Increasing graph connectivity from 1 to 2

1
Increasing graph connectivity from 1 to 2

• Guy Kortsarz
• Joint work with Even and Nutov

2
Augmenting edge connectivityfrom 1 to 2

• Given undirected graph G(V,E)
• And a set of extra legal for
• addition edges F
• Required a subset F? F of minimum
• size so that G(V,EF) is
• 2-edge-connected

3
Bi-Connected Components
G
A
H
F
B
D
E
C
4
The tree augmentation problem

• Input A tree T(V,E) and a separate set
• edge F
• Output Add minimum amount of edges
• F from F so there will be no
• bridges (GF is 2EC)

5
Shadow Completion

• Part of the shadows added

6
Shadows-Minimal Solutions

• If a link in the optimum can be
• replaced by a proper shadow and
• the solution is still feasible, do it.
• Claim in any SMS, the leaves have degree 1

7
Example
Hence the leaf to leaf links in OPT form a
matching
8
Simple ratio 2 minimally leaf-closed trees
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Covering minimally leaf-closed trees

• Let up(l) be the highest link
• (closest to the root) for l, after
• shadow completion.
• Let T be a minimally leaf-closed tree
• Then up(l) l? T covers T
• Given that, we spent L links in covering T. The
  optimum spent at least L/2
• A ratio of 2 follows

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Proof

• If an edge e?T is not covered then we found a
  smaller leaf-closed tree T

e
T
v
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Problematic structure Stem

• A link whose contraction creates a leaf

STEM
Twin Link
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The lower bound for 1.8

• Compute a maximum matching M among matching not
  containing stem links
• Let B be the non-leaf non-stems
• Let U be the unmatched leaves in M
• Let t be the number of links touching the twin of
  a stem with exactly one matched leaf in M
• For this talk let call a unique link touching a
  twin a special matched link

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Example
M2, U3, t1, B2
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The leaf-stem lower bound for 1.8
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Coupons and tickets

• Every vertex in U gets 1
• Every non-special matched link in M
• gets 1.5 coupons.
• Every special matched link gets 2 coupons.
• Every vertex in B touched by OPT gets
  degopt(v)/2 coupons
• This term is different, depends on OPT

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Example
?OPT
1
1
1.5
2
1
The blue link mean the actual bound is larger by
½ than what we know in advance
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1-greedy and 2-greedy

• If a link closes a path that has
• 2 coupons, the link can be contracted
• This is a 1-greedy step

1
Unmatched leaf has 1 coupon
Unmatched leaf has 1 coupon
1
1
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A stem with 2 matched links an example of
2-greedy

• A stem with two matched pairs

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The algorithm exahusts all 1,2 greedy all stems
are contracted

• Stems enter compound nodes
• Note that we may assume it has exactly one
  matched twin

1
z
y
s
2
x
z
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If no 1,2-greedy applies then the contraction of
any e?M never create a new leaf

• The paths covered by e,e are disjoint as no
  2-greedy
• Now say that later contracting e ? M creates a
  leaf

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Why not find minimum leaf-closed tree and add
up(leaves)?

• There is not enough credit
• Every unmatched leaf (vertex in U) does have a
  coupon needed to pay for the up link
• Unfortunately, every matched pair has only 3/2together, so it does not work

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Main idea

• Find a tree with k1 coupons that
• can be covered with k links

1
K1
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The tree that we find

• Method
• Let I be the edges added so far
• Compute T/(M? I)
• No new leaves are created
• Find a minimally leaf-closed tree Tv
• in T/(M? I)
• Let Aup(leaf) in T/(M? I)
• M?A covers Tv

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In picture
v
x
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Basic cover and the extra

• M?A is called the basic cover of Tv
• After M is contracted, T/(I?M) has only unmatched
  leaves
• Every l?A being an unmatched leaf can pay with
  its coupon for up(l )
• Every e?M has 1.5 coupons. Pays for
• its contraction with ½ to spare

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A trivial case

• The problem is that we need to leave 1 coupon in
  the created leaf (every unmatched leaf has one
  coupon)
• If T has two matched leaves or more
• the 2 ½1 spare can be left on the leaf

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Less than 2 matched pairs

• If there is a matched pair Remember that every
  non-leaf non-stem touched by opt has ½ a coupon
  so together it would be a full coupon which is
  enough
• First treat the case of no matched pairs.
• If only one leaf, solved like the DFS case

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No matched pairs at least two leaves

• We may assume that there is no compound node that
  is not a leaf, otherwise we have an extra coupon
  on the compound node
• The two unmatched leaves l, l are covered by
  links to two other nodes say x and y (note no
  1-greedy applies x and y are not leaves)
• x,y belong to Tv because Tv l, l closed
• Since x and y are not leaves they are nodes in
  the middle of the tree and non-compound
• Hence they belong to B 2 tickets, a coupon.

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At least four leaves one matched pair

• The only vertices not in B that can be linked to
  the (at least) two unmatched leaves l, l are the
  matched pair leaves say b and b
• Recall, b and b have degree 1 in OPT
• Thus l, l and b and b must form a perfect
  matching

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A ticket follows to cover the root

• The matched pair b and b have no more
• links in OPT as matched to l, l and have
• degree 1 in OPT
• There must be a link going out of Tv covering v
  (unless vr and we are done)
• This link does not come out of l, l because Tv
• is closed with respect to unmatched leaves
• And by the above it can not come out of b or b

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Covering v

• Therefore, the link comes out of a non-leaf
  internal node
• There are no compound internal nodes
• Thus v is covered by a vertex in B?T
• This means that we have the extra ½ needed. We
  use the basic cover and leave a coupon

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Remarks

• The case of one matched pair and 3 leaves gets a
  special treatment
• In the 1.5 ratio algorithm the stems do not
  disappear after 1,2-greedy
• Getting 1.5 requires 3 (more complex that what
  was shown here) extra new ideas and some
  extensive case analysis

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Only one open question

• The weighted case
• Cannot use leaf-closed trees
• In my opinion the usual LP does not suffice. BTW
  known to have IG 1.5
• Due to J. Cheriyan, H. Karloff, R. Khandekar,
  and J. Könemann
• We have stronger LP that we think has integrality
  gap less than 2
• We (all) failed badly in proving it (so far?)