a primal–dual schema based approximation algorithm for the element connectivity problem

Journal of Algorithms 45 (2002) 1–15

www.academicpress.com

A primal–dual schema based approximationalgorithm for the element connectivity problem

Kamal Jain,a,1 Ion Mandoiu,b,2 Vijay V. Vazirani,c,3 andDavid P. Williamsond,∗

a Microsoft Research, One Microsoft Way, Redmond, WA 98052, USAb University of California at San Diego, Computer Science and Engineering Department,

La Jolla, CA 92093, USAc Georgia Institute of Technology, College of Computing, 801 Atlantic Drive,

Atlanta, GA 30332, USAd IBM Almaden Research Center, 650 Harry Rd., San Jose, CA 95120, USA

Received 5 April 1999

Abstract

The element connectivity problem falls in the category of survivable network designproblems—it is intermediate to the versions that ask for edge-disjoint and vertex-disjointpaths. The edge version is by now well understood from the view-point of approximationalgorithms [Williamson et al., Combinatorica 15 (1995) 435–454; Goemans et al., in:SODA ’94, 223–232; Jain, Combinatorica 21 (2001) 39–60], but very little is known aboutthe vertex version. In our problem, vertices are partitioned into two sets: terminals andnonterminals. Only edges and nonterminals can fail—we refer to them aselements—andonly pairs of terminals have connectivity requirements, specifying the number of element-disjoint paths required. Our algorithm achieves an approximation guarantee of factor 2Hk ,wherek is the largest requirement andHn = 1+ 1

2 + · · · + 1n . Besides providing possible

* Corresponding author.E-mail addresses:[email protected] (K. Jain), [email protected] (I. Mandoiu),

[email protected] (V.V. Vazirani), [email protected] (D.P. Williamson).1 Microsoft Research. This research was performed while the author was a student at the Georgia

Institute of Technology and supported by NSF Grant CCR 9627308.2 This research was performed while the author was a student at the Georgia Institute of

Technology and supported by NSF Grant CCR 9627308.3 Supported by NSF Grant CCR 9627308.

0196-6774/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0196-6774(02)00222-5

2 K. Jain et al. / Journal of Algorithms 45 (2002) 1–15

insights for solving the vertex-disjoint paths version, the element connectivity problem isof independent interest, since it models a realistic situation. 2002 Elsevier Science (USA). All rights reserved.

1. Introduction

Given an undirected graphG = (V ,E) with non-negative costsce for edgese ∈ E, and a valueruv for each pair of verticesu,v ∈ V , thesurvivable networkdesign problem(SNDP) is that of finding a minimum-cost subgraph such thatthere areruv disjoint paths between each pair of verticesu and v. The pathscan be required to be either edge-disjoint or vertex-disjoint; we refer to theformer as theedge-connectivity SNDP(EC-SNDP) and the latter as thevertex-connectivity SNDP(VC-SNDP). The survivable network design problem is anatural generalization of the Steiner tree problem, and captures the problemof designing a minimum-cost network such thatu and v are still connectedin the network after up toruv − 1 links fail (for EC-SNDP) or up toruv − 1links or nodes fail (VC-SNDP). The survivable network design problem arisesfrom problems in the telecommunications industry (cf. [7,11]) and has beenstudied from many different approaches including polyhedral combinatorics [7,14], interchange heuristics [12], min-max relations [1] (in the unweighted case),approximation algorithms [5,13,16], and implementations thereof [11]. In thispaper, we consider approximation algorithms for the SNDP. Aρ-approximationalgorithm for the SNDP runs in polynomial time and finds a solution of value nomore thanρ times the value of an optimal solution.

There appears to be a qualitative difference in difficulty between EC-SNDPand VC-SNDP. For example, although an exact min-max formula is known forthe number of edges needed to add to a graph to have it satisfy the edge-disjointpaths constraint [1], no similar formula is known in the case of the vertex-disjointpaths.

