On linear and semidefinite programming relaxations for hypergraph matching

1
On linear and semidefinite programming
relaxations for hypergraph matching

• (work appeared in SODA 10)
• Yuk Hei Chan (Tom)
• joint work with Lap Chi Lau _at_ CUHK

2
Hypergraph Matching
Vertex set V V n
Hyperedge set E

• Hypergraph matching find a largest subset of
  disjoint hyperedges
• Known approximation results T(vn) Halldórsson,
  Kratochvíl, Telle 98
• k-Set Packing each hyperedge has k vertices

Hazan, Safra, Schwartz 03 O ( k / log(k) )
hardness
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Special Cases of k-Set Packing
e1
e1
e2
e2

• Bounded degree independent set

e4
e3
e3
e4

• k-Dimensional Matching

• Latin square completion

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Previous Work Local Search
Improve add t edges in, remove fewer edges
t 2
t 3

• Local optimal t-opt solution
• Greedy solution 1-opt solution
• Greedy solution is k-approximate
• Running time and performance guarantee depends on
  t

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Previous Work Local Search
Unweighted Ratio
Hurkens, Schrijver 89

Weighted
Arkin, Hassin 97
Chandra, Halldórsson 99
Berman 00
Berman, Krysta 03
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Previous Work Linear Programming Relaxation
Füredi 81 integrality gap k - 1 1/k
(unweighted)
Füredi, Kahn, Seymour 93 integrality gap k -
1 1/k (weighted)

• No projective plane as a sub-hypergraph
  integrality gap k - 1
• Non-algorithmic, do not directly imply
  approximation algorithm

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Previous Work Integrality Gap Examples

• Projective plane (of order k 1)
• k2 - k 1 hyperedges
• Degree k on each vertex
• Pairwise intersecting
• Exists when k - 1 is a prime power
• LP solution 1/k on every edge gives k - 1 1/k
• Integral solution 1

k 3 Fano plane
"order 3 projective plane"...
Integrality gap k - 1 1/k
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Overview of New Results

• Tight algorithmic analysis of the standard LP
  relaxation
• Strengthening of LP by local constraints
• Fano LP Sherali-Adams relaxation
• Improvement but not much
• Strengthening of LP by global constraints
• Clique LP SDP
• Improve by a constant factor over local
  constraints
• New connection between local search and LP/SDP

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Standard LP Relaxation
Tight algorithmic analysis of the standard LP
relaxation

• Algorithmic proof of gap k - 1 for
  k-Dimensional Matching

k - 1 1/k for k-Set Packing
Theorem 1 A 2-approximation algorithm for
weighted 3-D Matching

• Improve the local search algorithms by e
• New technique iterative rounding local ratio

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Better LP?
Can we write a better LP?

• For unweighted 3-Set Packing,

add Fano plane constraint
1

• Main proof idea in this Fano LP, any basic
  solution has no Fano plane!
• Then apply Füredis result directly

Theorem 2 Fano LP integrality gap 2
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Better LP?
Can we improve further by adding more local
constraints?

• Sherali-Adams will add all local constraints on
  edges after rounds

Simplify by linearizing and projecting
where are disjoint edge subsets with

• Capture all local constraints on hyperedges
• No integrality gap for any set of
  hyperedges
• e.g. 7 rounds to get Fano plane constraint

1
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Bad example for Sherali-Adams hierarchy

• A modified projective plane
• Still an intersecting family optimal
  1
• Fractional solution k 2

Theorem 3 SA gap is at least k - 2 after O(n /
k3) rounds
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Global Constraints
Clique constraint for a set of intersecting
edges, allow sum of values 1
Theorem 4 Clique LP integrality gap (k 1)
/ 2
Some new connections between local search and
LP/SDP relaxations
(k 1) / 2
OPT
Local OPT
Clique LP
Extend local search analysis
Non-constructive no rounding algorithm
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Clique LP
Clique LP has exponentially many constraints and
no separation oracle is known
(k 1) / 2
OPT
Local OPT
Clique LP
Theorem 5 Clique LP has a compact representation
when k is a constant

• Use a result in extremal combinatorics

There is polynomial size LP with smaller
integrality gap than SA relaxations
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SDP
Indirect way of bounding SDP gap
(k 1) / 2
OPT
Local OPT
Clique LP
SDP
Lovász theta function is an SDP formulation for
the independent set problem.
Grötschel, Lovász, SchrijverSDP captures the
clique constraints
A way to improve k-Set Packing?
Theorem 6 Lovász theta function has integrality
gap (k 1) / 2
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Details explained...

