Approximation algorithms for connectivity augmentation problems
AApproximation algorithms for connectivityaugmentation problems
Zeev Nutov
The Open University of Israel, [email protected]
Abstract In Connectivity Augmentation problems we are given a graph H = ( V, E H ) and an edge set E on V , and seek a min-size edge set J ⊆ E such that H ∪ J has larger edge/node connectivity than H .In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by 1.In the
Block-Tree Augmentation problem H is connected and H ∪ S should be 2-connected.In Leaf-to-Leaf Connectivity Augmentation problems every edge in E connects minimaldeficient sets. For this version we give a simple combinatorial approximation algorithm with ratio5 /
3, improving the 1 .
91 approximation of [6] (see also [23]), that applies for the general case. Wealso show by a simple proof that if the
Steiner Tree problem admits approximation ratio α thenthe general version admits approximation ratio 1 + ln(4 − x ) + (cid:15) , where x is the solution to theequation 1 + ln(4 − x ) = α + ( α − x . For the currently best value of α = ln 4 + (cid:15) [7] this givesratio 1 . .
91 of [6], but has the advantage of using
SteinerTree approximation as a “black box”, giving ratio < . α ≤ .
35 can be achieved.In the
Element Connectivity Augmentation problem we are given a graph G = ( V, E ), S ⊆ V , and connectivity requirements r = { r ( u, v ) : u, v ∈ S } . The goal is to find a min-size set J of new edges on S (any edge is allowed and parallel edges are allowed) such that for all u, v ∈ S thegraph G ∪ J contains r ( u, v ) uv -paths such that no two of them have an edge or a node in V \ S incommon. The problem is NP-hard even when r max = max u,v ∈ S r ( u, v ) = 2. We obtain approximationratio 3 /
2, improving the previous ratio 7 / S we obtainthe same ratio with just +1 degree violation, which is tight, since deciding whether there exists afeasible solution is NP-hard even when r max = 2. A similar result is shown for the more generalproblem of covering a skew-supermodular set function by a min-size set of edges. Theory of computation → Design and analysis of algorithms
Keywords and phrases connectivity augmentation, approximation algorithm, element connectivity
Digital Object Identifier
A graph is k -connected if it contains k internally disjoint paths between every pair of nodes;if the paths are only required to be edge disjoint then the graph is k -edge-connected . In Connectivity Augmentation problems we are given an “initial” graph G = ( V, E ) andan edge set E on V , and seek a min-size edge set J ⊆ E such that G ∪ J = ( V, E ∪ J ) haslarger edge/node connectivity than G .In the Edge-Connectivity Augmentation problem we seek to increase the edgeconnectivity by one, so G is k -edge-connected and G ∪ J should be ( k + 1)-edgeconnected.In the 2 -Connectivity Augmentation problem we seek to make a connected graph2-connected, so G is connected and G ∪ J should be 2-connected.A cactus is a “tree-of-cycles”, namely, a 2-edge-connected graph in which every blockis a cycle (equivalently - every edge belongs to exactly one simple cycle). By [8], the Edge-Connectivity Augmentation problem is equivalent to the following problem: © Zeev Nutov;licensed under Creative Commons License CC-BYLeibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany a r X i v : . [ c s . D S ] N ov X:2 Approximation algorithms for connectivity augmentation problems
Cactus Augmentation
Input:
A cactus T = ( V, E T ) and an edge set E on V . Output:
A min-size edge set J ⊆ E such that T ∪ J is 3-edge-connected.It is also known (c.f. [16]) that the 2 -Connectivity Augmentation problem isequivalent to the following problem: Block-Tree Augmentation
Input:
A tree T = ( V, E T ) and an edge set E on V . Output:
A min-size edge set F ⊆ E such that T ∪ F is 2-connected.A more general problem than Cactus Augmentation is as follows. Two sets
A, B cross if A ∩ B = ∅ and A ∪ B = V . A set family F on a groundset V is a crossing family if A ∩ B, A ∪ B ∈ F whenever A, B ∈ F cross; F is a symmetric family if V \ A ∈ F whenever A ∈ F . The 2-edge-cuts of a cactus form a symmetric crossing family, with theadditional property that whenever A, B ∈ F cross and A \ B, B \ A are both non-empty, theset ( A \ B ) ∪ ( B \ A ) is not in F ; such a symmetric crossing family is called proper [9]. Dinitz,Karzanov, and Lomonosov [8] showed that the family of minimum edge cuts of a graph G can be represented by 2-edge cuts of a cactus. Furthermore, when the edge-connectivity of G is odd, the min-cuts form a laminar family and thus can be represented by a tree. Dinitzand Nutov [9, Theorem 4.2] (see also [20, Theorem 2.7]) extended this by showing that anarbitrary symmetric crossing family F can be represented by 2-edge cuts and specified 1-nodecuts of a cactus; when F is a proper crossing family this reduces to the cactus representationof [8]. We say that an edge f covers a set A if f has exactly one end in A . The followingproblem combines the difficulties of the Cactus Augmentation and the
Block-TreeAugmentation problems, see [23].
Crossing Family Augmentation
Input:
A graph G = ( V, E ) and a symmetric crossing family F on V . Output:
A min-size edge set J ⊆ E that covers F .In this problem, the family F may not be given explicitly, but we require that certainqueries related to F can be answered in polynomial time, see [23]. Block-Tree Augment-ation and
Crossing Family Augmentation admit ratio 2 [25, 11], that applies also forthe min-cost versions of the problems.The inclusion minimal members of a set family F are called leaves . In the Leaf-to-LeafCrossing Family Augmentation problem, every edge in E connects two leaves of F . Inthe Leaf-to-Leaf Block-Tree Augmentation problem, every edge in E connects twoleaves of the input tree T . (cid:73) Theorem 1.