The first approximation algorithm for EC-SNDP, achieving a guarantee offactor 2k, where k = maxu,v ruv , was given by Williamson et al. [16]. Thispaper also formalized a basic mechanism for using the primal–dual schema.The algorithm of [16] picks edges for the desired subgraph ink phases. Theaugmentation problem for each phase can be written as an integer program, theLP-relaxation of which is solved within factor 2 by the primal–dual processalluded to above. A clever reordering of the augmentations led to a 2Hk-approximation algorithm due to Goemans et al. [5], whereHn = 1+ 1

2+· · ·+ 1n≈

lnn. Jain has given a 2-approximation algorithm for the problem [8], using arounding-based algorithm. For a detailed overview of these algorithmic ideasand the primal–dual schema, we refer the reader to the survey article [6] or thebook [15].

K. Jain et al. / Journal of Algorithms 45 (2002) 1–15 3

In the case of VC-SNDP, however, very little is known in terms of ap-proximation algorithms. Using the basic algorithmic outline established in [5,16], Ravi and Williamson [13] give a 3-approximation algorithm in the casethat ruv ∈ {0,1,2}. For the problem in whichruv = k for all u,v ∈ V , alsoknown as the minimum-costk-vertex-connected subgraph problem, there is also a(2+2(k − 1)/n)-approximation algorithm due to Khuller and Raghavachari [10]in the case that edge costs obey the triangle inequality. However, no non-trivialapproximation algorithm is known for the vertex-connectivity survivable networkdesign problem in its full generality.

In this paper we make progress on this important problem by considering anatural problem intermediate to EC-SNDP and VC-SNDP. We call it theelementconnectivity survivable network design problem(ELC-SNDP). In this versionof the problem, the vertices are partitioned intoterminals and nonterminals.Nonterminals and edges can fail; these are theelements. On the other hand,terminals cannot fail. Further, for each pair of terminals,u, v, we are given aconnectivity requirementruv . The problem is to find a minimum-cost subgraphsuch that for each pair of terminals,u, v, despite the failure of anyruv − 1elements, there is still a path left connectingu andv; that is, there areruv element-disjoint paths between each pair of terminalsu andv. Notice that nonterminalsdo not have any connectivity requirements. This model is realistic, since in manypractical situations, the terminals are robust and do not fail, whereas intermediatenodes, which do not have connectivity requirements, do fail. In the VC-SNDP, allvertices and all edges are allowed to fail. The EC-SNDP is a special case of theELC-SNDP with an empty set of nonterminals.

Our central result is a 2Hk-approximation algorithm for this problem, wherek =maxu,v ruv is the largest connectivity requirement. Our algorithm also followsthe basic algorithmic and proof outline established in [5,16]. The main difficultyin solving VC-SNDP is that there is no known way of writing the augmentationproblem for each phase as an integer program. (An example of the kinds ofconstraints one gets using the usual way of breaking the problem into phasesis: pick either edgee1 or bothe2 ande3.) The problem ELC-SNDP is definedprecisely in a way to get around this difficulty. However, even after the integerprogram is written, the ideas of [5,16] do not apply in a straightforward manner.

The remainder of the paper is structured as follows. In Section 2 we give theinteger programming formulation and its LP-relaxation. In Section 3 we showhow the problem is decomposed into phases and we prove the approximationguarantee, assuming the correct implementation of a phase. In Section 4 we givethe phase algorithm. In the last section we give a tight example, thereby showingthat no better guarantee can be established for our algorithm.

Since the appearance of an extended abstract of this result [9], several relatedresults have appeared. Cheriyan et al. [2] give a 6Hk-approximation algorithmfor minimum-costk-vertex connected subgraph that contain at least 6k2 vertices.A 2Hk-approximation algorithm for ELC-SNDP is also obtained as a special case


of an algorithm by Zhao et al. [17]. Fleischer [3] has obtained a 2-approximationalgorithm for VC-SNDP whenrij ∈ {0,1,2}, and Fleischer et al. [4] havedeveloped a 2-approximation algorithm for ELC-SNDP. The latter two resultsdraw upon the rounding algorithm and analysis developed by Jain [8] for EC-SNDP.

