Transcript Yi Wu IBM Almaden Research Joint work with Alina Ene and Jan Vondrak.

Yi Wu
IBM Almaden Research
Joint work with Alina Ene and Jan Vondrak
Definition of Problems
Graph Multiway Cut
Input: a graph
with 𝑘 terminals.
Graph Multiway Cut
Input: a graph
with 𝑘 terminals.
Goal: remove minimum number of edges to
disconnect the terminals.
Graph Multiway Cut
Input: a graph
with 𝑘 terminals.
Equivalent Goal: divide the graph into 𝑘 parts to
minimize number of cross edges.
Constraint Satisfaction Problem (CSP)
with “≠” constraint
𝑥6
𝑥5
𝑥3
1
𝑥2
≠
𝑥4
2
≠
≠
≠
≠
≠
≠
≠
≠
𝑥7
𝑥8
≠ ≠
𝑥1
≠
3
1 ≠ 𝑥3
1 ≠ 𝑥2
𝑥3 ≠ 𝑥4
𝑥2 ≠ 𝑥4
𝑥4 ≠ 𝑥5
𝑥4 ≠ 𝑥8
𝑥5 ≠ 𝑥8
𝑥8 ≠ 𝑥1
𝑥8 ≠ 3
𝑥1 ≠ 3
𝑥5 ≠ 𝑥6
𝑥7 ≠ 𝑥5
𝑥6 ≠ 2
𝑥7 ≠ 2
Equivalent Problem: assign 𝑥𝑖 ∈ 𝑘 to minimize the satisfied inequality.
Approximability of Graph Multiway Cut
 Upper bound
1
 1.5 − -approximation
𝑘
by [Calinescu-Karloff-
Rabani,1998]
 1.3438 − 𝑜𝑘 (1)-approximation by
[Karger-Klein-Stein-Thorup-Young, 1999]
 Lower bound: assuming Unique Games Conjecture,
 NP-hard to get better than
12
-approximation.
11
 an earth mover Linear Programming is optimal
polynomial time approximation (the ratio is unknown).
[Manokaran-Naor-Raghavendra-Schwartz,2008]
Variant:
Node Weighted Multiway Cut
Goal: remove minimum number (weights) of
nodes to disconnect the terminals.
Variant:
Hypergraph Multiway Cut (HMC)
 Given a hypergraph 𝐺(𝑉, 𝐸) and 𝑘-terminals
𝑡1 , 𝑡2 , … , 𝑡𝑘 ∈ 𝑉. Remove the minimum number of
edges to disconnect 𝑡1 , 𝑡2 , … , 𝑡𝑘 .
 Approximation equivalent to Node Weighted Multiway
Cut [Zhao-Nagamochi-Ibaraki 2005].
 Min-CSP with NAE (Not All Equal) constraint on the
edges.
Generalization:
Submodular Multiway Partition
 Given a ground set 𝑉 and some submodular function
𝑓: 2𝑉 → 𝑅 + and 𝑘 terminals 𝑡1 , 𝑡2 , … , 𝑡𝑘 ∈ 𝑉. Find set
𝑆1 , 𝑆2 , … , 𝑆𝑘
 𝑆1 ∪ 𝑆2 ∪ ⋯ 𝑆𝑘 = V
 𝑆𝑖 ∩ 𝑆𝑗 = ∅
A function is submodular if
𝑓 𝐴∪𝐵 ≤𝑓 𝐴 +𝑓 𝐵 .
 𝑡𝑖 ∈ 𝑆𝑖
Goal: minimize
𝑘
𝑖=1 𝑓
𝑆𝑖 .
Hypergraph Multiway Cut is a special case[ZhaoNagamochi-Ibaraki 2005].
Another interesting SMP:
Hypergraph Multiway Partition
 Given a hypergraph graph with 𝑘-terminals.
 partition the graph into 𝑘 parts.
 The cost on each edge is the number of different parts it
falls in.
Relationship
Submodular Multiway Partition
Hypergraph Multiway cut
= Node Weighted Multiway Cut
Graph Multiway
Cut Cut
Graph
Multiway
Hypergraph Multiway Partition.
Our Results
Our Results (1)
 There is a 2
4/3-approximation for 3way submodular partion.
2
−
𝑘
-approximation for the general
submodular multiway partition.
 Previous Work:
 2-approximation by [Chekuri-Ene, 2011]
 (2 −
2
)-approximation
𝑘
for node weighted/hypergraph
multiway cut [Garg-Vazirani-Yannakakais,1994]
Based on the half
integrality of an LP.
Overview of the algorithm
 Lovasz Relaxation of Submodular Function:
 Variables 𝑥𝑣 ∈ Δ𝑘 (𝑘 dimensional probability simplex)
for any 𝑣 ∈ 𝑉.
𝑘
𝑖=1 𝑓
𝑆𝑖 for
the following construction of 𝑆𝑖 : choosing random 𝜃 ∈
(0,1], assign 𝑣 to 𝑆𝑖 if 𝑥𝑣,𝑖 > 𝜃.
 The rounding is not necessarily feasible as 𝑣 can be
assigned to multiple 𝑆𝑖 .
 We can efficiently minimize
 Lovasz Relaxation is the expected value of
min E𝜃 [ 𝑓(𝑆𝑖 )]
𝑥𝑉
The rounding algorithm
1.
Choose a random 𝜃 ∈
1
,1
2
.
2. For every 𝑣, 𝑖, set 𝑥𝑣,𝑖 = 1 for 𝑥𝑣,𝑖 > 𝜃 (i.e, assign 𝑣 to
the 𝑖-th terminal.
3. Randomly set all the undecided terminals to a
partition.
To improve from 2 to (2 −
analyzing step 3.
2
),
𝑘
the main technicality is
Our result (2)
matching UG-hardness
 It is Unique Games-hard to get better than (2
2
− )𝑘
approximation for hypergraph multiway cut.
 Previous work: 2 − 𝜖 UG-hardness (from vertex cover).
 We prove that assuming UGC,