The leaf-to leaf versions of
Crossing Family Augmentation and
Block-Tree Augmentation admit ratio / . Better ratios are known for two special cases. In the
Tree Augmentation problem thefamily F is laminar, namely, any two sets in F are disjoint or one contains the other; thisproblem can be also defined in connectivity terms - make a spanning tree 2-edge-connectedby adding a min-size edge set J ⊆ E . This problem was vastly studied; see [1, 15, 18, 10, 22]and the references therein for additional literature on the Tree Augmentation problem.In the
Leaf-to-Leaf Tree Augmentation problem, every edge in E connects two leavesof the tree; this problem admits ratio 17 /
12 [19]. The
Cycle Augmentation problem is aparticular case of the
Cactus Augmentation problem when the cactus is a cycle; in this eev Nutov XX:3 case the leaves are the singleton nodes. The
Cycle Augmentation problem admits ratio + (cid:15) [14]; our algorithm from Theorem 1 uses some ideas from [14].Byrka, Grandoni, and Ameli [6] showed that Cactus Augmentation admits ratio2 ln 4 − + (cid:15) < Crossing Family Augmentation and
Block Tree Augmentation in [23].In the
Steiner Tree problem we are given a graph G = ( V, E ) with edge costs and a set R ⊆ V of terminals, and seek a min-cost subtree of G that spans R . We prove the following. (cid:73) Theorem 2. If Steiner Tree admits ratio α then Crossing Family Augmentation and
Block-Tree Augmentation admit ratio − x ) + (cid:15) , where x is the solution tothe equation − x ) = α + ( α − x . Currently, α = ln 4 + (cid:15) [7]; in this case we have ratio 1 .
942 for the problems in the theorem.This is slightly worse than the ratio 1 .
91 of [6] (see also [23]), but our algorithm is verysimple and has the advantage of using
Steiner Tree approximation as a “black box”. E.g.,if ratio α = 1 .
35 can be achieved, then we immediately get ratio 1 . < . Element Connectivity Augmentation
Input:
An undirected graph G = ( V, E ), a set S ⊆ V of terminals, and connectivityrequirements { r ( u, v ) : u, v ∈ S } on pairs of terminals. Output:
A minimum size set J of new edges on S (any edge is allowed and parallel edgesare allowed) such that the graph G ∪ J contains r ( u, v ) uv -paths such that no two of themhave an edge or a node in V \ S in common.A particular case when the graph G is bipartite with sides S and V \ S is known as the Hypergraph Edge-Connectivity Augmentation problem; here S is the set of nodes ofthe hypergraph and V \ S is the set of the hyperedges. This problem is solvable in polynomialtime for uniform requirements when r ( u, v ) = k for all u, v ∈ S [2] (see also [4] and [5] for asimpler algorithm and proof), and when r max = 1, where r max is the maximum requirement.See also [12, 13, 5] for additional polynomially solvable cases. The non-uniform version ofthe problem is NP-hard even when the initial graph G is connected and r max = 2 [17]. Theprevious best approximation ratio for the general version was 7 /
4, and 3 / r max = 2[21].In the degree bounded version of the problem we also have degree bounds { b ( v ) : v ∈ S } and require that d J ( v ) ≤ b ( v ) for all v ∈ S , where d J ( v ) is the degree of v w.r.t. J . We showthat Element Connectivity Augmentation admits ratio 3 /
2, and that this ratio can beachieved also for the degree bounded version with only additive +1 degree violation; a betterdegree approximation is unlikely, since deciding whether there exists a feasible solution isNP-hard even when r max = 2 and b max = 1 [17]. (cid:73) Theorem 3.
Element Connectivity Augmentation admits approximation ratio / . Moreover, the degree bounded version admits a bicriteria approximation algorithm thatcomputes a solution J of size at most / times the optimal such that d J ( v ) ≤ b ( v ) + 1 forall v ∈ S . The proof of this theorem is based on a generic algorithm for covering a skew-supermodularset function, as is explained in Section 5.
X:4 Approximation algorithms for connectivity augmentation problems
We prove Theorem 1 for the
Crossing Family Augmentation problem, and later indicatethe changes needed to adopt the proof for the
Block-Tree Augmentation problem. Weneed some definition to describe the algorithm. Let F be a set family on V . We say that A ∈ F separates u, v ∈ V if | A ∩ { u, v }| = 1; u, v are F -separable if such A exists and u, v are F -inseparable otherwise. Similarly, A separates edges f, g if one of f, g has bothends in A and the other has no end in A ; f, g are F -separable if such A ∈ F exists, and F -inseparable otherwise. The relation { ( u, v ) ∈ V × V : u, v are F -inseparable } is anequivalence, and we call its equivalence classes F -classes . W.l.o.g. we will assume that all F -classes are singletons and that no edge in E has both ends in the same class; in particular,the leaves of F are singletons, and we denote the leaf set of F by L . We will also oftenabbreviate the notation for singleton sets and write v, e instead of { v } , { e } . Given J ⊆ E ,the residual instance (( V J , E J ) , F J ) is defined as follows.The residual family F J of F w.r.t. J consists of all members of F that are uncoveredby the edges in J . It is known that F J is crossing (and symmetric) if F is. V J is the set of F J -classes (w.l.o.g, each of them can be shrunk into a single element). E J is obtained from E \ J by removing all edges that have both ends in the same F J -class.In addition, given a set R ⊆ V of terminals, the residual set of terminals R J is the set of F J -classes that contain some member of R . For illustration see Fig. 1(a,b,c).For any edge e = uv , there is an F e -class that contains both u and v ; denote this class by C ( F , e ). Given a set R of terminals (a subset of F -classes), the ( R, E, F ) -incidence graph H = ( U, E H ) has node set U = E ∪ R and edge set E H = { ee : e, e ∈ E are F -inseparable } ∪ { er : r ∈ R, e ∈ E, r ∈ C ( F , e ) } . Let R ⊆ V and let H be the ( R, E, F )-incidence graph. Note that R is an independentset in H . It was shown in [23] that for R = L being the set of leaves of F , an edge set J ⊆ E is a feasible solution to Crossing Family Augmentation if and only if the subgraph H [ J ∪ R ] of H induced by J ∪ R is connected. The proof in [23] extends to any R ⊆ V thatcontains L . This implies that Crossing Family Augmentation admits an approximationratio preserving reduction to the following problem (see [23, 3] for more details).