2. The problem, its integer programming formulation and LP-relaxation

Let G = (V ,E) be an undirected graph with non-negative costsce on edges.The setV is partitioned into two disjoint setsR and S. R is the set ofterminals; there is a non-negative connectivity requirementruv between eachpair of terminals. We assume that these vertices are reliable. On the other handvertices inS, also known as nonterminals, and all the edges are unreliable.We call the members ofS ∪ E elements. We define the element connectivityproblem as choosing a minimum-cost set of edgesE′ ⊆E, so that in the subgraphH = (V ,E′) for every pairu andv, there still remains a path between them in caseruv−1 elements fail. In other words there areruv element disjoint paths betweenuandv. For convenience we extend the definition ofruv to any pairu andv ofvertices by assuming thatruv = 0 if at least one ofu andv is a nonterminal.

Let H be a feasible solution to this problem. Suppose that a set of elementsA ⊆ S ∪ E has failed. Now the surviving graph,H − A, should have at leastruv − |A| element-disjoint paths for every pairu andv. Note that as far as theinteger program is concerned, for a pairu andv, it is sufficient to considerA’sof cardinality ruv − 1 only, and write cut constraints that ensure the existenceof a path fromu to v. However, it is easy to show that the LP-relaxation ofthis integer program has a bad integrality gap. Consider an unweighted completegraph onn vertices from which we want to select a low-costk-edge-connectedsubgraph. Since the degree of every vertex in the optimal solution should beat leastk, the cost of optimal solution is at leastnk/2. If we form the linearprogram only forA’s of cardinalityk− 1 then picking every edge to the extent of1/(n− k) will be a valid feasible solution. This gives us the integrality gap of atleastk(n− k)/(n− 1). Choosingk = n/2 makes the integrality gap at leastn/4.

Hence, we include in the integer program constraints corresponding to setsA

of all cardinalities. As a corollary of our algorithm, we show that the LP-relaxationof this integer program has integrality gap bounded by 2Hk.

Let f (B)=maxu∈B,v /∈B ruv . The integer program is:

min∑e∈E

cexe (1)

subject to:

∀A⊆ S ∪E,B ⊆ V −A:∑

e∈δG−A(B)

xe � f (B)− |A|,

∀e ∈E: xe ∈ {0,1},


whereδG−A(B) denotes the set of edges with one endpoint inB after removingAfrom graphG.

To get the linear program we further relax the conditionxe ∈ {0,1} to xe � 0.The following is the dual of the above linear program:

max∑

A⊆S∪E,B⊆V−A

(f (B)− |A|)yBA (2)

subject to:

∀e ∈E:∑

B,A: A⊆S∪E,B⊆V−A,e∈δG−A(B)

yBA � ce,

∀A⊆ S ∪E,B ⊆ V −A: yBA � 0.

Let OPT be the optimal cost of IP (1). By weak duality theorem any solutionto (2) will be a lower bound on OPT.

We will use the fact thatf is weakly supermodular[5], i.e.,f (V )= 0 and, foreveryA,B ⊆ V , at least one of the following holds

• f (A)+ f (B) � f (A−B)+ f (B −A),• f (A)+ f (B) � f (A∩B)+ f (A∪B).

3. High level description of the algorithm

Given an infeasible solutionH to IP (1), define thedeficiency of a constraintasthe difference between the right-hand side and the left-hand side of the constraint.Only unsatisfied constraints will have positive deficiency. The deficiency of a setB ⊆ V is defined to be the maximum deficiency of a constraint in whichB isinvolved.

As in [5] our algorithm hask phases numbered fromk to 1. We design thealgorithm so that at the start ofpth phase, the deficiency of a set can be at mostp

and at the end of the phase it can be at mostp − 1. So at the end of the 1st phasewe have a feasible solution and we output that.

LetH be the partial solution constructed at the beginning of thepth phase. Leth be defined by

h(B)={

1, if deficiency ofB is p,0, otherwise.

Let ΓH(B) be the set of nonterminals which are not inB but have a neighborwith respect toH in B. LetρH (B) be the set of those edges ofH which have oneendpoint inB and the other inR−B. Finally, we define theelement neighborhoodof set B w.r.t. Hto beεH(B)= ΓH(B)∪ ρH (B).