The integrality gap of a basic LP (generalizing the earth mover
LP) is the approximation threshold.
2
𝑘

The integrality gap is 2 −

The LP is also optimal for any CSPs contains " ≠ “ constraint.
.
The LP for Hypergraph Multiway Cut
For hypergraph 𝐺(𝑉, 𝐸) with terminals 𝑡1 , 𝑡2 , … , 𝑡𝑘 ∈ 𝑉:
 Variables: 𝑥𝑣,𝑖 for every 𝑣 ∈ 𝑉 and 𝑖 ∈ [𝑘] , 𝑥𝑒,𝛼 for
every 𝑒 ∈ 𝐸 and 𝛼 ∈ 𝑘 |𝑒| .
 Goal: min 𝑥𝑒,𝛼 ⋅ I ¬(𝛼1 = 𝛼2 … = 𝛼 𝑒 )
 Constraint:
 For every 𝑒,
𝛼 𝑥𝑒,𝛼
=1
 For every 𝑒 = (𝑣1 , 𝑣2 , … , 𝑣 𝑒 ) and 𝑣𝑚 ∈ 𝑒
𝑥𝑒,𝛼 = 𝑥𝑣𝑚 ,𝑗
𝛼𝑚 =𝑗
 For the 𝑖-th 𝑡𝑖 terminal 𝑥(𝑡𝑖 ,𝑖 ) = 1
The LP for general Min-CSP
For hypergraph 𝐺(𝑉, 𝐸) and cost function
𝑐𝑜𝑠𝑡𝑒 : 𝑘 𝑒 → [0,1] on each 𝑒 ∈ 𝐸.
 Variables: 𝑥𝑣,𝑖 for every 𝑣 ∈ 𝑉 and 𝑖 ∈ [𝑘] , 𝑥𝑒,𝛼 for
every 𝑒 ∈ 𝐸 and 𝛼 ∈ 𝑘 |𝑒| .
 Goal: min 𝑥𝑒,𝛼 ⋅ cost e 𝛼
 Constraint:
 For every 𝑒,
𝛼 𝑥𝑒,𝛼
=1
 For every 𝑒 = (𝑣1 , 𝑣2 , … , 𝑣 𝑒 ) and 𝑣𝑚 ∈ 𝑒
𝑥𝑒,𝛼 = 𝑥𝑣𝑚 ,𝑗
𝛼𝑚 =𝑗
Optimal LP if the
constraint contains " ≠ “.
Our Results (3):
matching oracle hardness
 For the submodular multiway partition problem, it
requires exponential number of queries on 𝑓 to get
better than 2 −
2
𝑘
approximation.
 We prove this by constructing symmetric gap of
hypergraph multiway cut.
Q: is it a coincident that the oracle hardness
is the same as the Unique Games hardness?
Symmetric gap for
Hypergraph Multiway Cut
 The graph is symmetric for any permutation 𝜋: 𝑉 → 𝑉
from a permutation group Π.
 Vertices 𝑣1 = 𝑣2 are equivalent if there exists some
permutation 𝜋 ∈ Π that 𝜋 𝑣1 = 𝑣2 .
solution
 A (fractional) solution is symmetric if itOptimum
is the same
(by independent rounding).
on all equivalent vertices.
 Symmetric gap: Let I be a instance
𝑜𝑝𝑡(𝐼)
max
𝐼 𝑜𝑝𝑡𝑠𝑦𝑚 (𝐼)
.
Optimum Symmetric solution
(by independent rounding).
Why study symmetric gap?
 Symmetric gap of a CSP implies an oracle hardness result
for the submodular generalization of that CSP.
1
2
 Symmetric gap for Max Cut  -hardness for non-montone
Submodular Function [Feige-Mirrokni-Vondrak, 2007]
 Symmetric gap for NAZ (not all zero)  1 −
1
𝑒
-hardness for
Monotone Submodular Function with cardinality constraint
[previously known Nemhauser-Wolsey, 1978]
 Symmetric gap for Hypergraph Multiway Cut 2 −
Submodular Multiway Partition
2
𝑘
-
Our Results (4)
 Q: is it a coincident that the oracle hardness is the same
as the Unique Games hardness?
 A: No. We prove that for any CSP instance,
symmetric gap = LP integrality gap.
Conclusion
 We have a 2 −
2
𝑘
-approximation for the general
submodular multiway partition problem.
 2−
2
𝑘
2
−
𝑘
oracle hardness and 2
UG-hardness for
the hypergraph multiway cut/Node Multiway Cut
problem
 Equivalence between LP gap and approximation
threshold as well as oracle hardness for general CSPs.
Open problem
 The integrality gap of hypergraph multiway partition?
 Between 1.5 −
1
𝑘
and
12
11
 It is corresponding to an oracle hardness result for
Symmetric submodular multiway partition.