Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al. prove an approximation guarantee of two for connectivity augmentation problems where the connectivity requirements can be specified by so-called uncrossable functions. They state: ``Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. This property characterizes uncrossable functions\dots\ A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.'' Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result. (1) A 16-approximation algorithm for augmenting a family of small cuts of a graph $G$. (2) A $16 \cdot {\lceil k/u_{min} \rceil}$-approximation algorithm for the Cap-$k$-ECSS problem which is as follows: Given an undirected graph $G = (V,E)$ with edge costs $c \in \mathbb{Q}_{\geq 0}^E$ and edge capacities $u \in \mathbb{Z}_{\geq 0}^E$, find a minimum-cost subset of the edges $F\subseteq E$ such that the capacity of any cut in $(V,F)$ is at least $k$; we use $u_{min}$ to denote the minimum capacity of an edge in $E$. (3) An $O(1)$-approximation algorithm for the model of $(p,2)$-Flexible Graph Connectivity.


Introduction
Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar . . . A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems. Handling all non-uncrossable functions is ruled out by the fact that there exist instances corresponding to non-uncrossable {0, 1} functions whose relative duality gap is larger than any constant.
Our main contribution in this work is a novel analysis of the WGMV primal-dual approximation algorithm applied to a class of functions that strictly contain the class of uncrossable functions; we show that the algorithm still yields an O(1) approximation guarantee for this larger class. This new class of functions captures some well-studied network design problems. An application of our main result provides improved approximation ratios for the capacitated k-edge connected subgraph problem, some instances of the flexible graph connectivity problem, and the problem of augmenting all small cuts of a graph. A detailed discussion of our results can be found in Section 1.1. For the benefit of the reader, in Section 2.1 we give an overview of WGMV's primal-dual algorithm and its analysis.
The primal-dual algorithm for solving network design problems follows the common strategy of starting with a graph that has no edges and then iteratively buying (i.e., including) a subset of edges into the infeasible solution until feasibility is attained. Within each iteration, the algorithm's goal is to buy a cheap edge-set that fixes some or all of the infeasibility of the current solution. Let F denote the edge-set that has been bought until some step in the algorithm. A set of nodes S is said to be violated if the number of F -edges in the cut of S is less than the prespecified connectivity requirement of S. The algorithm deems an edge to be useful if it is in the cut of a violated set S. Clearly, the family of violated sets is important for the design and analysis of these algorithms, especially the inclusion-wise minimal violated sets. A family F of sets is called uncrossable if the following holds: Informally speaking, the uncrossability property ensures that the the minimal sets within the family can be considered independently. Formally, a minimal violated set A in an uncrossable family F cannot cross another set S ∈ F; otherwise, we get a contradiction since A, S ∈ F implies that either A ∩ S or A \ S is in F. This key property is one of the levers used in the design of O(1)-approximation algorithms for some network-design problems. Unfortunately, there are important problems in network design where the family of violated sets does not form an uncrossable family. For instance, see the instance described in Appendix B. This leads us to define a new class of set families that contains all uncrossable families.
Call a family F pliable if the following holds: In the full version of our paper, we show that the WGMV primal-dual algorithm has a superconstant approximation ratio for pliable families. Nevertheless, by enforcing an additional property on the given pliable family, we can establish that the WGMV algorithm yields an O(1) approximation. We call this additional assumption property (γ); see Section 1.1.1 for the formal definition. From a structural standpoint, this property still allows a minimal violated set to cross another violated set, but, crucially, it does not allow them to cross an arbitrary number of violated sets in arbitrary ways. As we show later, the fact that disparate network design problems can be captured by pliable families with property (γ) hints that this property is "just right".
The above connectivity augmentation problems can be understood in a general framework called f -connectivity. In this problem, we are given an undirected graph G = (V, E) on n vertices with nonnegative costs c ∈ Q E ≥0 on the edges and a requirement function f : 2 V → {0, 1} on subsets of vertices. We are interested in finding an edge-set F ⊆ E with minimum cost c(F ) := e∈F c e such that for all cuts δ(S), S ⊆ V , we have |δ(S)∩F | ≥ f (S). This problem can be formulated as the following integer program where binary variables x e model inclusion of edge e in F : We remark that in its most-general form, f -connectivity is hard to approximate within a logarithmic factor. This can be shown via a reduction from the hitting set problem. 1 Thus, research on f -connectivity has focused on instances where f has some nice structural properties.
Note that the problem of augmenting an uncrossable (pliable) family can be seen as an f -connectivity problem whose requirement function is an uncrossable (pliable) function.