Observation 3.1. The deficiency ofB is f (B) − |εH (B)| and is the deficiencyof the constraint corresponding to the set pairεH(B) andB. Moreover inclusionof any edge ofE −H which decreases the deficiency of this constraint will alsodecrease the deficiency ofB.

So the integer program we want to solve for thepth phase is:

min∑

e∈E−Hcexe (3)

subject to:

∀B ⊆ V :∑

e∈δG−εH (B)(B)

xe � h(B),

∀e ∈E −H : xe ∈ {0,1}.Let I be a feasible solution of this integral program. By Observation 3.1 the

deficiency of any set with respect toI ∪H will be at mostp− 1.By relaxingxe ∈ {0,1} to xe � 0 we get a linear program, whose dual is:

max∑B⊆V

h(B)yB (4)

subject to:

∀e ∈E −H :∑

B: e∈δG−εH (B)(B)

yB�0

yB � ce.

In the next section we will prove:

Theorem 3.2. We can find in polynomial time a feasible solution,F , for IP (3)and a feasible solution,y, for LP (4) such thatcost(F ) � 2

∑B⊆V h(B)yB .

Let y be as in the above theorem. Let

yBA ={yB, if A= εH (B),0, otherwise.

It is easy to verify thaty is a feasible solution for LP (2).

Lemma 3.3.

cost(F ) � 2

p·OPT.

Proof. By definition,h(B)= 1 iff f (B)− |εH (B)| = p. So,


∑B⊆V

h(B)yB =∑B⊆V

h(B)yBεH (B) = 1

p

∑B⊆V

(f (B)− ∣∣εH (B)

∣∣)yBεH (B)

� 1

p·OPT.

The claim follows immediately from Theorem 3.2.✷Corollary 3.4. The cost of edges chosen by the algorithm in allk phases is atmost

2

(1

k+ 1

k − 1+ · · · + 1

)·OPT= 2Hk ·OPT.

4. The algorithm for a phase, and its analysis

4.1. The algorithm

We now present the algorithm for a phase, which will augment sets whosedeficiency isp. Our augmentation algorithm follows those in [5,16]. We first givethe algorithm, then turn to stating and proving Theorem 3.2.

Let I be an infeasible solution to IP (3). A set isviolatedwith respect toI ifthe constraint corresponding to it in IP (3) is unsatisfied. A violated set isactiveif none of its proper subsets is violated. Our algorithm for a phase is as follows:

Algorithm for phase p.

I ←−∅, y← 0, i← 0While there are violated sets w.r.t.I

i← i + 1Find all active sets(an algorithm will be discussed below).IncreaseyB uniformly for all active setsB until some dual constraint becomes

tight, i.e., until∑

B:e∈δG−εH (B)(B) yB = cei for some edgeei /∈ I .

I ← I ∪ {ei}For l← i downto1

If there are no violated sets w.r.t.I − {el}I ← I − {el}

4.2. Proof of Theorem 3.2

To prove that this algorithm satisfies Theorem 3.2, we follow the general proofframework of [16]. From now onH is fixed as the partial solution obtained at theend of(p+ 1)st phase and we fix an iterationi of the while loop of the algorithmabove. We will call this the “current iteration.” LetI be the partial solution at


the beginning of this iteration. LetF be the final set of edges returned at the endof the phase. We claim that Theorem 3.2 follows from the theorem below; afterstating the theorem, we will establish the claim.

Theorem 4.1. If C is the set of active sets in this iteration, then∑C∈C

∣∣F ∩ δG−εH (C)(C)∣∣ � 2|C|.

Proof of Theorem 3.2. We wish to show that

cost(F ) � 2∑B⊆V

h(B)yB.

Note that by the properties of the algorithmce =∑B : e∈δG−εH (B)(B) yB for each

e ∈ F . Thus the theorem follows if∑e∈F

∑B: e∈δG−εH (B)(B)

yB � 2∑B⊆V

h(B)yB.

We can rewrite the lefthand side as∑

B⊆V |F ∩ δG−εH (B)(B)|yB , so that thetheorem follows if∑

B⊆V

∣∣F ∩ δG−εH (B)(B)∣∣yB � 2

∑B⊆V

h(B)yB.

