Approximability of Multiway Partition

Yi WuIBM Almaden Research

Joint work with Alina Ene and Jan Vondrak

Definition of Problems

Graph Multiway CutInput: a graph with terminals.

Graph Multiway Cut

Goal: remove minimum number of edges to disconnect the terminals.

Input: a graph with terminals.

Graph Multiway CutInput: a graph with terminals.

Equivalent Goal: divide the graph into parts to minimize number of cross edges.

Constraint Satisfaction Problem (CSP) with “” constraint

1

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𝑥3

𝑥4

𝑥5

𝑥6

𝑥7

2

3𝑥1

𝑥8

≠

≠ ≠

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Equivalent Problem: assign to minimize the satisfied inequality.

Approximability of Graph Multiway CutUpper bound

-approximation by [Calinescu-Karloff-Rabani,1998]-approximation by

[Karger-Klein-Stein-Thorup-Young, 1999]Lower bound: assuming Unique Games

Conjecture,NP-hard to get better than -approximation.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 and -terminals . Remove

the minimum number of edges to disconnect

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 PartitionGiven a ground set and some submodular

function and terminals . Find set = V

Goal: minimize

Hypergraph Multiway Cut is a special case[Zhao-Nagamochi-Ibaraki 2005].

A function is submodular if

Another interesting SMP: Hypergraph Multiway Partition

Given a hypergraph graph with -terminals.partition the graph into parts.The cost on each edge is the number of

different parts it falls in.

RelationshipSubmodular Multiway PartitionHypergraph Multiway cut

= Node Weighted Multiway Cut

Hypergraph Multiway Partition.

Graph Multiway CutGraph Multiway Cut

Our Results

Our Results (1) There is a -approximation for the general

submodular multiway partition.

Previous Work:-approximation by  [Chekuri-Ene, 2011]-approximation for node weighted/hypergraph

multiway cut [Garg-Vazirani-Yannakakais,1994]

4/3-approximation for 3-way submodular

partion.

Based on the half integrality of an LP.

Overview of the algorithmLovasz Relaxation of Submodular Function:

Variables ( dimensional probability simplex) for any

Lovasz Relaxation is the expected value of for the following construction of : choosing random , assign to if

The rounding is not necessarily feasible as can be assigned to multiple

We can efficiently minimize]

The rounding algorithm1. Choose a random 2. For every , set for (i.e, assign to the -th

terminal.3. Randomly set all the undecided terminals to

a partition.

To improve from to , the main technicality is analyzing step 3.

Our result (2)matching UG-hardness

It is Unique Games-hard to get better than -approximation for hypergraph multiway cut.Previous work: 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.

The integrality gap is . The LP is also optimal for any CSPs contains

constraint.

The LP for Hypergraph Multiway CutFor hypergraph with terminals :Variables: for every and , for every and .Goal: Constraint:

For every For every and

For the -th terminal

The LP for general Min-CSPFor hypergraph and cost function on each Variables: for every and , for every and .Goal: Constraint:

For every For every and

Optimal LP if the constraint contains .

Our Results (3): matching oracle hardnessFor the submodular multiway partition

problem, it requires exponential number of queries on to get better than 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 from a permutation group .

Vertices are equivalent if there exists some permutation that

A (fractional) solution is symmetric if it is the same on all equivalent vertices.

Symmetric gap: Let I be a instance .

Optimum Symmetric solution (by independent

rounding).

Optimum 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.Symmetric gap for Max Cut -hardness for non-

montone Submodular Function [Feige-Mirrokni-Vondrak, 2007]

Symmetric gap for NAZ (not all zero) -hardness for Monotone Submodular Function with cardinality constraint [previously known Nemhauser-Wolsey, 1978]

Symmetric gap for Hypergraph Multiway Cut -Submodular Multiway Partition

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.

ConclusionWe have a -approximation for the general

submodular multiway partition problem.

oracle hardness and 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 problemThe integrality gap of hypergraph multiway

partition?Between and It is corresponding to an oracle hardness result

for Symmetric submodular multiway partition.