Subset Steiner Connected Dominating Set ( SS-CDS ) Input:
A graph H = ( U, E H ) and a set R ⊆ U of independent terminals. Output:
A min-size node set S ⊆ U \ R such that H [ S ] is connected and S dominates R .Given a SS-CDS instance and s ∈ S = U \ R let R ( s ) = R H ( s ) denote the set of neighborsof s in H that belong to R . Let opt be the optimal solution value of a problem instance athand. Before describing the algorithm, we will prove the following lemma. (cid:73) Lemma 4.
Let I = ( H, R ) be a SS-CDS instance such that | R ( s ) | = 2 for all s ∈ S = U \ R .Then one of the following holds: (i) There are adjacent a, b ∈ S with R ( a ) ∩ R ( b ) = ∅ . (ii) opt ≥ | R | − . Proof.
Assume that (i) does not hold for I ; we will prove that then (ii) holds. The proof isby induction on | R | . In the base case | R | = 2 (ii) holds. Assume that the statement is truefor | R | − ≥
2. Let T be an optimal solution tree and S the set of non-terminals in T . Root T at some node and let s ∈ S be a non-terminal farthest from the root. The children of s are eev Nutov XX:5 x yx z tvu wu zy twxv e gh f d (b)(a) (d)(c) g h f g du th yzf gduv wteh e Figure 1
Illustration of definitions for a
Crossing Family Augmentation instance where F is represented by a cactus. Here A ∈ F if and only if A is a connected component obtained byremoving a pair of edges that belong to the same cycle of the cactus. The edges in E are shown bydashed arcs and the terminals in R are shown by gray circles. The cactus of the residual familyw.r.t. to a single edge is obtained by “squeezing” the cycles along the path of cycles between theends of the edge. (a) The original instance. (b) The residual instance w.r.t. e . (c) The residualinstance w.r.t. f . (d) The ( R, E, F )-incidence graph of the instance in (a). (a) (b) vu vu ’ s ’ s a a bbs s Figure 2
Illustration to the proof of Lemma 4.
X:6 Approximation algorithms for connectivity augmentation problems terminal leaves, and assume w.l.o.g. that R ( s ) = { u, v } is the set of children of s ; if s hasjust one child in T , then it has another terminal neighbor in H , that can be attached to s .Consider the residual instance I = ( G = ( V , E ) , R ) and the tree T obtained bycontracting R ( s ) into the new terminal s , and deleting any z ∈ U \ ( R + s ) with R ( z ) = R ( s ).Then | R | = | R | − | R ( z ) | = 2 for all z ∈ R , T is an optimal solution for I , and S = S − s is the set of non-terminals of T .If (i) does not hold for the new instance I then (ii) holds for I , by the inductionhypothesis. Then | S | = | S | + 1 ≥ ( | R | −
1) + 1 = | R | −
1, and we get that (ii) holds for I .Assume henceforth that (i) holds for I . We obtain a contradiction by showing that then (i)holds for I . Let a, b ∈ V \ R be such that R ( a ) ∩ R ( b ) = ∅ , see Fig. 2. If s / ∈ R ( a ) ∪ R ( b )then clearly (ii) holds for I . Otherwise, if say s ∈ R ( a ), then we have two cases. If oneof u, v , say v , is a neighbor of a in G (see Fig. 2(a)) then R ( a ) ∩ R ( b ) = ∅ . Otherwise (seeFig. 2(a)), R ( a ) ∩ R ( s ) = ∅ . In both cases, we obtain a contradiction to the assumption that(i) does not hold for I . (cid:74) We also need the following known lemma. (cid:73)
Lemma 5.
Any inclusion minimal cover J of a set family F is a forest. Proof.