We prove this by induction on the value ofy during the course of the algorithm.Initially y = 0, so the inequality holds. In any iteration, ifC is the set of activesets in that iteration, and eachyC for C ∈ C is increased byγ , then the left-handside increases by

γ∑C∈C

∣∣F ∩ δG−εH (C)(C)∣∣,

while the right-hand side increases by 2γ |C|. The theorem then follows byTheorem 4.1. ✷4.3. Proof of polynomial time

We now turn to proving that the algorithm can be implemented in polynomialtime. To do this, we will need some definitions and a theorem, which we providebelow.

Definition 4.2. Two setsA,B ⊆ V εH∪I -cross if A �⊆ B, B �⊆ A, A ∩ (B ∪εH∪I (B)) �= ∅ andB ∩ (A∪ εH∪I (A)) �= ∅.

Note that sinceA,B ⊆ V andεH∪I ⊆ E ∪ V , we could have replacedεH∪Iwith ΓH∪I in the definition above.


Definition 4.3. A family of subsets ofV is εH∪I -laminar if no two sets in thefamily εH∪I -cross.

Observation 4.4. Let X ⊆ Y ⊆ V ∪ E, andA,B ⊆ V . If A andB εX-cross,they alsoεY -cross. Similarly, if a family of sets isεY -laminar, then they are alsoεX-laminar.

Note that this is a stronger notion than the usual notion of laminarity, whichis reproduced below. A laminar family according to this notion is also laminaraccording to the usual notion.

Definition 4.5. Two setsA,B ⊆ V crossif A �⊆ B, B �⊆A, andA∩B �= ∅.

Definition 4.6. A family of subsets ofV is laminar if no two sets in the familycross.

Theorem 4.7. If A,B ⊆ V are violated sets with respect toI , then eitherA ∩ B

andA∪B are also violated orA−B− εH∪I (B) andB−A− εH∪I (A) are alsoviolated.

We defer the proof of this theorem for the moment. The theorem implies thefollowing corollary.

Corollary 4.8. No violated set with respect toI εH∪I -crosses any active set withrespect toI .

Proof. If a violated setA εH∪I -crosses an active setB, then by the theorem,eitherA∩B orB−A−εH∪I (A) is also violated. This contradicts the minimalityof B. ✷

Theorem 4.7 thus implies that the algorithm can be implemented in polynomialtime.

Theorem 4.9. The algorithm for a phase can be implemented in polynomial time.

Proof. It follows from Corollary 4.8 that the active sets are disjoint. Hence eachvertex can be in at most one active set. Consider a network onH ∪ I , wherethe capacity of each edge and each nonterminal is one and the capacity of eachterminal is unbounded. Consider a vertexu: it will be in an active set if thereexists a vertexv such that the minimumu–v cut with respect to edgesH ∪ I isof capacityruv − p. Let v be one such vertex. The active set in whichu lies isthe minimal (inclusion-wise)u–v min-cut. This can be found in polynomial timeusing a max-flow subroutine.✷


We now prove Theorem 4.7. To prove this theorem we need the followingdefinitions and lemmas.

Definition 4.10. Letϕ : 2V → Z+. We say thatϕ is εH∪I -submodularif ϕ(V )= 0

and, for everyA,B ⊆ V , the following two conditions hold:

(1) ϕ(A)+ ϕ(B) � ϕ(A∩B)+ ϕ(A∪B),(2) ϕ(A)+ ϕ(B) � ϕ(A−B − εH∪I (B))+ ϕ(B −A− εH∪I (A)).

Lemma 4.11. |εH∪I | is εH∪I -submodular.

Proof. We need to prove the two inequalities in the definition ofεH∪I -submodu-larity.

(1) One can easily verify that the contribution of any element to the left-hand sideof the inequality is at least the contribution of the element to the right-handside of the inequality. This proves the first condition ofεH∪I -submodularity.