Our Contributions
In this work, we introduce the class of pliable functions and study the approximation ratio of WGMV's algorithm on f -connectivity instances arising from pliable functions. To the best of our knowledge, we are the first to investigate the f -connectivity problem beyond uncrossable functions. As mentioned before, the algorithm of WGMV can perform poorly on an arbitrary instance with a pliable function f . In the full version [3, Section 6], we present an instance where the solution returned by the WGMV algorithm costs Ω( √ n) times the optimal cost.

Pliable Functions and Property (γ)
As alluded to in the introduction, the analysis of WGMV relies on the property that for any inclusion-wise minimal violated set C and any violated set S, either C is a subset of S or C is disjoint from S ([33, Lemma 5.1(3)]). This property does not hold when we apply the primal-dual method to augment a pliable function; see the instance described in Appendix B. Nevertheless, we carve out a subclass of pliable functions -still containing all uncrossable functions -for which the WGMV algorithm yields an O(1)-approximate solution. This subclass is characterized by the following structural property that allows for minimal violated sets to cross other violated sets, but in a limited way.

Property (γ)
: For any edge-set F ⊆ E and for any violated sets (w.r.t. f and F ) C, S 1 , S 2 , with S 1 ⊊ S 2 , the following conditional proposition holds: (C is inclusion-wise minimal) and (C crosses both S 1 and S 2 ) In the next three sections, we introduce the network-design applications where Theorem 3 gives new/improved approximation algorithms. In each of these applications, we setup an f -connectivity problem where the function f is a pliable function with property (γ).

Application 1: Augmenting a Family of Small Cuts
Our first application is on finding a minimum-cost augmentation of a family of small cuts in a graph. Formally, in an instance of the AugSmallCuts problem we are given an undirected capacitated graph G = (V, E) with edge-capacities u ∈ Q E ≥0 , a set of links L ⊆ V 2 with costs c ∈ Q L ≥0 , and a threshold λ ∈ Q ≥0 . A subset F ⊆ L of links is said to augment a node-set S if there exists a link e ∈ F with exactly one end-node in S. The objective is to find a minimum-cost F ⊆ L that augments all non-empty S ⊊ V with u(δ(S) ∩ E) < λ.
We remark that some special cases of the AugSmallCuts problem have been studied previously, and, to the best of our knowledge, there is no previous publication on the general version of this problem. Let λ(G) denote the minimum capacity of a cut of G, thus, λ(G) := min{u(δ(S) ∩ E) : ∅ ⊊ S ⊊ V }. Assuming u is integral and λ = λ(G) + 1, we get the well-known connectivity augmentation problem for which constant-factor approximation algorithms are known [13,23]. On the other hand, when λ = ∞, a minimum-cost spanning tree of (V, L), if one exists, gives an optimal solution to the problem.
Our main result here is an O(1)-approximation algorithm for the AugSmallCuts problem that works for any choice of λ. The proof of the following theorem is given in Section 4.

▶ Theorem 4.
There is a 16-approximation algorithm for the AugSmallCuts problem.
As an aside, we refer the reader to Benczur & Goemans [4] and the references therein for results on the representations of the near-minimum cuts of graphs; they do not study the problem of augmenting the near-minimum cuts.
In Appendix B, we give a small instance of the AugSmallCuts problem that illustrates some of the technical challenges which arise while working with the f -connectivity problem for a pliable function with property (γ). The instance described has bizarre properties that do not arise when working with uncrossable functions. First, it has a minimal violated set which crosses another violated set. Second, none of the optimal solutions to the dual LP of the f -connectivity problem are supported on a laminar family. The latter was believed to be a major hindrance to developing constant-factor approximation algorithms for general network-design problems.

Application 2: Capacitated k-Edge-Connected Subgraph Problem
In the capacitated k-edge-connected subgraph problem (Cap-k-ECSS), we are given an undirected graph G = (V, E) with edge costs c ∈ Q E ≥0 and edge capacities u ∈ Z E ≥0 . The goal is to find a minimum-cost subset of the edges F ⊆ E such that the capacity across any cut in (V, F ) is at least k, i.e., u(δ F (S)) ≥ k for all non-empty sets S ⊊ V . Let u max and u min , respectively, denote the maximum capacity of an edge in E and the minimum capacity of an edge in E. We may assume (w.l.o.g.) that u max ≤ k.
We mention that there are well-known 2-approximation algorithms for the special case of the Cap-k-ECSS problem with u max = u min = 1, which is the problem of finding a minimum-cost k-edge connected spanning subgraph. Khuller & Vishkin [24] presented a combinatorial 2-approximation algorithm and Jain [20] matched this approximation guarantee via the iterative rounding method.
Goemans et al. [17] gave a 2k-approximation algorithm for the general Cap-k-ECSS problem. Chakrabarty et al. [6] gave a randomized O(log |V (G)|)-approximation algorithm; note that this approximation guarantee is independent of k but does depend on the size of the underlying graph. Recently, Boyd et al. [5] improved on these results by providing a min(k, 2u max )-approximation algorithm. In this work, we give a (16·⌈k/u min ⌉)-approximation algorithm, which leads to improved approximation guarantees when both u min and u max are sufficiently large. In particular, in the regime when k ≥ u max ≥ u min ≥ 32 and u min · u max ≥ 16k.
The proof of Theorem 5 can be found in Section 5.

