Julia Chuzhoy (TTI-C)
Yury Makarychev (TTI-C)
Aravindan Vijayaraghavan (Princeton)
Yuan Zhou (CMU)
Cut minimization
• Min st-cut: delete the min #edges to disconnect s, t
t
s
Duality:
Maxflow(s, t) = Mincut(s, t)
=2
Multicut
• Given r pairs (si, ti), delete min #edges to disconnect all
(si, ti) pairs
t
1
•Upper bound on max
multicommodity flow
t3
s3 •Identifies bottlenecks
in the graph
s1
t2
s2
•O(log r) approximation
algorithm [GVY95]
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
t3
s3
s1
t2
s2
For example, when k = 2,
OPT = 1.
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
• VC-kRC: vertex connectivity version, remove min. wt. of
edges so that for each i, (si, ti)-vertex-connectivity < k
Motivation
• Multiroute generalization : a fault tolerant setting
• st-k-route flow: a fractional combination of elementary
k-route st-flows [Kis96, KT93, AO02]
– Flow is resilient to (k-1) failures
Maxflow/
Mincut
multiroute
generalization
st-k-route
flow
multicommodity
flow
k-route
multicommodity
flow
multicut
k-route cut
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
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
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
(both EC-2RC and VC-2RC)
• [Kolman-Scheideler'11]
– O(log3r)-approximation for k=3
(EC-2RC)
• No sub-polynomial approx. algorithm known for k > 3
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
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
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
arbitrary k,
general
O(k log1.5 r)
(1+ε, (1/ε)log1.5 r)
(2, log2.5 r loglog r)
in time nO(k)
(log r, log3 r)
in poly(n, k) time
A comparison : VC-kRC
k=2
arbitrary k
Previous results
Our results
O(log2 r) [BC10]
O(log1.5 r)
multicut hardness:
2.5r loglog r)
(2,
dklog
APX-hard [DJP+94]
alg in time nO(k)
superconstant
assuming UGC
ε)-hardness
Ω(k
[KV05, CKK+06]
The rest of this talk...
• O(k log1.5 r)-approximation algorithm for unweighted ECkRC
• (2, log2.5 r loglog r)-bicriteria approx. algorithm for
general EC-kRC (sketch)
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
s
1
• Key observation.
the red edge cannot
provide extra connectivity
for s1, t1
t1
graph G
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
s
graph G
1
• Key observation.
the red edge cannot
provide extra connectivity
for s1, t1
t1
a bad example for k = 3
• No longer true for k = 3 (or more)
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
O(k log1.5 r)-approximation algorithm for
unweighted EC-kRC
Cut sparsity, and unweighted EC-kRC
• Let
d(v) = #source-sink pairs that v participates in
d (v )
d(S) =

vS
• Define uniform sparsity to be
( S ) 
edges( S , S )
min{d ( S ), d ( S )}
,
 (G )  min ( S )
S
• Theorem.[ARV04] O(log0.5 r)-approx. for Φ(G).
• Lemma.  (G ) 
k  OPT
r
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 (S , S )
• Step 2. Delete all the edges across the cut
• Step 3. Recurse into the subinstances defined by each
side of the cut
k  OPT
• Lemma.  (G ) 
r
• Fact. #cut edges deleted in Step 2 is at most
edges ( S , S )  log r   (G )  min{d ( S ), d ( S )}
min{d ( S ), d ( S )}
 log r  k 
 OPT
r
• Corollary. #edges deleted in total is at most k log1.5 r  OPT
Proof of
k  OPT
• Lemma.  (G ) 
r
• Consider H = G \ OPT
• For every (si, ti) pair,
mincutH(si, ti) = |edges(Si, Ti)| < k
(a witness cut)
Si
• Claim. The witness cuts are laminar
si
ti
Ti
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
Proof of
k  OPT
• Lemma.  (G ) 
r
• Consider H = G \ OPT
• For every (si, ti) pair,
mincutH(si, ti) = |edges(Si, Ti)| < k
(a witness cut)
S2
• Claim. The witness cuts are laminar S1
• Let S1, S2, ..., Sm be the maximal
witness cuts
S3
Proof of
k  OPT
• Lemma.  (G ) 
r
• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
S2
1. d(S1) + d(S2) + ... + d(Sm) >= r
2. edges OPT ( S i , S i )  1
edges H ( S i , S i )  k  1
therefore
S1
edges G ( S i , S i )  k edges OPT ( S i , S i )
S3
k  OPT
• Lemma.  (G ) 
r
Proof of
• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
S2
1. d(S1) + d(S2) + ... + d(Sm) >= r
2. edges G ( S i , S i )  k edges OPT ( S i , S i )
m
 edges (S , S )  2k  OPT
i 1
G
i
i
(since each edge is shared
by at most 2 maximal cuts)
S1
S3
k  OPT
• Lemma.  (G ) 
r
Proof of
• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT
S2
1. d(S1) + d(S2) + ... + d(Sm) >= r
2. edges G ( S i , S i )  k edges OPT ( S i , S i )
m
 edges (S , S )  2k  OPT
G
i 1
i
S1
i
3. by expansion
m
m
 edges (S , S )   (G)d (S )
i 1
G
i
i
i 1
 r  (G)
In all: r  (G)  2k  OPT
i
S3
(2, log2.5 r loglog r)-bicriteria approx. for
general EC-kRC (sketch)
(2, log2.5 r loglog r)-bicriteria approx. for
general EC-kRC (sketch)
• k-route non-uniform sparsity
(k)
wt
(S , S )
(k )
 (S ) 
,
D( S , S )

S
where
: total wt of all the edges across the cut
wt (S , S )
but the most expensive (k-1) ones
(k)
D(S , S )
: #source-sink pairs across the cut
• Corollary. (of [ALN05]) O(log0.5 r loglog r) approx. in
nO(k) time
• Lemma. 
( 2 k 1)

 ( k ) (G )  min  ( k ) ( S )
OPT
(G )  log r 
r
(2, log2.5 r loglog r)-bicriteria approx. for
general EC-kRC (sketch) (cont'd)
• Lemma. 
( 2 k 1)
OPT
(G )  log r 
r
• 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 (2k2) 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
Open questions
• Algorithm side.
– Better true approximation algorithm for general ECkRC (and VC-kRC)
• Hardness side.
– Is EC-kRC (for large k) strictly harder than multicut?
– Understand the simplest case: st-EC-kRC.
Thank you!