1. 2-approximation for 3-D matching
2. Integrality gap (k 1) / 2 for clique LP

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Approximation Algorithm for k-D Matching
Theorem 1 A 2-approximation algorithm for
weighted 3-D Matching

1. Compute a basic solution
2. Find a good ordering iteratively with small
   neighborhood
3. Use local ratio to compute an approximate solution

Same algorithm for k-Set Packing gives k - 1 1/k
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1. Basic Solution
Only degree constraints can be tight.
Delete edges with xe 0.
Basic solution variables tight
constraints in a basic solution
Lemma in a basic solution, there is a vertex
with degree at most 2
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Basic Solution
Lemma in a basic solution, there is a vertex
with degree at most 2
Let E' be the set of non-zero edges, i.e. edges
s.t. xe gt 0

• Suppose not, then
• Since each edge consists of 3 vertices, so
• In a basic solution, , so

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Basic Solution

• Every edge in E' consist of vertices in T only

• Since the graph is 3-partite,

• Constraints are not linearly independent, i.e.
  solution is not basic

Lemma in a basic solution, there is a vertex
with degree at most 2
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2. Small (fractional) Neighborhood
Lemma in a basic solution, there is a vertex
with degree at most 2
xb
xa
1 - xb
1 - xb
2
xb
xb
) (
) (
) (
(
)
This gives 2 approx. for unweighted case.
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Weighted Case
The same algorithm does not work in the weighted
case.
we 80xe 0.2
we 2xe 0.8

• Pick the green edgeGain 2, lose (up to) 91

we 10xe 0.2
we 1xe 0.2
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Weighted Case
Strategy Write fractional solution as a linear
combination of matchings.
xe 0.3
0.3
0.3
xe 0.7
0.3
0.4
xe 0.4
If sum of coefficients is small,
by averaging, there is a matching of large weight.
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Finding Good Ordering
Lemma in a basic solution, there is a vertex
with degree at most 2
Idea Use Lemma to find a good ordering, then
apply greedy coloring
xb
xa

• Ordering Procedure
• Repeat
• Find an edge e with x(Ne) 2,add it to the
  ordering.
• Remove e from the graph
• Until the graph is empty

1 - xb
? xe 1 - xb
? xe 2 (xa 1 - xb)
Lemma there is an ordering of edge e1, e2, ,em
s.t. x(Nei n ei, ei1, em) 2
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Apply greedy coloring
Lemma there is an ordering of edge e1, e2, ,em
s.t. x(Nei n ei, ei1, em) 2
e4
Use greedy coloring, color the edges in reverse
order
e3
e2
Decompose the fractional solution x as a linear
combination of matchings Mi
, where
e1
e5

• By averaging argument, there is a matching with
  weight at least half of the optimum
• Implies integrality gap at most 2

Not an efficient algorithm yet
Need local ratio
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3. Fractional Local Ratio
Split the weight vector into 2.
(Number denotes weight)
Step 1 pick an edge with ? xe 2 in the closed
neighborhood (by Lemma)
Step 2 make a copy of the graph in the
neighborhood of the blue edge
Step 3 distribute the weight
Step 4 remove non-positive edges and solve the
residue instance
Step 5 join the solution
10
0
10
10
20
10
10
0
10
7
10
-3
7
25
13
10
3
13
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? xe 2 pick any edge here 2-approx.
Obtain a 2-approximate solution by induction
This is a 2-approximate solution
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Clique LP has integrality gap (k 1) / 2
Strategy Fix a 2-local optimal matching M, bound
the ratio of any fractional solution
Extend local search analysis
M
Not rounding algorithm
F set of non-zero edges
F2
F1
Want to show x(F) (k 1) M / 2
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Clique LP has integrality gap (k 1) / 2
Let
Claim F1(e) is an intersecting family for
x(F1(e)) 1
M
Otherwise, exists disjoint f1, f2 in F1(e)
e
Replace e by f1, f2
x(F1) M
x(F1(e)) 1
f1
f2
F1(e)
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Clique LP has integrality gap (k 1) / 2

• There are k M vertices in M
• By degree constraint, x(F2) k M
• Each edge intersect 2 edges in M

M
x(F2) k M / 2
F2
Theorem 4 Clique LP integrality gap (k 1)
/ 2
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Open Problems
More Iterative Rounding Local Ratio rounding
algorithms?
What is the integrality gap of the SDP?

• o(k) ?

• Lower bound on the integrality gap?

What is the approximability of k-Set Packing?

• Between O(k / log k) to (k 1) / 2

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End