Application 3: (p, 2)-Flexible Graph Connectivity
Adjiashvili, Hommelsheim and Mühlenthaler [1] introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur [5] introduced a generalization of FGC. Let p ≥ 1 and q ≥ 0 be integers. In an instance of the (p, q)-Flexible Graph Connectivity problem, denoted (p, q)-FGC, we are given an undirected graph G = (V, E), a partition of E into a set of safe edges S and a set of unsafe edges U, and nonnegative edge-costs c ∈ Q E ≥0 . A subset F ⊆ E of edges is feasible for the (p, q)-FGC problem if for any set F ′ consisting of at most q unsafe edges, the subgraph Boyd et al. [5] presented a 4-approximation algorithm for (p, 1)-FGC based on the WGMV primal-dual method, and they gave an O(q log n)-approximation algorithm for general (p, q)-FGC and a (q + 1)-approximation for (1, q)-FGC. Concurrently with our work, Chekuri and Jain [8] obtained O(p)-approximation algorithms for (p, 2)-FGC, (p, 3)-FGC and (2p, 4)-FGC; in particular, they present a (2p + 4)-approximation ratio for (p, 2)-FGC. Chekuri and Jain have several other results for network design in non-uniform fault models; [7] have results on the flexible graph connectivity problem that arises from the classical survivable network design problem, which they call (p, q)-Flex-SNDP.
Our main result here is an O(1)-approximation algorithm for the (p, 2)-FGC problem.
▶ Theorem 6. There is a 20-approximation algorithm for the (p, 2)-FGC problem. Moreover, for even p, the approximation ratio is 6.
Note that in comparison to [8], Theorem 6 yields a better approximation ratio when p > 8 or p ∈ {2, 4, 6, 8}. For p = 1, the approximation ratio of 3 from [5] is better than the guarantees given by [8] and Theorem 6. The proof of Theorem 6 can be found in Section 6.

Related work
Goemans & Williamson [18] introduced the notion of proper functions with the motivation of designing approximation algorithms for problems in network design. They formulated several of these problems as the f -connectivity problem where f is a proper function. A symmetric function f : Jain [20] designed the iterative rounding framework for the setting when f is weakly supermodular and presented a 2-approximation algorithm. A function f is said to be weakly One can show that proper functions are weakly supermodular. We mention that there are examples of uncrossable functions that are not weakly supermodular, see [5].

Preliminaries
This section has definitions and preliminary results. Our notation and terms are consistent with [10,30], and readers are referred to those texts for further information.
For a positive integer k, we use [k] to denote the set {1, . . . , k}. For a ground-set V and is a subset of the other one. A family of sets L ⊆ 2 V is said to be laminar if for any two sets A, B ∈ L either A and B are disjoint or one of them is a subset of the other one.
We may use abbreviations for some standard terms, e.g., we may use "(p, q)-FGC" as an abbreviation for "the (p, q)-FGC problem". In some of our discussions, we may use the informal phrasing "we apply the primal-dual method to augment a pliable function" instead of the phrasing "we apply the primal-dual method to an f -connectivity problem where the function f is a pliable function".

Graphs, Subgraphs, and Related Notions
Let G = (V, E) be an undirected multi-graph (possibly containing parallel edges but no loops) with non-negative costs c ∈ R E ≥0 on the edges. We take G to be the input graph, and we use n to denote |V (G)|. For a set of edges F ⊆ E(G), c(F ) := e∈F c(e), and for a subgraph G ′ of G, c(G ′ ) := c(E(G ′ )). For any instance G, we use opt(G) to denote the minimum cost of a feasible subgraph (i.e., a subgraph that satisfies the requirements of the problem). When there is no danger of ambiguity, we use opt rather than opt(G).
Let G = (V, E) be any multi-graph, let A, B ⊆ V be two disjoint node-sets, and let F ⊆ V be an edge-set. We denote the multi-set of edges of G with exactly one end-node in each of A and B by E (A, B), . By a p-cut we mean a cut of size p. We use G[A] to denote the subgraph of G induced by A, G − A to denote the subgraph of G induced by V \ A, and G − F to denote the graph (V, E \ F ). We may use relaxed notation for singleton sets, e.g., we may use We use the following observations. ▶ Fact 7. Let A, B ⊆ V be a pair of crossing sets. For any edge-set F ⊆ V 2 and any ., if f (S) = 1 and there are no F -edges in the cut δ(S). We drop f and F when they are clear from the context. The next observation states that the violated sets w.r.t. any pliable function f and any "augmenting" edge-set F form a pliable family.

