1. Approximating k-route cuts
Julia Chuzhoy (TTI-C)
Yury Makarychev (TTI-C)
Aravindan Vijayaraghavan (Princeton)
Yuan Zhou (CMU)

2. Cut minimization Min st-cut: delete the min #edges to disconnect s, t
Duality:
Maxflow(s, t) = Mincut(s, t) s
= 2

3. Multicut
Given r pairs (si, ti), delete min #edges to disconnect all (si, ti) pairs t1
Upper bound on max multicommodity flow
Identifies bottlenecks in the graph
O(log r) approximation algorithm [GVY95] t3 s3 s2 t2 s1

4. Min k-route cuts
Unweighted version. Given r pairs (si, ti), delete min #edges to k-disconnect all (si, ti) pairs
i.e. for all i, (si, ti)-edge-connectivity < k
General version. Given a weighted graph and r pairs, delete min wt. of edges to k-disconnect all (si, ti) pairs t1
For example, when k = 2, OPT = 1. t3 s3 s2 t2 s1

5. Min k-route cuts: variants and specal cases
EC-kRC: edge connectivity version, remove min. wt. of edges so that for each i, (si, ti)-edge-connectivity < k
Unweighted case: all edge weights = 1
k = 1: Minimum multicut
s-t EC-kRC: single source-sink pair version
VC-kRC: vertex connectivity version, remove min. wt. of edges so that for each i, (si, ti)-vertex-connectivity < k

6. multiroute generalization
Motivation
Multiroute generalization
st-k-route flow: a fractional combination of elementary k-route st-flows [Kis96, KT93, AO02]
Flow is resilient to (k-1) failures
: a fault tolerant setting
multiroute generalization
Maxflow/
Mincut
st-k-route
flow
multicommodity
flow
k-route
multicommodity
flow
multicut
k-route cut

7. Motivation (cont'd)
Multiroute generalization: a fault tolerant setting
As standard multicut, k-route cut also reveals network bottleneck, and in particular measures resilience of the network
multicut
k-route cut

8. Approximation algorithms
α-approximation: delete edges of wt. αOPT such that all the pairs are k-disconnected
(β,α)-bicriteria approximation: delete edges of wt. αOPT such that all the pairs are βk-disconnected

9. Previous work [Chekuri-Khanna'08] O(log2n log r)-approximation for k=2
(both EC-2RC and VC-2RC)
[Barman-Chawla'10]
O(log2r)-approximation for k=2
NP-Hardness for s-t EC-kRC
[Kolman-Scheideler'11]
O(log3r)-approximation for k= (EC-2RC)
No sub-polynomial approx. algorithm known for k > 3

10. Our results : algorithms for EC-kRC
Unweighted EC-kRC
O(k log1.5 r)-approximation
(1+ε, (1/ε)log1.5 r)-bicriteria approximation
General EC-kRC
O(log1.5 r)-approximation for k = 2
(2, log2.5 r loglog r)-bicriteria approx. in nO(k) time
(log r, log3 r)-bicriteria approx. in poly(n, k) time

11. Our results : VC-kRC Algorithms O(log1.5 r)-approximation for k = 2
(2, d k log2.5 r loglog r)-bicriteria approx. in nO(k) time, where each node belongs to at most d source-sink pairs
Harndess for VC-kRC
NP-Hard to approximate VC-kRC within Ω(kε) for some specific ε > 0
Hardness for st-VC-kRC
Superconstant hardness assuming random k-AND hypothesis of [Feige'02]
Ω(ρ0.5) hardness assuming ρ-inapproximability of Densest k-Subgraph

12. A comparison : EC-kRC Previous results Our results k = 2
O(log2 r) [BC10]
O(log1.5 r)
k = 3
O(log3 r) [KS11]
arbitrary k,
unweighted
O(k log1.5 r)
(1+ε, (1/ε)log1.5 r)
general
(2, log2.5 r loglog r)
in time nO(k)
(log r, log3 r)
in poly(n, k) time

13. superconstant assuming UGC
A comparison : VC-kRC
Previous results
Our results
k = 2
O(log2 r) [BC10]
O(log1.5 r)
arbitrary k
multicut hardness:
APX-hard [DJP+94]
superconstant assuming UGC
[KV05, CKK+06]
(2, dklog2.5r loglog r)
alg in time nO(k)
Ω(kε)-hardness

14. Why only bicriteria algorithm for large k?
The problem might be hard even for single s-t version
st-VC-kRC: no sub-polynomial approx. if assuming no sub-polynomail approx. for Densest k-Subgraph
st-EC-kRC: no sub-polynomial approx. known
(there is a (2, 2)-bicriteria approx. alg.)
Embarrassing situation for st-EC-kRC: even APX-hardness is not known
Recall the problem:
A weighted graph G,
source s, sink t
Goal: remove k edges in G to minimize min s-t cut

15. The rest of this talk...
O(k log1.5 r)-approximation algorithm for unweighted EC-kRC
O(log1.5 r)-approximation algorithm for general EC-2RC
(2, log2.5 r loglog r)-bicriteria approx. algorithm for general EC-kRC (sketch)

16. The difficulty for large k (> 2)
Simple recursion (used in [BC10]) for k = 2
Find a balanced cut (by region growing)
Remove all the cut edges
but the most expensive one
recurse into both sides
Key observation.
the red edge cannot
provide extra connectivity
for s1, t1
graph G s1 t1

17. The difficulty for large k (> 2)
Simple recursion (used in [BC10]) for k = 2
Find a balanced cut (by region growing)
Remove all the cut edges
but the most expensive one
recurse into both sides
Key observation.
the red edge cannot
provide extra connectivity
for s1, t1
No longer true for k = 3 (or more)
graph G s1 t1
a bad example for k = 3

