1. On linear and semidefinite programming relaxations for hypergraph matching
(work appeared in SODA 10’)
Yuk Hei Chan (Tom)
joint work with Lap Chi CUHK

2. 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
No constant factor approximation for set packing. Not approximable within m^(1/2-\epsilon), where m is # sets, or m^(1-\epsilon) unless NP=ZPP.
Hypergraph matching: also known as set packing
k-set packing
2-set packing = matching
Focus on k-set packing
[Hazan, Safra, Schwartz 03’]: Ω ( k / log(k) ) hardness

3. Special Cases of k-Set Packing
Bounded degree independent set e4 e3 e3 e4
k-Dimensional Matching 1 4 2 k 3
row i, column j
row i, color k
column j, color k
row i
column j
Bounded degree independent set: degree upper bound k -> k-Set Packing
k-dimensional matching: the set of vertices is partitioned into k parts, each edge contain one vertex from each part.
Latin square extension can be seen as a 3-D Matching by the reduction on the left. (approximation ratio is preserved)
Latin square completion

4. Previous Work: Local Search
Improve: add ≤ t edges in, remove fewer edges
t = 2
Algorithmic results based on local search.
Idea: try all possible subsets of at most t edges, so time depends on t.
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

5. Previous Work: Local Search
Unweighted
Ratio
Hurkens, Schrijver 89’
Weighted
Arkin, Hassin 97’
Chandra, Halldórsson 99’
Berman 00’
Berman, Krysta 03’

6. 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

7. 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

8. 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
Local constraints only give slight improvements.
Global: Improve by a constant factor, better than local
Analysis: a new connection between local search and LP/SDP that are seemingly different.

9. Standard LP Relaxation
Tight algorithmic analysis of the standard LP relaxation
Algorithmic proof of gap k − 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

10. add Fano plane constraint:
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 Füredi’s result directly
Theorem 2: Fano LP integrality gap = 2

11. Simplify by linearizing and projecting
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
Linearizing and projecting: introduce new variables
Capture all local constraints: essentially “the best” you can do by adding local constraints
Capture all local constraints on hyperedges
No integrality gap for any set of hyperedges
e.g. 7 rounds to get Fano plane constraint
≤ 1

12. Bad example for Sherali-Adams hierarchy
A modified projective plane
Still an intersecting family optimal = 1
Fractional solution ≥ k – 2
Bad example is an intersecting family (i.e. clique), SA cannot capture this.
Add clique constraints to tackle the highly intersecting family.
Theorem 3: SA gap is at least k − 2 after Ω(n / k3) rounds

13. 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
Explain why is this global: the constraint may include a lot of edges.
Unlike usual LP analysis, this is not constructive (no rounding algorithm)
We extended the local search analysis, so actually the ratio is between a local optimal solution and the fractional solution.
Local OPT
OPT
Clique LP
Extend local search analysis
Non-constructive; no rounding algorithm

14. Use a result in extremal combinatorics
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

15. 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

16. Details explained... 2-approximation for 3-D matching
Integrality gap ≤ (k + 1) / 2 for clique LP

17. Approximation Algorithm for k-D Matching
Theorem 1: A 2-approximation algorithm for weighted 3-D Matching
Compute a basic solution
Find a good ordering iteratively with small neighborhood
Use local ratio to compute an approximate solution
Same algorithm for k-Set Packing gives k − 1 + 1/k

18. 1. Basic Solution Only degree constraints can be tight.
Delete edges with xe = 0.
Basic solution: few non-zero variables
Basic solution: # variables ≤ # tight constraints in a basic solution
Lemma: in a basic solution, there is a vertex with degree at most 2

19. 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
Suppose not, then
Since each edge consists of 3 vertices, so
In a basic solution, , so
|E'| = |T| means equality holds everywhere on the second to last line. Hence deg(v) = 3 for every v \in T.

20. Lemma: in a basic solution, there is a vertex with degree at most 2
Every edge in E' consist of vertices in T only
Since the graph is 3-partite,
(The graph is now 3-regular and 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

21. 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 )
≤ 2
This gives 2 approx. for unweighted case.

22. The same algorithm does not work in the weighted case.
we = 80 xe = 0.2
we = 2 xe = 0.8
Pick the green edge: Gain 2, lose (up to) 91
we = 10 xe = 0.2
we = 1 xe = 0.2

23. by averaging, there is a matching of large weight.
Weighted Case
Strategy: Write fractional solution as a linear combination of matchings.
xe = 0.3
× 0.3
× 0.3
xe = 0.7
× 0.4
× 0.3
xe = 0.4
If sum of coefficients is small,
by averaging, there is a matching of large weight.

24. 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(N[e]) ≤ 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(N[ei] ∩ {ei, ei+1, …em}) ≤ 2

25. Not an efficient algorithm yet
Apply greedy coloring
Lemma: there is an ordering of edge e1, e2, …,em s.t. x(N[ei] ∩ {ei, ei+1, …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

26. 3. Fractional Local Ratio
Split the weight vector into 2.
(Number denotes weight)
Step 3: distribute the weight
Step 4: remove non-positive edges and solve the residue instance
Step 2: make a copy of the graph in the neighborhood of the blue edge
Step 5: join the solution
Step 1: pick an edge with ∑ xe ≤ 2 in the closed neighborhood (by Lemma) 10 10 10 20 10 10 10 7 -3 7 10
In another scenario where no edge in the neighborhood is selected, we simply add back the original edge 25 3 13 13 10 20
∑ xe ≤ 2: pick any edge here = 2-approx.
Obtain a 2-approximate solution by induction
This is a 2-approximate solution

27. 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
After fixing M, every other edges have to intersect with at least one edge in M.
F: set of non-zero edges F2 F1
Want to show: x(F) ≤ (k + 1) |M| / 2

28. Clique LP has integrality gap ≤ (k + 1) / 2
Let
Claim: F1(e) is an intersecting family for
x(F1(e)) ≤ 1
Otherwise, exists disjoint f1, f2 in F1(e) M e
Replace e by f1, f2
x(F1(e)) ≤ 1
x(F1) ≤ |M| f1 f2
F1(e)

29. 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
Here x(F) <= (k+2) |M| / 2, but we can show (with a more detail analysis) that x(F) <= (k+1) |M| / 2
x(F2) ≤ k |M| / 2 F2
Theorem 4: “Clique” LP integrality gap ≤ (k + 1) / 2

30. What is the approximability of k-Set Packing?
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 Ω(k / log k) to (k + 1) / 2

31. End