The WGMV Primal-Dual Algorithm for Uncrossable Functions
In this section, we give a brief description of the primal-dual algorithm of Williamson et al. [33] that achieves approximation ratio 2 for an f -connectivity problem where the function f is an uncrossable function.
▶ Theorem 9 (Lemma 2.1 in [33]). The algorithm and its analysis are based on the following LP relaxation of (f -IP) (stated on the left) and its dual.
The algorithm starts with an infeasible primal solution F = ∅, which corresponds to x = χ F = 0 ∈ {0, 1} E , and a feasible dual solution y = 0. At any time, we say that S ∈ S is violated if δ F (S) = ∅, i.e., the primal-covering constraint for S is not satisfied. We call inclusion-wise minimal violated sets as active sets. An edge e ∈ E is said to be tight if S∈S:e∈δ(S) y S = c e , i.e., the dual-packing constraint for e is tight. Throughout the algorithm, the following conditions are maintained: (i) integrality of the primal solution; (ii) feasibility of the dual solution; (iii) y S is never decreased for any S; and (iv) y S may only be increased for S ∈ S that are active.
The algorithm has two stages. In the first stage, the algorithm iteratively improves primal feasibility by including tight edges in F that are incident to active sets. If no such edge exists, then the algorithm uniformly increases y S for all active sets S until a new edge becomes tight. The first stage ends when x = χ F becomes feasible. In the second stage, called reverse delete, the algorithm removes redundant edges from F . Initially F ′ = F . The algorithm examines edges picked in the first stage in reverse order, and discards edges from F ′ as long as feasibility is maintained. Note that F ′ is feasible if the subroutine in the hypothesis of Theorem 9 does not find any (minimal) violated sets.
The analysis of the 2-approximation ratio is based on showing that a relaxed form of the complementary slackness conditions hold on "average". Let F ′ and y be the primal and dual solutions returned by the algorithm. By the design of the algorithm, S∈S:e∈δ(S) y S = c e holds for any edge e ∈ F ′ . Thus, the cost of F ′ can be written as e∈F ′ S∈S:e∈δ(S) y S = S∈S y S · |δ F ′ (S)|. Observe that the approximation ratio follows from showing that the algorithm always maintains the following inequality: Consider any iteration and recall that the dual variables corresponding to active sets were uniformly increased by an ε > 0 amount, until some edge became tight. Let C denote the collection of active sets during this iteration. During this iteration, the left-hand side of (1) increases by ε · S∈C |δ F ′ (S)| and the right-hand side increases by 2 · ε · |C|. Thus, (1) is maintained if one can show that the average F ′ -degree of active sets in any iteration is ≤ 2, and this forms the crux of the WGMV analysis.
We refer the reader to [19] for a detailed discussion of the primal-dual method for network design problems.