18. Algorithms for k > 2
[Kolman-Scheideler'11] O(log3r)-approximation for k=3, by multi-level region growing (based on the same LP used in [BC10])
Our method
Idea 1. Relate k-route cut to the value of sparest cut
Idea 2. Solve the problem iteratively rather than recursively

19. O(k log1.5 r)-approximation algorithm for unweighted EC-kRC

20. Cut sparsity, and unweighted EC-kRC
Let
d(v) = #source-sink pairs that v participates in
d(S) =
Define uniform sparsity to be
Intuition. Given a cut , is small when
the cut size is small
the cut separates many terminals (or, the cut is balanced in terms of d)

21. Cut sparsity, and unweighted EC-kRC
Let
d(v) = #source-sink pairs that v participates in
d(S) =
Define uniform sparsity to be
Theorem.[ARV04] O(log0.5 r)-approx. for Φ(G).
Lemma.

22. Algorithm for unweighted EC-kRC
Step 0. Assume source-sink pairs are not k-disconnected
Step 1. Use the algorithm in [ARV04] to find an approximate sparse cut
Step 2. Delete all the edges across the cut
Step 3. Recurse into the subinstances defined by each side of the cut
Fact. #cut edges deleted in Step 2 is at most
Lemma.

23. A standard charging argument
Fact. #cut edges deleted in Step 2 is at most
Charge this cost to the smaller part among
At one step, each terminal is charged by
Each terminal can be in "small parts"
In total, each terminal is charged by
Since there are r terminals, total cost:

24. Proof of Lemma. Consider H = G \ OPT For every (si, ti) pair,
mincutH(si, ti) = |edges(Si, Ti)| < k
(a witness cut)
Claim. The witness cuts are laminar Si si ti Ti

25. Proof of Claim: witness cuts are laminar
Gomory-Hu Tree. (exists for every graph)
A weighted tree that consists of edges representing all pairs minimum s-t cuts in the graph.
mincutH(s, t) = mincutT(s, t)
All s-t mincuts in the tree
are laminar
==> All mincuts in H are laminar
==> All witness cuts are laminar H:
Gomory-Hu tree T

26. Proof of Lemma. Consider H = G \ OPT For every (si, ti) pair,
mincutH(si, ti) = |edges(Si, Ti)| < k
(a witness cut)
Claim. The witness cuts are laminar
Let S1, S2, ..., Sm be the maximal
witness cuts
(the smaller parts) S2 S1 S3

27. Lemma.
Proof of
Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
1. d(S1) + d(S2) d(Sm) >= r 2.
therefore S2 S1 S3

28. (since each edge is shared by at most 2 maximal cuts)
Lemma.
Proof of
Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
1. d(S1) + d(S2) d(Sm) >= r 2. S2 S1
(since each edge is shared by at most 2 maximal cuts) S3

29. Lemma.
Proof of
Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
1. d(S1) + d(S2) d(Sm) >= r 2.
3. by expansion
In all: S2 S1 S3

30. O(log1.5 r)-approx. algorithm for EC-2RC

31. Another definition of sparsity
the most expensive edge
2-route uniform sparsity
Corollary of [ARV04].
2-route uniform sparisty can
be efficiently approximated within O(log0.5 r) factor
Proof. Guess the red edge, remove it, and run ARV. S S

32. Another definition of sparsity
the most expensive edge
2-route uniform sparsity
Corollary of [ARV04].
2-route uniform sparisty can
be efficiently approximated within O(log0.5 r) factor S S
Lemma.

33. Algorithm for EC-2RC Step 1. Find an approximate 2-route sparse cut
Step 2. Delete all but the most expensive edge across the cut
Step 3. Recurse into the subinstances defined by each side of the cut
Claim. The algorithm outputs a valid 2-route cut.
Proof. By the key observation we made before.

34. Algorithm for EC-2RC Step 1. Find an approximate 2-route sparse cut
Step 2. Delete all but the most expensive edge across the cut
Step 3. Recurse into the subinstances defined by each side of the cut
Fact. wt of edges deleted in Step 2 is at most
Corollary. wt of edges removed in total
Lemma.

35. Proof of Lemma. Consider H = G \ OPT For every (si, ti) pair,
mincutH(si, ti) = |edges(Si, Ti)| < 2
(a witness cut)
Claim. The witness cuts are laminar
Let S1, S2, ..., Sm be the maximal
witness cuts
(the smaller parts) S2 S1 S3

36. Lemma.
Proof of
Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
1. d(S1) + d(S2) d(Sm) >= r 2. S2 S1
, therefore, S3

37. Lemma.
Proof of
Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
1. d(S1) + d(S2) d(Sm) >= r 2.
In all,
thus, there exists i:

38. (2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)

39. (2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)
k-route non-uniform sparsity
where
Corollary. (of [ALN05]) O(log0.5 r loglog r) approx. in nO(k) time
: total wt of all the edges across the cut but the most expensive (k-1) ones
: #source-sink pairs across the cut
Lemma.

40. (2, log2. 5 r loglog r)-bicriteria approx
(2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch) (cont'd)
The iterative algorithm. (Applying Idea 2)
Step 1. Use the algorithm in [ALN05] to find an approximate sparse cut
Step 2. Delete all the edges across the cut but the (2k-2) most expensive ones
Step 3. Remove all the source-sink pairs that are (2k-1)-disconnected
Step 4. Repeat Step 1~3 until no source-sink pair remains
Theorem. Wt. of removed edges <= log2.5 r loglog r OPT
Lemma.

41. Open questions Algorithm side.
Better true approximation algorithm for general EC-kRC (and VC-kRC)
Hardness side.
Is EC-kRC (for large k) strictly harder than multicut?
Understand the simplest case: st-EC-kRC.

42. Thank you!