(2) The proof of this inequality is similar to the proof of the first inequality exceptfor the case when there is an edgers, r ∈ R, s ∈ S ∩ A ∩ B and eitherr ∈ A − B − εH∪I (B) or r ∈ B − A − εH∪I (A). In this cases contributesto the right-hand side of the inequality but does not contribute to the left-hand side. But to counteract the contribution ofs, edgers contributes only tothe left-hand side of the inequality.✷

Definition 4.12. Let ϕ : 2V → Z. We say thatϕ is weaklyεH∪I -supermodularifϕ(V ) = 0 and, for everyA,B ⊆ V , at least one of the following two conditionshold:

(1) ϕ(A)+ ϕ(B) � ϕ(A∩B)+ ϕ(A∪B),(2) ϕ(A)+ ϕ(B) � ϕ(A−B − εH∪I (B))+ ϕ(B −A− εH∪I (A)).

Lemma 4.13. f (B)=maxu∈B,v /∈B ruv is weaklyεH∪I -supermodular.

Proof. SinceεH∪I (B) does not contain any terminals,f (A− B − εH∪I (B))=f (A − B). Similarly, f (B − A − εH∪I (A)) = f (B − A). The lemma followsfrom the fact thatf is weakly supermodular [5]. ✷

Now we are ready to prove Theorem 4.7.

Proof of Theorem 4.7. SinceA andB are violated sets, their deficiency isp,which is the maximum deficiency for this phase. Hence,

2p = (f (A)− ∣∣εH∪I (A)

∣∣)+ (f (B)− ∣∣εH∪I (B)

∣∣)= (

f (A)+ f (B))− (∣∣εH∪I (A)

∣∣+ ∣∣εH∪I (B)∣∣).


Sincef is weaklyεH∪I -supermodular, either

f (A)+ f (B) � f (A∩B)+ f (A∪B) or

f (A)+ f (B) � f(A−B − εH∪I (B)

)+ f(B −A− εH∪I (A)

).

Suppose the former holds. ByεH∪I -submodularity, we also have∣∣εH∪I (A)∣∣+ ∣∣εH∪I (B)

∣∣ �∣∣εH∪I (A∩B)

∣∣+ ∣∣εH∪I (A∪B)∣∣.

Hence,

2p �(f (A∩B)+ f (A∪B)

)− (∣∣εH∪I (A∩B)∣∣+ ∣∣εH∪I (A∪B)

∣∣)= (

f (A∩B)− ∣∣εH∪I (A∩B)∣∣)+ (

f (A∪B)− ∣∣εH∪I (A∪B)∣∣)

� 2p.

The last inequality results from the fact that no set has deficiency more thanp.Since(

f (A∩B)− ∣∣εH∪I (A∩B)∣∣)+ (

f (A∪B)− ∣∣εH∪I (A∪B)∣∣)

is at most as well as at least 2p, it is 2p. This is possible only ifA∩B andA∪B

are violated.Similarly, if

f (A)+ f (B) � f(A−B − εH∪I (B)

)+ f(B −A− εH∪I (A)

),

thenA−B − εH∪I (B) andB −A− εH∪I (A) are violated. ✷4.4. Proof of Theorem 4.1

Finally, we turn to the proof of Theorem 4.1. LetC be the collection of activesets in the current iteration. LetY be the set of edgese ∈ F for which there existsC ∈ C such thate ∈ δG−εH (C)(C). For each edgee ∈ Y we define awitness set,We ⊆ V , as a set that meets the following conditions:

(1) |εH∪I∪F (We)| = f (We)− p+ 1;(2) |εH∪I∪F−{e}(We)| = f (We)− p.

To see that a witness setWe must exist for everye ∈ Y , observe that byconstruction of the algorithmI ∪ F − {e} is not a feasible solution; thus theremust be some violated setWe which is a witness set. Awitness familyfor Y is afamily of subsets ofV , so that it exactly contains one witness for each edge inY .Note that it cannot be the case that the same setW is a witness for two differentedgese andf .

Theorem 4.14. There is aεH -laminar witness family forY .