Extending the WGMV Primal-Dual Method to Pliable functions
In this section, we prove our main result, Theorem 3: we show that the primal-dual algorithm outlined in Section 2.1 is a 16-approximation algorithm for the f -connectivity problem where f is a pliable function with property (γ). Our analysis follows the same high-level plan as that of Williamson et al. [33] which was outlined in Section 2.1. We will show that, in any iteration of the first stage of the primal-dual algorithm, C∈C |δ F ′ (C)| ≤ 16|C|, where C is the collection of active sets in that iteration, and F ′ is the set of edges output by the algorithm at termination, after the reverse delete stage. For the remainder of this proof we assume that the iteration, and thus C, is fixed. We define H := ∪ C∈C δ F ′ (C). (Informally speaking, H is the subset of F ′ that is relevant for the analysis of our fixed iteration.) Additionally, to ease notation when discussing a laminar family of sets, we say that two sets A, B overlap if A \ B, A ∩ B  Proof. Suppose that two sets C 1 , C 2 ∈ C intersect. Then due to property (i) of pliable functions, one of the sets C 1 ∩ C 2 , C 1 \ C 2 , or C 2 \ C 1 is violated; thus, a proper subset of either C 1 or C 2 is violated. This is a contradiction because C 1 and C 2 are minimal violated sets and no proper subset of C 1 (respectively, C 2 ) is violated. ◀ Let L be the laminar family of witness sets together with the node-set V . Let T be a rooted tree that represents L; for each set S ∈ L, there is a node v S in T, and the node v V is taken to be the root of T. The edges of T are oriented away from the root; thus, T has an oriented edge (v Q , v S ) iff Q is the smallest set of L that properly contains the set S of L. Let ψ be a mapping from C to L that maps each active set C to the smallest set S ∈ L that contains it. If a node v S of T has some active set mapped to its associated set S, then we call v S active and we assign the color red to v S . Moreover, we assign the color green to each of the non-active nodes of T that are incident to three or more edges of T; thus, node v S of T is green iff deg T (v S ) ≥ 3 and v S is not active. Finally, we assign the color black to each of the remaining nodes of T; thus, node v S of T is black iff deg T (v S ) ≤ 2 and v S is not active.
Let the number of red, green and black nodes of T be denoted by n R , n G and n B , respectively. Clearly, n R + n G + n B = |T| = |F ′ | + 1. Let n L denote the number of leaf nodes of T.
▶ Lemma 13. The following are true: Proof. The first claim follows by repeating the argument in [33, Lemma 5.3]. Next, by (i), we have n L ≤ n R . Moreover, we have n G ≤ n L because the number of leaves in any tree is at least the number of nodes that are incident to three or more edges of the tree. ◀ Observe that each black node of T is incident to two edges of T; thus, every black non-root node of T has a unique child.
Let us sketch our plan for proving Theorem 3. Clearly, the theorem would follow from the inequality C∈C |δ F ′ (C)| ≤ O(1) · |C|; thus, we need to prove an upper bound of O(|C|) on the number of "incidences" between the edges of F ′ and the cuts δ(C) of the active sets C ∈ C. We start by assigning a token to T corresponding to each "incidence". In more detail, for each edge e ∈ F ′ and cut δ(C) such that C ∈ C and e ∈ δ(C) we assign one token to the node v Se of T that represents the witness set S e of the edge e. Thus, the total number of tokens assigned to T is C∈C |δ F ′ (C)|; moreover, after the initial assignment, it can be seen that each node of T has ≤ 2 tokens (see Lemma 14 below). Then we redistribute the tokens according to a simple rule such that (after redistributing) each of the red/green nodes has ≤ 8 tokens and each of the black nodes has no tokens. Lemma 15 (below) proves this key claim by applying property (γ). The key claim implies that the total number of tokens assigned to T is ≤ 8n R + 8n G ≤ 16n R ≤ 16|C| (by Lemma 13). This concludes our sketch.
We apply the following two-phase scheme to assign tokens to the nodes of T.
In the first phase, for C ∈ C and e ∈ δ F ′ (C), we assign a new token to the node v Se corresponding to the witness set S e for the edge e. At the end of the first phase, observe that the root v V of T has no tokens (since the set V cannot be a witness set).
In the second phase, we apply a root-to-leaves scan of T (starting from the root v V ). Whenever we scan a black node, then we move all the tokens at that node to its unique child node. (There are no changes to the token distribution when we scan a red node or a green node.) ▶ Lemma 14. At the end of the first phase, each node of T has ≤ 2 tokens.
Proof. Consider a non-root node v Se of T. This node corresponds to a witness set S e ∈ L and e is the unique edge of F ′ in δ(S e ). The edge e is in ≤ 2 of the cuts δ(C), C ∈ C, because the active sets are pairwise disjoint (in other words, the number of "incidences" for e is ≤ 2 Let C ∈ C be an active set such that e 1 ∈ δ(C).
▷ Claim 16. C is not a subset of S 1 .
For the sake of contradiction, suppose that C is a subset of S 1 . Since e 1 has (exactly) one end-node in C and b 1 ̸ ∈ S 1 , we have a 1 ∈ C. Let W be the smallest set in L that contains C. Then W ⊆ S 1 , and, possibly, W = S 1 . Thus, we have a 1 ∈ W and b 1 ̸ ∈ W , hence, e 1 ∈ δ(W ). Then we must have W = S 1 (since e 1 is in exactly one of the cuts δ(S), S ∈ L). Then the mapping ψ from C to L maps C to W = S 1 , hence, v S1 is colored red. This is a contradiction.
▷ Claim 17. C crosses each of the sets S 2 , S 3 , S 4 . First, observe that e 1 has (exactly) one end-node in C and has both end-nodes in S 2 . Hence, both S 2 ∩ C and S 2 \ C are non-empty. Next, using Claim 16, we can prove that C is not a subset of S 2 . Otherwise, S 2 would be the smallest set in L that contains C, hence, v S2 would be colored red. Repeating the same argument, we can prove that C is not a subset of S 3 , and, moreover, C is not a subset of S 4 . Finally, note that V \ (C ∪ S 4 ) is non-empty. Otherwise, at least one of C \ S 4 or C ∩ S 4 would be violated, since f is a pliable function, and that would contradict the fact that C is a minimal violated set. Observe that S 2 crosses C because all four sets S 2 ∩ C, Similarly, it can be seen that S 3 crosses C, and S 4 crosses C.
For the sake of contradiction, suppose that both sets S 3 \ (C ∪ S 2 ), S 4 \ (C ∪ S 3 ) are empty. Then C ⊇ S 4 \ S 3 and C ⊇ S 3 \ S 2 . Consequently, both end-nodes of e 3 are in C (since a 3 ∈ S 3 \ S 2 and b 3 ∈ S 4 \ S 3 ). This leads to a contradiction, since e 3 ∈ F ′ is incident to an active set in C, call it C 3 (i.e., e 3 ∈ δ(C 3 )), hence, one of the end-nodes of e 3 is in both C and C 3 , whereas the active sets are pairwise disjoint.