Suppose to the contrary that J contains a cycle C . Since P = C \ { e } is a uv -path,then for any A covered by e , there is e ∈ P that covers A . This implies that J \ { e } alsocovers F , contradicting the minimality of J . (cid:74) The algorithm starts with a partial solution J = ∅ and has two phases. Phase 1 consistsof iterations. At the beginning of each iteration, construct the ( E, R J , F J )-incidence graph H J , where initially R is the set of leaves of F . Then, do one of the following: If H J has a node e ∈ E with | R J ( e ) | ≥
3, then add e to J . Else, if there are e, f ∈ E with R J ( e ) ∩ R J ( f ) = ∅ , then add both e, f to J .If none of the above two cases occurs, then we apply Phase 2, in which we add to J aninclusion minimal cover of F J ; note that all edges in E J have both endnodes in R J . A moreformal description is given in Algorithm 1.We show that the algorithm achieves ratio 5 /
3. Note that:Adding an edge e as in step 4 reduces the number of terminals by at least 2.Adding an edge pair e, f as in step 5 reduces the number of terminals by at least 3. Algorithm 1: ( G = ( V, E ) , F , R ) J ← ∅ repeat let H J be the ( E J , R J , F J )-incidence graph if H J has a node e ∈ E with | R J ( e ) | ≥ do J ← J ∪ { e } else if H J has node pair e, f ∈ E with R J ( e ) ∩ R J ( f ) = ∅ then do J ← J ∪ { e, f } until no edge e or an edge pair e, f as above exists; find an inclusion minimal F J -cover and add it to J return J Hence the reduction in the number of terminals per added edge is at least 3 /
2. Let ‘ = | L | be the initial number of terminals. Let ‘ = | R J | be the number of terminals at the end ofPhase 1 (steps 2-6 in Algorithm 1). Let k be the number of edges added during Phase 1.Then ‘ ≤ ‘ − k , hence k ≤ ( ‘ − ‘ ). The number of edges added at the second phase is atmost ‘ −
1, by Lemma 5; note that every edge in E J has both ends in R J and that | R J | = ‘ . eev Nutov XX:7 On the other hand, opt ≥ ‘ , and opt ≥ ‘ −
1, by part (ii) of Lemma 4. Summarizing, wehave the following:The solution size is at most k + ‘ − ≤ ( ‘ − ‘ ) + ‘ − ‘ + ‘ − / opt ≥ ‘/ opt ≥ ‘ − (2 ‘ + ‘ − / { ‘/ ,‘ − } . If ‘/ ≥ ‘ − ‘ + ‘ − / { ‘/ , ‘ − } ≤ (2 ‘ + ( ‘/ − / ‘/ ‘/ − / ‘/ < . Else, ‘/ < ‘ −
1, and then(2 ‘ + ‘ − / { ‘/ , ‘ − } < (4( ‘ −
1) + ‘ − / ‘ − ‘ − / ‘ − < . In both cases the ratio is bounded by 5 / Block-Tree Augmentation problem. Let G = ( V, E )be a connected graph. A node v is a cutnode of G if G \ { v } is disconnected; an inclusionmaximal node subset whose induced subgraph is connected and has no cutnodes is a block of G ; equivalently, B is a block if it is the node set of an inclusion maximal 2-connectedsubgraph or of a bridge. The block-tree T of G has node set C G ∪ B G , where C G is the setof cutnodes of G and B G is the set of blocks of G ; T has an edge for each pair of a block anda cutnode that belongs to that block. It is known that every v ∈ V \ C G belongs to a uniqueblock, and that T is a tree. The block-tree mapping ψ : V → C G ∪ B G of G is defined by ψ ( v ) = v is v ∈ C G and ψ ( v ) is the block that contains v if v ∈ V \ C G .Given a Block-Tree Augmentation instance ( T = ( V, E T ) , E ) and J ⊆ E , the residual instance ( T J = ( V J , E JT ) , E J ) is defined as follows. T J is the block tree of T ∪ J . E J = { ψ ( u ) ψ ( v ) : uv ∈ E \ J, ψ ( u ) = ψ ( v ) } , where ψ is the the block-tree mapping of T ∪ J .For a set R ⊆ V of terminals, the residual set of terminals is R J = ψ ( R ) = ∪ r ∈ R ψ ( r ).For an edge e = uv let T e denote the unique uv -path in T . We say that e, f ∈ E are T -inseparable if the paths T e , T f have an edge in common. The ( R, E, T ) -incidence graph H = ( U, E H ) has node set U = E ∪ R and edge set E H = { ef : e, f ∈ E are T -inseparable } ∪ { er : r ∈ R, e ∈ E, r ∈ T e } . It was shown in [23] that for R = L being the set of leaves of F , an edge set J ⊆ E is afeasible solution to Block-Tree Augmentation if and only if the subgraph H [ J ∪ R ] of H induced by J ∪ R is connected. The proof in [23] extends to any R ⊆ V that contains L . Thisimplies that Crossing Family Augmentation admits an approximation ratio preservingreduction to
SS-CDS , see [23] for details. Lemma 5 also extends to this case, as it is knownthat an if J is an inclusion minimal edge set whose addition makes a connected graph2-connected, then J is a forest.With these definitions and facts, the rest of the proof for the Block-Tree Augmenta-tion coincides with the proof given for
Crossing Family Augmentation , concluding theproof of Theorem 1.
Recall that each of the problems
Crossing Family Augmentation and
Block-TreeAugmentation admits an approximation ratio preserving reduction to the
SS-CDS problem
X:8 Approximation algorithms for connectivity augmentation problems with R = L being the set of terminals. The SS-CDS instances that arise from this reductionhave the following property, see [6, 23]:( ∗ ) The neighbors of every r ∈ R induce a clique.In fact, SS-CDS with property ( ∗ ) is equivalent to the Node Weighted Steiner Tree problem with property ( ∗ ) with unit node weights for non-terminals (the terminals haveweight zero). Clearly, any SS-CDS solution is a feasible
Node Weighted Steiner Tree solution; for the other direction, note that if property ( ∗ ) holds, then the set of non-terminalsin any feasible Node Weighted Steiner Tree solution is a feasible
SS-CDS solution.The relation to the ordinary
Steiner Tree problem is given in following lemma. (cid:73)
Lemma 6 ([6]) . Let S be a SS-CDS solution and T = ( U, J ) a Steiner Tree solutionon instance ( G, R ) with unit edge costs. Then: (i) If ( ∗ ) holds then T can be converted into a SS-CDS solution S J with | S J | = | J | − | R | + 1 . (ii) S can be converted into a Steiner Tree solution T S = ( U S , J S ) with | J S | = | S | + | R | − . Proof.
