approximating node-weighted survivable networks
DESCRIPTION
Approximating Node-Weighted Survivable Networks. Zeev Nutov The Open University of Israel. 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 - PowerPoint PPT PresentationTRANSCRIPT
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
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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
21 2
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?