15:12 Generalizing the WGMV Primal-Dual Method
To conclude the proof of the lemma, suppose that S 4 \(C ∪S 3 ) is non-empty (by Claim 18); the other case, namely, S 3 \ (C ∪ S 2 ) ̸ = ∅, can be handled by the same arguments. Then, by property (γ), S 4 \ (C ∪ S 3 ) is a violated set, therefore, it contains a minimal violated set, call it C. Clearly, the mapping ψ from C to L maps the active set C to a set S C . Either S C = S 4 or else S C is a subset of of S 4 \ S 3 . Both cases give contradictions; in the first case, S 4 is colored red, and in the second case, v S4 has ≥ 2 children in T so that S 4 is colored either green or red. Thus, we have proved the first part of the lemma.
The second part of the lemma follows by Lemma 13 and the sketch given below Lemma 13. In more detail, at the start of the second phase, each node of T has ≤ 2 tokens, by Lemma 14. In the second phase, we redistribute the tokens such that each (non-root) black node ends up with no tokens, and each red/green node v S receives ≤ 6 redistributed tokens because there are ≤ 3 black ancestor nodes of v S that could send their tokens to v S (by the first part of the lemma). Hence, each non-root non-black node has ≤ 8 tokens, after token redistribution. ◀

O(1)-Approximation Algorithm for Augmenting Small Cuts
In this section, we give a 16-approximation algorithm for the AugSmallCuts problem, thereby proving Theorem 4. Our algorithm for AugSmallCuts is based on a reduction to an instance of the f -connectivity problem on the graph H = (V, L) for a pliable function f with property (γ).
Recall the AugSmallCuts problem: we are given an undirected graph G = (V, E) with edge-capacities u ∈ Q E ≥0 , a set of links L ⊆ V 2 with costs c ∈ Q L ≥0 , and a threshold λ ∈ Q ≥0 . A subset F ⊆ L of links is said to augment a node-set S if there exists a link e ∈ F with exactly one end-node in S. The objective is to find a minimum-cost F ⊆ L that augments all non-empty S ⊊ V with u(δ E (S)) < λ.
Proof of Theorem 4. Define f : 2 V → {0, 1} such that f (S) = 1 if and only if S / ∈ {∅, V } and u(δ E (S)) < λ. We apply Theorem 3 for the f -connectivity problem on the graph H = (V, L) with edge-costs c ∈ Q L ≥0 to obtain a 16-approximate solution F ⊆ L. By our choice of f , there is a one-to-one cost-preserving correspondence between feasible augmentations for AugSmallCuts and feasible solutions to the f -connectivity problem. Thus, it remains to argue that the assumptions of Theorem 3 hold.
Next, we prove the two claims. Note that the sets C ∩ S 1 , C \ S 1 , S 1 \ C, V \ (C ∪ S 1 ) are non-empty, and note that f Last, we describe a polynomial-time subroutine that for any F ⊆ L gives the collection of all minimal violated sets w.r.t. f , F . Assign a capacity of λ to all edges in F , and consider the graph G ′ = (V, E ′ ) where E ′ := E ∪ F . Then, the family of minimal violated sets is given by We use the notion of solid sets to find all such minimally violated sets; see Naor, Gusfield, and Martel [29] and see Frank's book [12]. A solid set of an undirected graph H = (V, E ′′ ) with capacities w ∈ R E ′′ ≥0 on its edges is a non-empty node-set Z ⊊ V such that w(δ E ′′ (X)) > w(δ E ′′ (Z)) for all non-empty X ⊊ Z. Note that the family of minimal violated sets of interest to us is a sub-family of the family of solid sets of G ′ . The family of all solid sets of a graph can be listed in polynomial time, see [29] and [12,Chapter 7.3]. Hence, we can find all minimal violated sets w.r.t. f , F in polynomial time, by examining the list of solid sets to check (1) whether there is a solid set S that is violated, and (2) whether every proper subset of S that is a solid set is not violated. This completes the proof of the theorem. ◀