We prove (i). Any
Steiner Tree solution T = ( U , J ) can be converted into asolution T = ( U, J ) such that | J | = | J | and R is the leaf set of T . For this, for each r ∈ R that is not a leaf of T , among the edges incident to r in T , choose one and replace the otheredges by a tree on the neighbors of r ; this is possible by ( ∗ ). The non-leaf nodes of such T form a a SS-CDS as required. For (ii), taking a tree on S and for each r ∈ R adding anedge from r to S gives a Steiner Tree solution as required. (cid:74)
Let J ∗ be an optimal and J an α -approximate Steiner Tree solutions. Let S J , S ∗ be SS-CDS solutions, where S J is derived from J and S ∗ is an optimal one. Then | S J | + R − | J | ≤ α | J ∗ | ≤ α | J S ∗ | = α ( | S ∗ | − | R | ) = α | S ∗ | + α ( | R | − . This implies that if
Steiner Tree admits ratio α then SS-CDS with property ( ∗ ) admits apolynomial time algorithm that computes a solution S of size | S | ≤ α opt + ( α − | L | andachieves ratio α + ( α − | L | opt = α + ( α − x , where x = | L | opt , 0 < x ≤
2. We will prove thefollowing. (cid:73)
Theorem 7.
Crossing Family Augmentation and
Block-Tree Augmentation admit ratio (cid:16) − | L | opt (cid:17) + (cid:15) . From Lemma 6 and Theorem 7 it follows that we can achieve ratiomax { α + ( α − x, − x ) } + (cid:15) where x = | L | opt . The worse case is when these two ratios are equal, which gives the Theorem 2 ratio. In thecase α = ln 4 + (cid:15) [7], we have x ≈ . L ≈ . opt and opt ≈ . L . The ratio inthis case is 1 + ln(4 − x ) + (cid:15) < . A set function f is increasing if f ( A ) ≤ f ( B ) whenever A ⊆ B ; f is decreasing if − f isincreasing, and f is sub-additive if f ( A ∪ B ) ≤ f ( A ) + f ( B ) for any subsets A, B of theground-set. Let us consider the following algorithmic problem: eev Nutov XX:9
Min-Covering
Input:
Non-negative set functions ν, τ on subsets of a ground-set U such that ν isdecreasing, τ is sub-additive, and τ ( ∅ ) = 0. Output: A ⊆ U such that ν ( A ) + τ ( A ) is minimal.We call ν the potential and τ the payment . The idea behind this interpretation andthe subsequent greedy algorithm is as follows. Given an optimization problem, the potential ν ( A ) is the (bound on the) value of some “simple” augmenting feasible solution for A . Westart with an empty set solution, and iteratively try to decrease the potential by adding a set B ⊆ U \ A of minimum “density” – the price paid for a unit of the potential. The algorithmterminates when the price ≥
1, since then we gain nothing from adding B to A . The ratio ofsuch an algorithm is bounded by 1 + ln ν ( ∅ ) opt (assuming that during each iteration a minimumdensity set can be found in polynomial time). So essentially, the greedy algorithm convertsratio α = ν ( ∅ ) opt into ratio 1 + ln α .Fix an optimal solution A ∗ . Let ν ∗ = ν ( A ∗ ), τ ∗ = τ ( A ∗ ), so opt = τ ∗ + ν ∗ . The quantity τ ( B ) ν ( A ) − ν ( A ∪ B ) is called the density of B (w.r.t. A ); this is the price paid by B for a unit ofpotential. The Greedy Algorithm (a.k.a.
Relative Greedy Heuristic ) for the problemstarts with A = ∅ and while ν ( A ) > ν ∗ repeatedly adds to A a non-empty augmenting set B ⊆ U that satisfies the following condition, while such B exists: Density Condition: τ ( B ) ν ( A ) − ν ( A ∪ B ) ≤ min (cid:26) , τ ∗ ν ( A ) − ν ∗ (cid:27) .Note that since ν is decreasing, ν ( A ) − ν ( A ∪ A ∗ ) ≥ ν ( A ) − ν ( A ∗ ) = ν ( A ) − ν ∗ ; hence if ν ( A ) > ν ∗ , then τ ( A ∗ ) ν ( A ) − ν ( A ∪ A ∗ ) ≤ τ ∗ ν ( A ) − ν ∗ and there exists an augmenting set B that satisfiesthe condition τ ( B ) ν ( A ) − ν ( A ∪ B ) ≤ τ ∗ ν ( A ) − ν ∗ , e.g., B = A ∗ . Thus if B ∗ is a minimum density setand τ ( B ∗ ) ν ( A ) − ν ( A ∪ B ∗ ) ≤
1, then B ∗ satisfies the Density Condition; otherwise, the density of B ∗ is larger than 1 so no set can satisfy the Density Condition. The following statement isknown, c.f. an explicit proof in [24]. (cid:73) Theorem 8.
The
Greedy Algorithm achieves approximation ratio τ ∗ opt ln ν ( ∅ ) − ν ∗ τ ∗ . This applies also in the case when we can only compute a ρ -approximate minimum densityaugmenting set, while invoking an additional factor ρ in the ratio.To use the framework of Theorem 8 we need to define τ and ν . Let J ⊆ E be an edge set.The payment τ ( J ) = | J | is just the size of J . The potential of J is defined by ν ( J ) = | R J | − R is a set of terminals such that L ⊆ R ⊆ V , defined in the following lemma. For anedge set F let F LL be the set of edges in F with both ends in L , and F L the set of edges in F that have exactly one end in L . (cid:73) Lemma 9.
