicity of the augmented digraph is requested. With the help of a similar approach another natural variant, when only the simplicity of the augmenting digraph is requested, can also be managed. Theorem 7 provided a complete answer to this latter problem in the special case .
Let denote the complementary digraph of arising from the complete digraph by removing , that is, . In these terms, our goal is to find a degree-specified subgraph of for which is -connected. Note that in the case when an arbitrary digraph for possible new arcs is specified instead of , the problem becomes NP-complete even in the special case and since, for the degree specification , it is equivalent to finding a Hamiltonian circuit of .
We will show that the problem can be embedded into the framework of Theorem 4 in such a way that the augmented digraph provided by this theorem will automatically be simple. In this way, we shall obtain a good characterization for the general case when the initial digraph is arbitrary. This characterization, however, will include independent families of bi-sets, and in this sense it is more complicated than the one given in Theorem 3 for the special case .
In the special case when has no arcs at all, that is, when the goal is to find a simple degree-specified -connected digraph, the general characterization will be significantly simplified in such a way that the use of independent bi-set families is completely avoided, and we shall arrive at a characterization which is a straight extension of the one in Theorem 3 concerning the special case . Recall that the simple digraph with node-set was defined in Section 1.1 so that was an arc for distinct precisely if or . Also, was introduced above to denote the set of non-trivial -one-way bi-sets.
Let be a simple digraph with in- and out-degree functions and , respectively. There is a digraph fitting for which is simple and -connected if and only if
holds for every pair of subsets and for every independent family of non-trivial bi-sets which are one-way with respect to , where for and denotes the number of arcs for which and .