O(k/u min )-Approximation Algorithm for the Capacitated k-Edge-Connected Subgraph Problem
In this section, we give a 16 · ⌈k/u min ⌉-approximation algorithm for the Cap-k-ECSS problem, thereby proving Theorem 5. Our algorithm is based on repeated applications of Theorem 4. Recall the capacitated k-edge-connected subgraph problem (Cap-k-ECSS): we are given an undirected graph G = (V, E) with edge costs c ∈ Q E ≥0 and edge capacities u ∈ Z E ≥0 . The goal is to find a minimum-cost subset of the edges F ⊆ E such that the capacity across any cut in (V, F ) is at least k, i.e., u(δ F (S)) ≥ k for all non-empty sets S ⊊ V .
Proof of Theorem 5. The algorithm is as follows: Initialize F := ∅. While the minimum capacity of a cut δ(S), ∅ ̸ = S ⊊ V, in (V, F ) is less than k, run the approximation algorithm from Theorem 4 with input G = (V, F ) and L = E \ F , to augment all cuts δ(S), ∅ ̸ = S ⊊ V, with u(δ(S)) < k and obtain a valid augmentation F ′ ⊆ L. Update F by adding F ′ , that is, F := F ∪ F ′ . On exiting the while loop, output the set of edges F . At any step of the algorithm, let λ denote the minimum capacity of a cut in (V, F ), i.e., λ := min{u(δ F (S)) : ∅ ⊊ S ⊊ V }.
The above algorithm outputs a feasible solution since, upon exiting the while loop, λ is at least k. Let F * ⊆ E be an optimal solution to the Cap-k-ECSS instance. Notice that F * \ F is a feasible choice for F ′ during any iteration of the while loop. Hence, by Theorem 4, c(F ′ ) ≤ 16 · c(F * ). We claim that the above algorithm requires at most ⌈ k umin ⌉ iterations of the while loop. This holds because each iteration of the while loop (except possibly the last iteration) raises λ by at least u min . (At the start of the last iteration, k − λ could be less than u min , and, at the end of the last iteration, λ could be equal to k). Hence, at the end of the algorithm, c(F ) ≤ 16 · ⌈ k umin ⌉c(F * ). This completes the proof. ◀ I C A L P 2 0 2 3

15:14 Generalizing the WGMV Primal-Dual Method
We remark that our new result (Theorem 4) is critical for the bound of ⌈ k umin ⌉ on the number of iterations of this algorithm. Earlier methods only allowed augmentations of minimum cuts, so such methods may require as many as Ω(k) iterations. (In more detail, the earlier methods would augment the cuts of (V, F ) of capacity λ but would not augment the cuts of capacity ≥ λ + 1; thus, cuts of capacity λ + 1 could survive the augmentation step.)