Proof. Given a witness family, suppose two setsWe and Wf εH -cross. Thenthey will εH∪F∪I -cross also. LetX and Y be the two sets obtained fromTheorem 4.7; that is, eitherWe ∩Wf andWe ∪Wf or We −Wf − εH∪F∪I (Wf )

andWf −We − εH∪F∪I (We). These sets do notεH∪F∪I -cross since neither setsWe ∩ Wf andWe ∪ Wf εH∪F∪I -cross nor setsWe −Wf − εH∪F∪I (Wf ) andWf −We−εH∪F∪I (We) εH∪F∪I -cross. We will replaceWe andWf by two othersetsXe andYf . We will show that these setsXe andYf will be witnesses for edgese andf . When the first case of Theorem 4.7 is valid, we will have that these twosets areWe ∩Wf andWe ∪Wf . When the second case of the theorem holds,Xe

andYf will be subsets ofWe−Wf −εH∪F∪I (Wf ) andWf −We−εH∪F∪I (We).Note that in either case the two sets do notεH -cross. Also notice that this processcannot continue indefinitely without decreasing the minimum cardinality of awitness in the family, so it must end with a laminar witness family.

For the sake of argument, we assume that the first case of Theorem 4.7 holds;the other case is similar, and we will explain the minor differences at the end ofthe proof.

SinceWe andWf are witnesses we get∣∣εH∪F∪I (We)∣∣+ ∣∣εH∪F∪I (Wf )

∣∣= f (We)+ f (Wf )− 2p+ 2.

By the arguments in the proof of Theorem 4.7 we can use the weakεH∪F∪I -supermodularity off (·) and theεH∪F∪I -submodularity of|εH∪F∪I (·)| to get that

∣∣εH∪F∪I (We ∩Wf )∣∣+ ∣∣εH∪F∪I (We ∪Wf )

∣∣� f (We ∩Wf )+ f (We ∪Wf )− 2p+ 2.

Note that this is possible, because the option which holds forf (·) does not dependupon theε function. SinceF ∪ I is a feasible solution to this phase,

∣∣εH∪F∪I (We ∩Wf )∣∣ � f (We ∩Wf )− p+ 1 and∣∣εH∪F∪I (We ∪Wf )∣∣ � f (We ∪Wf )− p+ 1.

Thus the above inequality implies that∣∣εH∪F∪I (We ∩Wf )

∣∣= f (We ∩Wf )− p+ 1 and∣∣εH∪F∪I (We ∪Wf )∣∣= f (We ∪Wf )− p+ 1.

Now considere. Applying the definition of witness fore we get∣∣εH∪F∪I−{e}(We)∣∣+ ∣∣εH∪F∪I−{e}(Wf )

∣∣ � f (We)+ f (Wf )− 2p+ 1.

Using the weakεH∪F∪I−{e}-supermodularity off (·) andεH∪F∪I−{e}-submodu-larity of |εH∪F∪I−{e}(·)|, we get that

∣∣εH∪F∪I−{e}(We ∩Wf )∣∣+ ∣∣εH∪F∪I−{e}(We ∪Wf )

∣∣� f (We ∩Wf )+ f (We ∪Wf )− 2p+ 1.


Now by the feasibility ofH we know that∣∣εH∪F∪I−{e}(We ∩Wf )∣∣ � f (We ∩Wf )− p and∣∣εH∪F∪I−{e}(We ∪Wf )∣∣ � f (We ∪Wf )− p.

Thus for at least one ofWe ∩Wf andWe ∪Wf , sayWe ∩Wf , |εH∪F∪I−{e}(We ∩Wf )| = f (We ∩ Wf ) − p. ThenWe ∩ Wf is a witness set fore and we setXe =We ∩Wf . A similar argument will show that the other set is a witness setfor edgef .

