SP graphs

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(1)

Maximum Series-Parallel Subgraph

Gruia C˘alinescu

Illinois Institute of Technology

Joint work with Cristina Fernandes and Hemanshu Kaul

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SP graphs

Simple graph is series-parallel:

• No K4 minor/subdivision

• Partial 2-tree. Maximal SP graph is a 2-tree

• Arises from a forest by adding parallel edges, subdividing edges, and at the end removing parallel edges to keep the graph simple.

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Treewidth decompostion of maximal SP graphs

s t

s b t

y b b t

s d a

c a s d d a

c g a

c g g a c h h a

s a t

a t

s c

s c a s e a a f t

e a

s e a f f t

c h a t

s

a

b c

d e

f g

h

s a

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The problem , and known results

On input (V, E), find maximum F ⊆ E such that (V, F) is a SP graph. Known to be NP-Hard, approximation 1/2 trivial as for SP graphs, |E| ≤ 2|V | − 3.

Also, weighted: Maximum Spanning Tree achieves 1/2 of OPT (from arboricity).

All our graphs from are simple (except when called multigraphs).

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New results

• 7/12 approximation using spruce structures

• Maximum spruce structure is a 2/3- approximation

• Computing maximum spruce structure NP- Hard

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(a) (b)

u

v

w z

A spruce has two bases and several tips. A tip is adjacent exactly to the two bases. If bases are adjacent, the spruce is complete.

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(a) (b)

u

v

w z

A spruce structure has blocks spruces or bridges.

If blocks of finite size, the approach fails.

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gain(S) := cyclomatic number.

For complete spruces, gain is the number of tips; adjusted gain gaind := gain.

For incomplete spruces, gain is one less then the number of tips; adjusted gain gaind := gain −1.

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Selecting spruces of large gain without discard- ing fails (a).

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. . . (a)

(a) (b)(b)

x

x yy

n tips

n tips

n 2 tips

n 2 tips

Degenerate spruces have gaind ≤ 0. Only two types (b).

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Local improvement examples

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(a)

(b)

(c)

x

x x

x

x=x y

y

y y

y

x

y y

2

2 2

2

2

2 2

TC

TC

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What we add

Spruces with tips isolated vertices.

What do we remove

We keep a weighted forest, with weights 1 be- tween each tip and one of the bases, and gaind be- tween bases. Trees: TC

indexQ(x, y) is an edge in TC of minimum weight in the path in TC from x to y. If x and y are in different components of Q, then let indexQ(x, y) be undefined and consider its weight to be zero.

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Construct-Spruce-Structure (G)

1 Q ← ∅

2 while there are x and y such that SQ(x, y) is defined and gain(d SQ(x, y)) > w(indexQ(x, y)) do

3 if indexQ(x, y) is undefined 4 then Q Q∪ {SQ(x, y)}

5 else let x and y be the endpoints of indexQ(x, y) 6 let S be the spruce in Q containing x and y 7 Q Q\ {S′ } ∪ {SQ(x, y)}

8 if x or y is a tip of S

9 then let z be between x, y, a tip of S 10 let {e, f} be the edges of S touching z

11 S S′ − {e, f}

12 if S is not degenerate nor single edge

13 then Q Q∪ {S}

14 add bridges to Q to obtain a connected spanning subgraph of G 15 return Q

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Running time analysis

Increasing gain would be good, but doesn’t quit work as we use gaind .

Define Φ(Q) = 3gain(Q) +c(Q), where c(Q) is the number of components of Q when Q is seen as a spanning subgraph of G.

Lemma 1 Every iteration of the algorithm in- creases the parameter Φ.

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Approximation ratio ideas - to beat 1 / 2

significant = constant fraction of n

1. If significantly many vertices in our structure, we win

2. If OPT has significantly less than 2n edges, we win

3. If none of the above, the spruces of OPT have significant gaind

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The spruces of OPT

s t

s b t

y b b t

s d a

c a s d d a

c g a

c g g a c h h a

s a t

a t

s c

s c a s e a a f t

e a

s e a f f t

c h a t

s

a

b c

d e

f g

h

s a

These spruces do not share tips. Some gaind is lost, when the tips are not isolated in our structure.

Still, enough left.

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Approximation ratio ideas - beating 1 / 2

From OP T, construct weighted SP graph with gaind on edges.

Compare to our weighted forest.

We have a maximum spanning forest in the union of the two graphs!

Thus our gaind is 1/2 of what that of OP T. Therefore we have significant gaind .

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7 / 12 tight

t

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Maximum spruce structure

The spruces of a SP graph have n − 2 tips. We get 1/3 of that by partitioning those into three spruce structures.

Turns out if no parent-child, no two siblings, it is a spruce structure.

Color root 1 and for every node, color its chil- dren with the two other colors.

s t

s b t

y b b t

s d a

c a s d d a

c g a

c g g a c h h a

s a t

a t

s c

s c a s e a a f t

e a

s e a f f t

c h a t

s

a

b c

d e

f g

h

s a

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Tight example for maximum spruce structure

replacemen

G0 G1 G2

x

y

z w

y

w

Contructed inductively.

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The NP reduction for Maximum Spruce Structure

(a) (b)

..

. r

x

x y

y

C1 C2 C3

x1 x¯1 x2 x¯2 x3 x¯3

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Conclusions

We do a very little better when we pick spruce maximizing gaind − index.

Completely new approach? Also for Maximum Planar Subgraph, at 4/9 since 1996.

Weighted?

Figure

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