Let F be an optimal solution to Crossing Family Augmentation instanceand c be a cost function on E defined by c ( e ) = 0 if e ∈ E LL , c ( e ) = 1 if e ∈ E L , and c ( e ) = 2 otherwise. Let J be a -approximate c -costs solution and let R be the set of ends of the edgesin J . Then | R | ≤ c ( J ) + L ≤ | F | − | L | = 4 opt − | L | . Proof.
Clearly, | R | ≤ c ( J ) + | L | . We show that c ( J ) ≤ | F | − | L | . Let F be the set ofedges in F that have no end in L . Since | F | = | F | − | F L | − | F LL | and 2 | F LL | + | F L | ≥ Lc ( F ) = | F L | + 2 | F | = | F L | + 2( | F | − | F L | − | F LL | ) = 2 | F | − ( | F L | + 2 | F LL | ) ≤ | F | − | L | . Since c ( J ) ≤ c ( F ), the lemma follows. (cid:74) X:10 Approximation algorithms for connectivity augmentation problems
It is easy to see that ν is decreasing and τ is subadditive. The next lemma shows that theobtained Min-covering instance is equivalent to the
Crossing Family Augmentation instance, and that we may assume that τ ∗ = opt and ν ∗ = 0. (cid:73) Lemma 10. If J is a feasible solution to Crossing Family Augmentation then ν ( J ) = 0 .If J is a feasible Min-Covering solution then one can construct in polynomial time afeasible
Crossing Family Augmentation solution of size ≤ τ ( J ) + ν ( J ) . In particular,both problems have the same optimal value, and Min-Covering has an optimal solution J ∗ such that ν ( J ∗ ) = 0 and τ ( J ∗ ) = opt . Proof. If J is a feasible Crossing Family Augmentation solution then | R J | = 1 and thus ν ( J ) = 0. Let I be a Min-Covering solution such that every edge in I has both ends in R ;e.g., I can be as in Lemma 9. Then I J is a feasible solution to the residual problem w.r.t. J and every edge in I J has both ends in R J . Let I ⊆ I J be an inclusion minimal edge set suchthat J ∪ I is a feasible solution. By Lemma 5, I is a forest, hence | I | ≤ | R J |−
1. Consequently, J ∪ I is a feasible solution of size at most | J | + | I | ≤ | J | + | R J | − τ ( J ) + ν ( J ). (cid:74) Recall also that ν ( ∅ ) ≤ opt − | L | , by Lemma 9. We will show how to find for any (cid:15) > (cid:15) )-approximate best density set in polynomial time. It follows therefore that we canapply the greedy algorithm to produce a solution of value 1 + (cid:15) times of1 + τ ∗ opt ln ν ( ∅ ) − ν ∗ τ ∗ = 1 + ln 4 opt − | L | opt = 1 + ln (cid:18) − | L | opt (cid:19) . In what follows note that if a , . . . , a q and b , . . . b q are positive reals, then by an averagingargument there exists an index 1 ≤ i ≤ q such that a i /b i ≤ P qj =1 a j / P qi =1 b j .Given a Crossing Family Augmentation instance, a set R ⊇ L of terminals, and F ⊆ E , consider the corresponding SS-CDS instance ( H = ( U, E H ) , R ) and the set ofnon-terminals Q that corresponds to F . The density of F is | F || R |−| R F | , and in the SS-CDS instance this is computed by taking a maximal forest in the graph induced by Q and theterminals that have a neighbor in Q ; then the density is | Q | over the number of treesin this forest. So in what follows we may speak of a density of a subforest of H . Let T i = ( S i ∪ R i , E i ), i = 1 , . . . , q , be the connected components of such a forest, ( R i is the setof terminals in T i ) and let s i = | S i | and r i = | R i | , where r i ≥
2. The density of the forest is P qi =1 s i / P qi =1 ( r i −
1) while the density of each T i is s i / ( r i − T i has density not larger than that of the forest. Consequently, we may assume thatthe minimum density is attained for a tree, say T .Let T = ( S ∪ R, E ) be a tree with leaf set R . The density of T is sr − , where r = | R | isthe number of terminals ( R -nodes) and s is the number of non-terminals ( S -nodes) in T .The usual approach is to show that for any k there exists a subtree T of T with k terminals(or k non-terminals) such that the density of T is at most 1 + f ( k ) times the density of T ,where lim k →∞ f ( k ) = 0. The decomposition lemma that we prove is not a standard one.The difficulty can be demonstrated by the following examples. Consider the case when T is a star with n leaves. Then the density of T is 1 / ( n − k leaveshas density 1 / ( k − T is a path with n non-terminals, then the density of T is n , whilea subtree with k < n non-terminals has density k/ ∞ . In both cases, the density ofthe subtree may be arbitrarily larger than that of T . To overcome this difficulty, we willdecompose T w.r.t. a certain subset P of the non-terminals.Let P ⊆ S . Let s = | S | , r = | R | , and p = | P | . For a subtree T of T let S ( T ), R ( T ),and P ( T ) denote the set of S -nodes, R -nodes, and P -nodes in T , respectively. We provethe following. eev Nutov XX:11 (cid:73) Lemma 11.
Let k ≥ . If p ≥ k + 1 then there exists subtrees T , . . . , T q of T such thatthe following holds. P qi =1 | S ( T i ) | ≤ s + q .Every R -node belongs to exactly one subtree, hence P qi =1 | R ( T i ) | = r . | P ( T i ) | ∈ [ k, k ] for all i and q ≤ pk − . Proof.
