Approximating Node-Weighted Survivable Networks

Zeev NutovThe Open University of Israel

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Talk Outline

• Problem Definition• History and Our Results• Greedy Algorithm for Node Weighted Steiner Trees• Reducing NWSN to Finding Minimum Weight

Edge-Cover of Uncrossable Set-Family• Spider-Cover Decomposition of Edge-Covers of

Uncrossable Set-Families• Algorithm for Covering Uncrossable Set-Families• Node-Weighted k-Flow is harder than

Densest -Subgraph

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Survivable Network (SN)

Instance: A graph G = (V,E), weight function w on edges/nodes, U V, and connectivity requirements r(u,v), u,v U.

Objective: A minimum weight spanning subgraph J of G containing U so that

λJ(u,v) ≥ r(u,v) for all u,v U λJ(u,v) = uv-edge-connectivity in J

Problem Definition

ApproximabilityEdge-weights: 2-approximable [Jain, FOCS 98], APX-hard Node-weights: for r(u,v) {0,1} O(log n)-approximable, Set-Cover hard [Klein and Ravi, IPCO 93]

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History and Our ResultsEdge-weights Year Node-weights

2 for r(u,v){0,1} [AKR] 1991

2rmax [WGMV] 1993 2H(n) for r(u,v){0,1} [KR]

2H(rmax) [GGPSTW] 1994

1996 1.35H(n) for r(u,v){0,1}[GK]

2 [J] 1998

Theorem 1NWSN admits a rmax ·3H(n)-approximation algorithm.

Theorem 2ρ-approximation for NWSN with |U|=2 implies 1/ρ2-approximation for Densest -Subgraph.

We do not have a polylogarithmic approximation for any rmax…

But this is not our fault!

What about node-weights and rmax =2?

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Node Weighted Steiner TreeInstance: A graph G=(V,E), a set U V of terminals,

weights w(v) for nodes in V−U.Objective: Find a min-weight subtree T of G containing U.

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a

2 c

31

4

b

d

The “deficiency” of a partial solution I: v(I) = # (components containing terminals in (V,I)) -1.

v(I) = 0w(I) = 5

v(I) = 0w(I) = 8

v(I) = 1w(I) = 7

The “node-weight” w(I) of a partial solution I E: w(I) = w(V(I)) = the weight of endnodes of I

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3

2

35 3

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The Greedy Algorithm

opt

w I

I I S I

Initialize: I While ν(I) > 0 do: Find S E – I so that I I FReturn I.

The Density Condition

optw S

I I S I

Theorem: If ν is decreasing and w is subadditive then the greedy algorithm has approximation ratio ρ ·H(ν()).

Objective:Find in polynomial time an “augmentation” S that satisfies the density condition for “small” ρ.

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A Lesson in Zoology

These are also spiders:

In general, a spider is a tree on at least 2 nodes, which has at most one node of degree ≥ 3.

This is a spider:

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Spider Decomposition of Trees

Center – The single node of degree ≥ 3. If there is no node of degree ≥ 3, any node can be a center.Leaves – The non-center nodes of degree 1.

Lemma: Every tree can be decomposed into node-disjoint spiders such that every leaf of the tree belongs to a unique spider.

1. Select a node v whose sub-tree is a spider.

2. Remove v and its sub-tree.3. Remove the path from v to its

closet ancestor of degree 3.4. Repeat.

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Finding the First Augmentation

Finding a spider S (in fact, a Shortest Path Tree) of optimal density: For each node s in the graph

1. Sort the paths from s to terminals in increasing weight order.2. Add the two lightest paths.3. Add paths in increasing weight order, till reaching minimum density.

23 74

Terminals: 2Weight: 8Density: 8 = 8/(2-1)

Terminals: 4Weight: 19Density: ~ 6.3=19/(4-1)

T = optimal tree; we may assume: terminals = leaves of T

iw T w S leaves / 2i iS S

By averaging, there is a spider Si such that:

2i

i

w S w T

S T

2 iT S

Terminals: 3Weight: 12Density: 6 = 12/(3-1)

{Si} – spider-decomposition of T. The spiders are disjoint

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The Complete Algorithm

Finding an augmentation with a general partial cover I:1. Contract every connected component of (V,I) into a

super-node; a super node is a super-terminal if it contains a terminal.

2. Find an augmentation in the new graph (the partial cover is now ).

The previous algorithm finds an augmentation obeying the Density Condition with ρ=2 if the current partial cover is I = .

The approximation ratio of the algorithm is 2H(|U|).

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Algorithm for NWSN

The algorithm has rmax iterations.In iteration k we find a 3H(n)-approximation for the problem:

Given: A graph J=Jk-1 with λJ(u,v) ≥ min{r(u,v),k-1} for all u,v UFind: An edge set I with w(V(I)) minimum so that λJ+I(u,v) ≥ min{r(u,v),k} for all u,v U

Hence after rmax iterations, a feasible solution of weight at most rmax ·3H(n)·opt is found.

Instance: A graph G = (V,E), weight function w on the nodes, U V, and connectivity requirements r(u,v), u,v U.

Objective: A minimum weight spanning subgraph J of G containing U so that λJ(u,v) ≥ r(u,v) for all u,v U.

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Covers of Uncrossable Set-Families

Node-Weighted Set-Family Edge-Cover (NWSFC)

The augmentation problem we want to solve is a particular case of the following problem:

Instance: A graph (V,E), node weights {w(v):v V}, and an uncrossable set-family on V.Objective: Find an -cover I ⊆ E of minimum node-weight (edge e covers set X if e has exactly one endnode in X)

is uncrossable if X,Y implies at least one of the following:

orX ∩Y, X Y

X − Y, Y −X X

Y V−Y

V−XNote: The inclusion minimal members of are pairwise disjoint.

