Increasing graph connectivity from 1 to 2

Guy Kortsarz

Joint work with Even and Nutov

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,E+F’) is 2-edge-connected

Bi-Connected Components

A

BC

DE

F

G

H

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 (G+F’ is 2EC)

Shadow Completion

Part of the shadows added

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

Example

Hence the leaf to leaf links in OPT form a matching

Simple ratio 2: minimally leaf-closed trees

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

Proof

If an edge eT’ is not covered then we found a smaller leaf-closed tree T’’

v

e

T’’

Problematic structure: Stem A link whose contraction creates a

leaf

STEM

Twin Link

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

Example

|M|=2, |U|=3, t=1, |B|=2

The leaf-stem lower bound for 1.8

Bv

opt vttUopt

t'

)(deg5.02'5.18.1

linksmatchedspecialnonof

numberthebeLet

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

Example

The blue link mean the actual bound is larger by ½ than what we know in

advance

211

1.51

OPTOPT

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

Unmatched leaf has 1 coupon

Unmatched leaf has 1 coupon

1 1

1

A stem with 2 matched links: an example of 2-greedy

A stem with two matched pairs:

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

2

s

1

x

y

z

z

If no 1,2-greedy applies then the contraction of any eM 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:

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 3/2<2 together, so it does not work

Main idea

Find a tree with k+1 coupons that can be covered with k links

K+1

1

The Algorithm

Let I be the edges added so far Exhaust 1 and 2 greedy Compute T/(M I) No new leaves are created Find a minimally leaf-closed tree Tv

in T/(M I) Let A=up(leaf) in Tv

Add to the solution (M Tv)A (covers Tv ) Iterate

In picture

v

x

Basic cover and the extra

MA is called the basic cover of Tv

After M is contracted, T/(IM) has only unmatched leaves

Every lA being an unmatched leaf can pay with its coupon for up(l )

Every eM has 1.5 coupons. Pays for its contraction with ½ to spare

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* ½=1 spare can be left on the leaf

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

No matched pairs at least two leaves

No such link No such link is possible is possible

as this as this means 1 means 1 greedygreedy

We can add We can add the up of the up of the two the two

leaves and leaves and leave 1 in leave 1 in

the the resulting resulting

leafleaf Not possible as Not possible as the tree is leaf the tree is leaf

closedclosed

The other endpoints The other endpoints belong to Q: 2 ticket, 1 belong to Q: 2 ticket, 1

couponcoupon

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

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 v=r 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’

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 BT This means that we have the extra

½ needed. We use the basic cover and leave a coupon

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

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. CheriyanJ. 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?)