We now return to the case that the result of Theorem 4.7 are the two setsWe −Wf − εH∪F∪I (Wf ) andWf −We − εH∪F∪I (We). The argument proceeds muchas above, except that it will show that one ofWe −Wf − εH∪F∪I−{e}(Wf ) andWf −We − εH∪F∪I−{e}(We) is a witness fore, and the corresponding oppositeof the pairWe −Wf − εH∪F∪I−{f }(Wf ) andWf −We − εH∪F∪I−{f }(We) is awitness forf . ✷

Given the laminar witness family, we can construct a rooted treeT , as follows.Let L be the set of sets in the witness family, plus the set of verticesV . We createa node of the tree for eachL ∈ L; the node corresponding toV will be the rootof T . The node corresponding toA ∈ L is a parent of the node corresponding toB ∈L if A is the smallest set inL strictly containingB. We associate each activesetC ∈ C with the node in the tree corresponding to the smallest set inL thatcontainsC; note that this notion is well-defined by Corollary 4.8.

In a moment, we will prove the following lemma.

Lemma 4.15. LetB be the set of active sets associated with a nodev in the treeT .Then the degree of nodev in the treeT is at least

∑C∈B |F ∩ δG−εH (C)(C)|.

First, let us show how the lemma implies Theorem 4.1.

Proof of Theorem 4.1. Let us color the nodes of the treeT ; we color a node blueif it has an active set associated with it, and red otherwise. Let Red and Blue bethe sets of red and blue nodes respectively, and let deg(v) be the degree of nodevin treeT . Note that every leaf of the tree is blue: since each leaf corresponds to aviolated set, it must contain some minimal violated set inside it. Then we have∑

C∈B

∣∣F ∩ δG−εH (C)(C)∣∣ �

∑v∈Blue

deg(v) (5)

=∑

v∈Blue∪Red

deg(v)−∑

v∈Red

deg(v)

� 2(|Blue| + |Red| − 1

)− 2(|Red| − 1

)(6)

� 2|Blue|� 2|C|, (7)


where (5) follows by Lemma 4.15, (6) by the fact thatT is a tree, and all rednodes (excepting perhaps the root) have degree at least two, and (7) follows sinceeach blue node has at least one member ofC associated with it. ✷

We now complete the proof.

Proof of Lemma 4.15. By the definition ofY ,

∑C∈B

∣∣F ∩ δG−εH (C)(C)∣∣=∑

C∈B

∣∣Y ∩ δG−εH (C)(C)∣∣.

Given a nodev of the treeT , and the set of active setsB corresponding to it,let YB be the set of edgesYB =⋃

C∈B(Y ∩ δG−εH (C)(C)). Note that there mustbe a witness set corresponding to each edge inYB , and at most one can be theset W corresponding tov itself. The proof follows if we can show that eachremaining edge ofYB can be mapped to a unique child ofv; that is, a uniquewitness setX such thatW is the smallest witness set containingX. Thus wesimply need to show that for an edgee ∈ YB , if its witness set is notW , it mustbe the case that for its witness setX, X ⊂W , and there is no witness setZ suchthatX ⊂Z ⊂W .

We prove these statements by contradiction. First assume thatX �⊂W . Sincethe witness sets are laminar, this implies thatX ∩ W = ∅. We know e ∈δG−εH (C)(C) with C ⊂W for some activeC ∈ B, ande ∈ δG−εH (X)(X) by thedefinition of a witness set. Ife ∈ δG−εH (W)(W), this contradicts the fact thatXis the unique witness set fore. The only way this cannot occur is if the endpointof e in X is in the element neighborhood ofW , εH (W), a possibility which iseliminated sinceX andW do notεH -cross. Next, suppose that there is a witnessset Z such thatX ⊂ Z ⊂ W . As above, we know thate ∈ δG−εH (X)(X) ande ∈ δG−εH (C)(C), whereC ⊆W , but C ∩ Z = ∅. Thus there is one endpoint ofe in C and the other inX. If e ∈ δG−εH (Z)(Z), this contradicts the fact thatX isthe unique witness set fore. The only way this cannot occur is if the endpointof e in C is in the element neighborhood ofZ, εH (Z), a possiblity which iseliminated since by Corollary 4.8 the active setC does notεH -cross the violatedsetZ. ✷

5. Tight example

For the special case when the set of nonterminals is empty, our algorithmreduces to the algorithm of [5] for edge connectivity. A 2Hk-approximation tightexample to this algorithm for edge-connectivity is given in the same paper.


Acknowledgments

We thank the two anonymous referees for useful comments.

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