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Spider-Covers of Uncrossable Set-Families• () = the family of inclusion minimal sets in (min-cores)• (C) = sets in that contain a unique min-core C (cores) • (s,C) = {X (C) : s V-X}• (s,) = {(s,C) : C }

Definition: Let ⊆ () and let sV. An edge set S is an (s,)-cover if: - S covers (s,C) for every C- if ={C} then no member of (C) contains sAn (s,)-cover S is a spider-cover if it can be partitioned into (s,C)-covers {SC:C } so that the node sets {V(SC) −s} are pairwise disjoint.

ss

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Spider-Coves Decompositions

Definition of a Spider-Cover Decomposition:A sub-partition S1,…,Sq of a cover I is a spider coverdecomposition of I if there exists a partition 1, …,q of () and centers s1,…,sq V so that:- Each Si is an (si,i)-cover - The node sets V(Si) are pairwise disjoint.

Spider-Cover Decomposition Theorem:Any uncrossable family cover has a spider-cover decomposition.

Proof: Later.

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Covering Uncrossable Families

S is a spider with leaves: Δ(S) ≥ /2 (tight for =2)S is an (s,)-cover with ||=: Δ(S) ≥ /3 (tight for =3)

= |()| = # (min-cores)Δ(S) = decrease in the deficiency caused by adding S to the partial solution

optw S

S

Density Condition (for I=)

If S is an (s,)-cover then Δ(S) ≥ (||-1)/2 if || ≥2 Δ(S) = 1 if || =1

The Spider-Cover Lemma

Tight Example

u0

v1

v2

u2

u1

s

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The Algorithm

Thus the Greedy Algorithm can be implemented in polynomial time with ρ=3.

Approximation ratio: 3H(()) = 3H(|()|) ≤ 3H(n)

The Spider-Cover Lemma implies that there exists a spider-cover that satisfies the Density Condition with ρ=3.

Such spider-cover can be found in polynomial time assuming we can compute in polynomial time:

- The family () of min-cores (max-flows)- Minimum weight (s,C)-cover (min-cost k-flows)

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The Spider-Cover Decomposition Thm – Proof Sketch

We may assume that I is an inclusion minimal -cover.Then for every eI there exists a witness set We , namely:

e is the unique edge in I that covers We.

A family = {We : e I} is called a witness family for I

(every eI has a unique witness set in We ).

Notation – uncrossable family (X,Y implies X ∩Y, X Y or X−Y,Y−X)I – an -cover (for any X there is eI with exactly one endnode in X)

Lemma:Let I be an inclusion minimal cover of an uncrossable family . Then there exists a witness family for I which is laminar.

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The Spider-Cover Decomposition Thm – Proof Sketch

For a min-core C() define:• LC = the maximal set in containing C• eC = the unique edge in I covering LC, eC=sCvC, vC LC

• SC = edges in I contained in LC plus eC

Assumptions:– Every member of is a core – Every minimal member (leaf) of is a min-core.

CL

eC C

C

C

s

v

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The Spider-Cover Decomposition Thm – Proof Sketch

Lemma:• The sets {LC : C in ()} are pairwise disjoint.• The sets {SC : C in ()} are pairwise disjoint.• SC covers all cores contained in LC. Corollary:Any partition 1, …, q of () induces a partition S1,…,Sq of I. We seek a partition so that S1,…,Sq is a spider-cover decomposition.

- A natural partition of () is by the stars of {eC : C()}.

- This approach fails for 1-edge stars; SC is not a spider-cover if there is a dangerous set MC containing LC+sC

- Every star with at least 2 edges indeed induces a spider-cover.

CL

eC

CL

MC

eC'eC

C

C

C

s

v

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The Spider-Cover Decomposition Thm – Proof Sketch

CL

MCeCeC'

s

How do we group dangerous cores? - group some together, or - assign to “non-dangerous” stars.

Assigning singleton classes:

Every singleton class {MC} of is assigned to the part of any edge eC’ covering MC .

Observation: Every dangerous MC is covered by some edge eC’ .

CL

MCeCeC'

s

Grouping dangerous cores together:

The relation ={(C,C’) : MC ∩ MC’ ≠ } is an equivalence, and its classes of size ≥2 induce spider-covers(center − any node in the intersection of MC’s)

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min : , .X X A B I X k

-appr f or NWSN

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max : ,I X X A B X

-appr f or D -S

min : , , 1..

.

min ,

k

k

X X A B I X k k I

k X

X X k

X A B I X k

1. Run the -approximation algorithm f or

f or all .

For every we have a (possibly empty) set

2. Set where is the largest integer so that

and .

2 2

2 2 2

opt .

'

1.

22 2

I X D S

X X X

I X I X I X I XX

Note that

3. Find with so that

B

A

s

t

( , ) ( , ) 0

, 1,

, : :

k r s t k r u v

J A B I w v v A B

s t sa a A bt b B

Node-Weighted k-Flow (NW F): and otherwise.

Given an instance , of bipartite D S, set

add nodes and edges of capacity each.

I

Reducing NWSN to bipartite DS

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Summary and Open QuestionsWhat did we do?Generalized the decomposition of a tree into spiders to covers of uncrossable families (looks easy after found…)What do we get?E.g., an rmax·3H(n)-approximation algorithm for NWSN.Any other applications?Probably YES.

Open Question:Node-Weighted k-Flow (NWkF) is a special case of

NWSN where r(s,t)=k and r(u,v)=0 otherwise. NWkF admits a k-approximation algorithm.Anything better, even for unit weights?