Root T at some node in S . For any v ∈ S chosen as a “local root”, the subtree T v rooted at v is a subtree of T that consist of v and its descendants. Let T v be an inclusionminimal rooted subtree of T such that | P ( T v ) | ≥ k + 1. Note that v ∈ P . Let B , . . . , B m be the branches hanging on v and let p j = | P ( B j ) | . By the definition of T v , each p j is inthe range [0 , k ] and P mj =1 p j ≥ k . We claim that { p , . . . , p m } can be partitioned such thatthe sum of each part plus 1 is in the range [ k, k ]. To see this, apply a greedy algorithmfor Multi-Bin Packing with bins of capacity 2 k ; at the end there is at most one bin withsum ≤ k − T v and the S -nodes on the path from v to its closestterminal ancestor, and apply the same procedure on the remaining tree. If the last rootedsubtree T v considered has | P ( T v ) | ≤ k −
1, then this tree can be joined to a subtree T i with | P ( T i ) | ≤ k derived in previous iteration. Finally, q ≤ p + qk by the construction and since | P ( T i ) | ≥ k for all i ; this implies q ≤ pk − . (cid:74) Now we let P = P ∪ P , where P is the set of nodes that have degree at least 3 in T and P is the of nodes that have a terminal neighbor in T . Note that | P | ≤ r and | P | ≤ r .Hence p ≤ r , and clearly p ≤ s . By an averaging argument and Lemma 11, the densityof some T i is bounded by s i / ( r i − ≤ P qj =1 s j / P qj =1 ( r j − ≤ ( s + q ) / ( r − q ). Thus for k ≥ s i r i − · r − s ≤ s + qr − q · rs ≤ s + p/ ( k − r − p/ ( k − · rs = 1 + 1 / ( k − − / ( k −
1) = kk − k − . This implies that we can find a (1 + (cid:15) )-approximate min-density tree by searching overall trees T with | P ( T ) | ∈ [ k, k ], where given (cid:15) > k = d /(cid:15) e + 3. Specifically, forevery P ⊆ S with | P | ∈ [ k, k ], we find an MST T in the metric completion of the currentincidence graph, and then add to T all the terminals that have a neighbor in P . Among allsubtrees we choose one of minimum density. The time complexity is n k which is polynomialfor any fixed (cid:15) > Block-Tree Augmentation is identical tothe one in the proof of Theorem 1. This concludes the proof of Theorem 7, and thus also theproof of Theorem 2 is complete.
Let p : 2 S → Z be a set function and J an edge set on a finite groundset S . We saythat J covers p if d J ( A ) ≥ p ( A ) for all A ⊆ S , where d J ( A ) denote the set of edgeswith exactly one end in A . p is symmetric if p ( A ) = p ( S \ A ) for all A ⊆ S , and p is skew-supermodular (a.k.a. weakly supermodular ) if for all A, B ⊆ S at least one ofthe following two inequalities holds: p ( A ) + p ( B ) ≤ p ( A ∩ B ) + p ( A ∪ B ) p ( A ) + p ( B ) ≤ p ( A \ B ) + p ( B \ A ) Element Connectivity Augmentation can be reduced to the following problem, withskew-supermodular set function p , c.f. [13, 21, 5]. X:12 Approximation algorithms for connectivity augmentation problems
Set Function Edge Cover
Input:
A set function p on a ground-set S . Output:
A minimum size set J of edges that covers p .In this problem, the function p may not be given explicitly, and a polynomial time implement-ation of algorithms requires that some queries related to p can be answered in polynomialtime. But the problem is also NP-hard for skew-supermodular p even if p given explicitly,specifically when p max = 1 and | A | = 3 for every set A with p ( A ) = 1 [21].In the degree bounded version of this problem we are also given degree bounds { b ( v ) : v ∈ S } and require that d J ( v ) ≤ b ( v ) for all v ∈ S . (cid:73) Definition 12.
A function g : S → Z + is a p -transversal if g ( A ) ≥ p ( A ) for all A ⊆ S .Let T g = { v ∈ S : g ( v ) ≥ } denote the support of g . We say that g is a minimal p -transversal if for any v ∈ T g reducing g ( v ) by results in a function that is not a p -transversal. The following was proved by Benczú and Frank in [4], see also [21, Lemmas 1.1 and 3.2]. (cid:73)
Lemma 13.
