On linear and semidefinite programming
relaxations for hypergraph matching
(work appeared in SODA 10’)
Yuk Hei Chan (Tom)
joint work with Lap Chi Lau @ CUHK
Hypergraph Matching
Vertex set V: |V| = n
Hyperedge set E
•
Hypergraph matching: find a largest subset of disjoint hyperedges
•
Known approximation results: Θ(√n) [Halldórsson, Kratochvíl, Telle 98’]
•
k-Set Packing: each hyperedge has k vertices
[Hazan, Safra, Schwartz 03’]: Ω ( k / log(k) ) hardness
Special Cases of k-Set Packing
e1
e1
e2
e2
e4
•
Bounded degree independent set
e3
e3
e4
•
k-Dimensional Matching
column j
1
row i
4
2
k
3
3
row i, column j
2
4
row i, color k
column j, color k
•
Latin square completion
Previous Work: Local Search
Improve: add ≤ t edges in, remove fewer edges
t=2
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Local optimal — t-opt solution
•
•
•
t=3
Greedy solution = 1-opt solution
Greedy solution is k-approximate
Running time and performance guarantee depends on t
Previous Work: Local Search
Unweighted
Hurkens, Schrijver 89’
Weighted
Arkin, Hassin 97’
Chandra, Halldórsson 99’
Berman 00’
Berman, Krysta 03’
Ratio
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
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Non-algorithmic, do not directly imply approximation algorithm
Previous Work: Integrality Gap Examples
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Projective plane (of order k – 1)
1. k2 − k + 1 hyperedges
2. Degree k on each vertex
3. Pairwise intersecting
4. Exists when k − 1 is a prime power
k = 3: Fano plane
"order 3 projective plane"...
•
LP solution: 1/k on every edge gives k − 1 + 1/k
•
Integral solution: 1
Integrality gap = k − 1 + 1/k
Overview of New Results
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Tight algorithmic analysis of the standard LP relaxation
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Strengthening of LP by local constraints
•
•
Fano LP & Sherali-Adams relaxation
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Improvement but not much
Strengthening of LP by global constraints
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“Clique” LP & SDP
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Improve by a constant factor over local constraints
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New connection between local search and LP/SDP
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 ε
•
New technique: iterative rounding + local ratio
Better LP?
Can we write a better LP?
•
For unweighted 3-Set Packing, add Fano plane constraint:
≤
1
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Main proof idea: in this Fano LP, any basic solution has no Fano plane!
•
Then apply Füredi’s result directly
Theorem 2: Fano LP integrality gap = 2
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
– No integrality gap for any set of
hyperedges
hyperedges
– e.g. 7 rounds to get Fano plane constraint
≤
1
Bad example for Sherali-Adams hierarchy
•
A modified projective plane
•
Still an intersecting family
optimal = 1
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Fractional solution ≥ k – 2
Theorem 3: SA gap is at least k − 2 after Ω(n / k3) rounds
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
Local OPT
Extend local search analysis
OPT
Clique LP
Non-constructive; no rounding algorithm
Clique LP
Clique LP has exponentially many constraints
and no separation oracle is known
≤ (k + 1) / 2
Local OPT
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
SDP
Indirect way of bounding SDP gap
≤ (k + 1) / 2
Local OPT
OPT
SDP
Clique LP
Lovász theta function is an SDP formulation for the independent set problem.
[Grötschel, Lovász, Schrijver]:
SDP captures the clique constraints
A way to improve k-Set Packing?
Theorem 6: Lovász theta function has integrality gap ≤ (k + 1) / 2
Details explained...
1. 2-approximation for 3-D matching
2. Integrality gap ≤ (k + 1) / 2 for clique LP
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
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
Basic Solution
Lemma: in a basic solution, there is a vertex with degree at most 2
Let T be the set of tight vertices, i.e. vertices s.t.
Let E' be the set of non-zero edges, i.e. edges s.t. xe > 0
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Suppose not, then
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Since each edge consists of 3 vertices, so
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In a basic solution,
, so
Basic Solution
Every edge in E' consist of vertices in T only
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Since the graph is 3-partite,
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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
2. Small (fractional) Neighborhood
Lemma: in a basic solution, there is a vertex with degree at most 2
xb
xa
( xb ) + ( ≤ xb ) + ( ≤ 1 − xb ) + ( ≤ 1 − xb )
This gives 2 approx. for unweighted case.
≤2
Weighted Case
The same algorithm does not work in the weighted case.
we = 80
xe = 0.2
we = 2
xe = 0.8
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Pick the green edge:
Gain 2, lose (up to) 91
we = 10
xe = 0.2
we = 1
xe = 0.2
Weighted Case
Strategy: Write fractional solution as a linear combination of matchings.
xe = 0.3
xe = 0.7
xe = 0.4
× 0.3
× 0.3
× 0.4
× 0.3
If sum of coefficients is small, by averaging, there is a matching of large weight.
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
xa
xb
Ordering Procedure
≤ 1 − xb
Repeat
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Find an edge e with x(N[e]) ≤ 2,
add it to the ordering.
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Remove e from the graph
∑ xe ≤ 1 − xb
Until the graph is empty
∑ xe ≤ 2 (xa ≤ 1 − xb)
Lemma: there is an ordering of edge e1, e2, …,em s.t. x(N[ei] ∩ {ei, ei+1, …em}) ≤ 2
Apply greedy coloring
Lemma: there is an ordering of edge e1, e2, …,em s.t. x(N[ei] ∩ {ei, ei+1, …em}) ≤ 2
e4
e3
Use greedy coloring, color the edges in reverse order
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
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Implies integrality gap at most 2
Not an efficient algorithm yet
Need local ratio
3. Fractional Local Ratio
Split the weight vector into 2.
(Number denotes weight)
Step 5:
1: join
2:
3:
4:
pick the
make
distribute
remove
ana edge
solution
copy
non-positive
the of
with
weight
the∑graph
xedges
inand
the
thesolve
neighborhood
closed
the
neighborhood
residue
of the
instance
blue
(by edge
Lemma)
e ≤ 2 in
10
0
10
20
10
10
10
0
7
-3
10
25
13
3
10
20
∑ xe ≤ 2: pick any edge here = 2-approx.
Obtain a 2-approximate solution by induction
This is a 2-approximate solution
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
F1
Want to show: x(F) ≤ (k + 1) |M| / 2
F2
Clique LP has integrality gap ≤ (k + 1) / 2
Let
x(F1(e)) ≤ 1
Claim: F1(e) is an intersecting family for
M
Otherwise, exists disjoint f1, f2 in F1(e)
e
Replace e by f1, f2
x(F1(e)) ≤ 1
x(F1) ≤ |M|
f1
f2
F1(e)
Clique LP has integrality gap ≤ (k + 1) / 2
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There are k |M| vertices in M
M
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By degree constraint, x(F2) ≤ k |M|
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Each edge intersect ≥ 2 edges in M
x(F2) ≤ k |M| / 2
F2
Theorem 4: “Clique” LP integrality gap ≤ (k + 1) / 2
Open Problems
More “Iterative Rounding + Local Ratio” rounding algorithms?
What is the integrality gap of the SDP?
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o(k) ?
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Lower bound on the integrality gap?
What is the approximability of k-Set Packing?
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Between Ω(k / log k) to (k + 1) / 2
End