O(1)-Approximation Algorithm for (p, 2)-FGC
In this section, we present a 20-approximation algorithm for (p, 2)-FGC, by applying our results from Section 3. Recall (from Section 1) that the algorithmic goal in (p, 2)-FGC is to find a minimum-cost edge-set F such that for any pair of unsafe edges e, f ∈ F ∩ U, the subgraph (V, Our algorithm works in two stages. First, we compute a feasible edge-set F 1 for (p, 1)-FGC on the same input graph, by applying the 4-approximation algorithm of [5]. We then augment the subgraph (V, F 1 ) using additional edges. Since F 1 is a feasible edge-set for (p, 1)-FGC, any cut δ(S), ∅ ⊊ S ⊊ V , in the subgraph (V, F 1 ) either (i) has at least p safe edges or (ii) has at least p + 1 edges (see below for a detailed argument). Thus the cuts that need to be augmented have exactly p + 1 edges and contain at least two unsafe edges. Let us call such cuts deficient. Augmenting all deficient cuts by at least one (safe or unsafe) edge will ensure that we have a feasible solution to (p, 2)-FGC.
The following example shows that when p is odd, then the function f in the f -connectivity problem associated with (p, 2)-FGC may not be an uncrossable function. In other words, the indicator function f : 2 V → {0, 1} of the sets S such that δ(S) is a deficient cut could violate the definition of an uncrossable function. Proof of Theorem 6. In the following, we use F to denote the set of edges picked by the algorithm at any step of the execution; we mention that our correctness arguments are valid despite this ambiguous notation; moreover, we use δ(S) rather than δ F (S) to refer to a cut of the subgraph (V, F ), where ∅ ̸ = S ⊆ V . Since F is a feasible edge-set for (p, 1)-FGC, any cut δ(S) (where ∅ ̸ = S ⊆ V ) either (i) has at least p safe edges or (ii) has at least p + 1 edges. Consider a node-set S that violates the requirements of the (p, 2)-FGC problem. We have ∅ ̸ = S ⊊ V and there exist two unsafe edges e, f ∈ δ(S) such that |δ F (S) \ {e, f }| ≤ p − 1. Since F is feasible for (p, 1)-FGC, we have |δ(S) \ {e}| ≥ p and |δ(S) \ {f }| ≥ p. Thus, |δ F (S)| = p + 1. In other words, the node-sets S that need to be augmented have exactly p + 1 edges in δ(S), at least two of which are unsafe edges. Augmenting all such violated sets by at least one (safe or unsafe) edge will result in a feasible solution to (p, 2)-FGC. Let f : 2 V → {0, 1} be the indicator function of these violated sets. Observe that f is symmetric, that is, f (S) = f (V \ S) for any S ⊆ V ; this additional property of f is useful for our arguments. We claim that f is a pliable function that satisfies property (γ), hence, we obtain an O(1)-approximation algorithm for (p, 2)-FGC, via the primal-dual method and Theorem 3.
Our proof of the following key lemma is presented in [3,Section 5].
▶ Lemma 20. f is a pliable function that satisfies property (γ). Moreover, for even p, f is an uncrossable function.
Lastly, we show that there is a polynomial-time subroutine for computing the minimal violated sets. Consider the graph (V, F ). Note that size of a minimum cut of (V, F ) is at least p since F is a feasible edge-set for (p, 1)-FGC. The violated sets are subsets S ⊆ V such that δ(S) contains exactly p + 1 edges, at least two of which are unsafe edges. Clearly, all the violated sets are contained in the family of sets S such that δ(S) is a 2-approximate min-cut of (V, F ); in other words, {S ⊊ V : p ≤ |δ(S)| ≤ 2p} contains all the violated sets. It is well known that the family of 2-approximate min-cuts in a graph can be listed in polynomial time, see [22,28]. Hence, we can find all violated sets and all minimally violated sets in polynomial time.
Thus, we have a 20-approximation algorithm for (p, 2)-FGC via the primal-dual algorithm of [33] based on our results in Section 3. Furthermore, for even p, the approximation ratio is 6(= 4 + 2) since the additive approximation-loss for the augmenting step is 2 when f is uncrossable (see Theorem 9). This completes the proof of Theorem 6. ◀ Figure 1 An instance of the AugSmallCuts problem where none of the optimal dual solutions have a laminar support.
Since each F ∈ C has cost 3, the optimal value for the instance is 3. Next, since L contains at least two links from every nontrivial cut, the vector x ∈ [0, 1] L with x e = 1 2 , ∀e ∈ L is a feasible augmentation for the fractional version of the instance, i.e., x is feasible for the primal LP given in (2.1), Section 2. Therefore, the optimal value of the primal LP is at most 5 2 . Now, consider the dual LP, which is explicitly stated below. The dual packing-constraints are listed according to the following ordering of the links: 12, 23, 34, 41. For notational convenience, we use the shorthand y 1 to denote the dual variable y {1} corresponding to the set {1}. We use similar shorthand to refer to the dual variables of the other sets; thus, y 234 refers to the dual variable y {2,3,4} , etc. Observe that adding all packing constraints gives 2 · S∈A (y S + y V \S ) ≤ 5, hence, the optimal value of the dual LP is at most 5/2. Moreover, a feasible dual solution with objective 5/2 must satisfy the following conditions: y 1 + y 234 = y 23 + y 14 = y 123 + y 4 = 1 2 and y 12 + y 34 = 1.
Clearly, there is at least one solution to the above set of equations, hence, by LP duality, the optimal value of both the primal LP and the dual LP is 5/2. Furthermore, any optimal dual solution y * satisfies max(y * S , y * V \S ) > 0 for all S ∈ A (by the above set of equations). We conclude by arguing that for any optimal dual solution y * , its support S(y * ) = {S ⊆ V : y * S > 0} is non-laminar, because some two sets A, B ∈ S(y * ) cross. Since the relation A crosses B is closed under taking set-complements (w.r.t. the ground-set V ), we may assume w.l.o.g. that the support contains each set in A = {{1}, {1, 2}, {2, 3}, {1, 2, 3}}. The support of y * is not laminar because {1, 2} and {2, 3} cross.