Let g be a transversal of a skew-supermodular set function p . Then: g ( S ) = max { P A ∈F p ( A ) : F is a subpartition of S } if g is minimal.There exists an optimal p -cover J such that every e ∈ J has both ends in T g . Let τ ( p ) denote the size of a minimal p -cover. As g = { d J ( v ) : v ∈ S } is a p -transversalfor any p -cover J , τ ( p ) ≥ g ( S ) / p -transversal g . Thus a natural approach tocompute a small p -cover is: repeatedly choose an edge uv with u, v ∈ T g , such that updating p and reducing g ( u ) and g ( v ) by 1, keeps g being a p -transversal. This approach worksfor many interesting special cases, c.f. [5], but in general such an edge uv may not exist.Formally, given u, v ∈ T g define p uv and g uv by: p uv ( A ) = max { p ( A ) − , } if | A ∩ { u, v }| = 1 and p uv ( A ) = p ( A ) otherwise; g uv ( w ) = g ( w ) − w = u or if w = v and g uv ( w ) = g ( w ) otherwise.It is easy to see that if p is (symmetric) skew-supermodular, so is p uv . However, g uv maynot be a p uv -transversal if g is. We say that a pair u, v ∈ T g is ( p, g ) -legal if g uv is a p uv -transversal; then replacing p, g by p uv , g uv is the splitting-off operation at u, v . Intuitively,splitting-off is an attempt to add the edge uv to a partial solution, and to consider theresidual problem of covering p uv with the residual lower bound d g uv ( S ) / e = d g ( S ) / e − (cid:73) Lemma 14 ([21]) . Let p be symmetric skew-supermodular and g a p -transversal. If p max ≥ then there exists a ( p, g ) -legal pair. Lemma 14 implies that if no ( p, g )-legal pair exists, then any inclusion minimal solutionon T g is a forest, and that any tree on T g is a feasible solution. In [21, 5] was considereda simple greedy algorithm which repeatedly splits-off legal pairs as long as such exist, andthen adds to the partial solution a tree (or any inclusion minimal solution) on T g . eev Nutov XX:13 Algorithm 2:
Greedy ( p, g )( p is symmetric skew-supermodular, g is a p -transversal) M ← ∅ while there exists a ( p, g )-legal pair u, v do g ← g uv , p ← p uv , M ← M + uv let F be a tree on T = { v ∈ S : g ( v ) = 1 } return M ∪ F In the degree bounded version we let g = { b ( v ) : v ∈ S } . If this g is not a p -transversal,then the problem has no feasible solution. To get a degree violation +1, at step 4 of thealgorithm we choose F to be a path on T .In [21] is was shown that for skew-supermodular p this algorithm achieves ratio 7 /
4, bycharacterizing those pairs p, g for which no ( p, g )-legal pair exists and deriving a lower boundon τ ( p ). We establish a better lower bound than that of [21] and prove the following. (cid:73) Theorem 15.
Algorithm
Greedy achieves approximation ratio / . Moreover, if F ischosen to be a path at step 4, then d J ( v ) ≤ g ( v ) + 1 for all v ∈ S . Theorem 15 second statement is obvious, so in the rest of this section we prove the firststatement. The following can be deduced from Lemma 14, see [21]. (cid:73)
Corollary 16 ([21]) . Let p be symmetric skew-supermodular and g a minimal p -transversalwith non-empty support T = { v ∈ V : g ( v ) ≥ } , and suppose that no ( p , g ) -legal pairexists. Then p max = g max = 1 , | T | ≥ , and for every A ⊆ T with | A | ∈ { , } there is A ⊆ V with p ( A ) = 1 such that A ∩ T = A . Furthermore, τ ( p ) ≥ | T | . We now describe the analysis of the 7 / k be the number of edgesaccumulated in M during the while-loop. Let t = g ( S ) be the initial value of the p -transversal g and let t = | T | be the the transversal value at the beginning of step 4. Note that t − t = 2 k and that | F | = | T | − t − k −
1. Consequently, | M ∪ F | ≤ k + ( t − k − ≤ t − k . Onthe other hand we have the lower bounds τ ( p ) ≥ t/ τ ( p ) ≥ | T | = ( t − k ). Thus theapproximation ratio is bounded by t − k max { t/ , t − k ) / } ≤ /
4, with k = t/ k ≥ t/ t/ ≥ ( t − k ), and thus in this case the ratio isbounded by t − kt/ ≤ / / = 3 /
2. To get ratio 3 / k ≤ t/ τ ( p ). For this, we need the following lemma. (cid:73) Lemma 17.
Let g be a minimal transversal of a skew-supermodular symmetric set function p . Then there exists an optimal p -cover J such that d J ( v ) ≥ g ( v ) if v ∈ T g and d J ( v ) = 0 otherwise. Proof.
By induction on τ ( p ). The base case τ ( p ) = 1 is trivial. For τ ( p ) ≥
2, let J be anoptimal p -cover such that every e ∈ J has both ends in T g ; such exists by Lemma 13. Choosesome e = uv ∈ J and let p = p uv . Let g u be obtained from g by decreasing g ( u ) by 1,and similarly g v is defined. Then one of { g, g u , g v , g uv } is a minimal p -transversal; denoteit by g . By the induction hypothesis there exists a p -cover J such that: d J ( w ) ≥ g ( w )if w ∈ T g and d J ( w ) = 0 otherwise. It is easy to see that J = J ∪ { e } has the requiredproperty. (cid:74) (cid:74)(cid:73) Lemma 18. τ ( p ) ≥ ( t − k ) . X:14 Approximation algorithms for connectivity augmentation problems
Proof.
Let J be a p -cover as in Lemma 17. Let X be the set of edges in J with both ends in T g \ T and Y the set of edges in J with exactly one end in T g \ T ; let x = | X | and y = | Y | .Since d J ( v ) ≥ g ( v ) for all v ∈ T , 2 x + y ≥ t − t = 2 k , hence y ≥ k − x . Let Q be the setof nodes in T that are uncovered by edges in X ∪ Y and let z be the number of edges in J with both ends in Q . Note that | Q | ≥ t − y = t − k − y . By Corollary 16, for every A ⊆ Q with | A | ∈ { , } there is A ⊆ V with p ( A ) > A ∩ T = A . This implies that z ≥ | Q | ≥ ( t − k − y ). Consequently, since | J | ≥ x + y + z and y ≥ k − x ) | J | ≥ x + y + 23 ( t − k − y ) = x + 13 y + 23 ( t − k ) ≥ x + 23 ( k − x ) + 23 ( t − k ) = 13 x + 23 ( t − k ) . Since x ≥ | J | ≥ ( t − k ). (cid:74) (cid:74) From Lemma 18 it follows that the approximation ratio of the algorithm is bounded by t − k max { t/ , t − k ) / } ≤ /
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