1. Iterative Methods in Combinatorial Optimization
Mohit Singh
McGill University
joint works with
L.C. Lau, F. Grandoni, T. Kiraly, S. Naor, R. Ravi and M. Salavatipour

2. Combinatorial Optimization
“Easy” Problems : polynomial time solvable (P)
Spanning Trees
Matchings
Matroid Intersection
“Hard” Problems : NP-hard
Survivable Network Design
Facility Location
Scheduling Problems
Natural dichotomy: easy and hard
Easy problems: LP has been the most powerful tool.
Hard problems: NP-hard. Unlikely they are solvable in polynomial time. One aims for an approximation algorithm for the problem: define rho approximation.
LP has been one of the most generic tools for these problems as well.
The general technique to is to formulate a linear programming relaxation for this problem. Do not expect this LP to be integral.
General techniques have been developed
In this talk, I will talk about the iterative rounding technique and its extensions that we have developed in our work. These extensions allow us to obtain strong results for a large class of problems.
We show that iterative methods are well-suited for poly time solvable problems. This will act as basic groundwork for analyzing hard problems as well.

3. Formulate Linear Program
Linear Programming
Formulate Linear Program P
NP-hard
Show Integrality
Round the fractional solution
to obtain approximation algorithm
Natural dichotomy: easy and hard
Easy problems: LP has been the most powerful tool.
Hard problems: NP-hard. Unlikely they are solvable in polynomial time. One aims for an approximation algorithm for the problem: define rho approximation.
LP has been one of the most generic tools for these problems as well.
The general technique to is to formulate a linear programming relaxation for this problem. Do not expect this LP to be integral.
General techniques have been developed
In this talk, I will talk about the iterative rounding technique and its extensions that we have developed in our work. These extensions allow us to obtain strong results for a large class of problems.
We show that iterative methods are well-suited for poly time solvable problems. This will act as basic groundwork for analyzing hard problems as well.
Randomized Rounding
Primal-Dual Schema
Iterative Rounding …

4. Iterative Rounding (Jain’98): ½-element ) 2-approximation
Typical Rounding:
LP Solver
Rounding
Procedure
Optimal
Fractional
Solution
Integer
Solution
Problem
Instance
Iterative Rounding (Jain’98):
½-element ) 2-approximation
LP Solver
Good
Part
Part
Integer
Optimal
Fractional
Solution
Problem
Instance
Iterative Rounding limitations (SNDP undirected and directed graphs)
Lots of techniques (uncrossing) were similar to techniques used in combinatorial optimization for analyzing LP formulations of exact problems.
Residual
Problem
Too
Fractional

5. Minimum Bounded Degree Spanning Tree
Design a spanning tree
Low cost
Low degree at all nodes
min c(T)
s.t. T is a spanning tree
deg (v) · B
Checking feasibility is NP-hard
problem is simple to state but captures the structure of more complicated problems. Our work shows that this will see that this is indeed true to some extent.
Feasibility problem is NP-hard.
8 v
11/12/2018
11/12/2018 5

6. Base Problem and Constrained Problem
MBDST problem
Spanning Tree Problem
min c(T) s.t. T is a spanning tree deg(v) · B 8 v2 V
11/12/2018

7. Iterative Rounding and Relaxation
Iterative Rounding works for “easy” problems.
Iterative proofs of integrality of spanning tree, arborescene, matroid, matching, matroid intersection …
Iterative Relaxation
Relax complicating constraints and bound violations.
Extend these integrality results to approximation algorithms.
mainly concentrate on degree bounded network design problems most well-studied being the MBDST problem.
Tighter results for 2-criteria problems.
Integral formulations.
11/12/2018

8. Result
Theorem [S., Lau ’07]: There exists a polynomial time algorithm for MBDST problem which returns a tree of
c(T) · c(OPT)
maximum degree · B+1.
OPT is the cheapest tree with maximum degree B.
Resolved a conjecture of Goemans ’91 and improved on Goemans’06.
Generalizes a result of Furer and Raghavachari ’92.

9. Outline Integrality of Spanning Tree B+1 result Extensions
Conclusion and Open Problems
Illustrate the iterative method on MBDST problem.
11/12/2018 9 9

10. Spanning Tree Polyhedron
Integer program
min e2 E ce xe
s.t. e2 E(V) xe= |V|-1
xe 2 {0,1}
Any tree has n-1 edges
Seperation
11/12/2018

11. Spanning Tree Polyhedron
Integer program
min e2 E ce xe
s.t. e2 E(V) xe= |V|-1
e2 E(S) xe· |S|-1 8 S½ V, |S|¸ 2
xe 2 {0,1}
E(S): set of edges with both endpoints in S.
Any tree has n-1 edges
Subtour elimination
constraints
Seperation
11/12/2018

12. Spanning Tree Polyhedron
Linear
Integer program
min e2 E ce xe
s.t. e2 E(V) xe= |V|-1
e2 E(S) xe· |S|-1 8 S½ V, |S|¸ 2
xe 2 {0,1} 0· xe ·1
(Edmonds ‘71)
Any tree has n-1 edges
Subtour elimination
constraints
Equivalent compact formulations [Wong ’80]
Polynomial time separation [Cunningham ’84]
Seperation
11/12/2018

13. (Iterative) Rounding Spanning Trees
Solve LP to obtain optimum extreme point x
Remove all edges s.t. xe = 0
Return E (the non-zero edges)
Claim: If |E|=|V|-1 then E is a MST.
Proof: E feasible ) x(E)=|V|-1
) xe=1 8 e2 E
) E is a tree.
Edges included in F will form a spanning tree in the end.
11/12/2018

14. Extreme Points and Uncrossing
Fact: x extreme ) d linearly independent tight constraints.
Claim: x is extreme ) |E| · n-1
Put References,
Put in 1-slide. (Looking Ahead)
No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction.
Put line later.
11/12/2018

15. Extreme Points and Uncrossing
Fact: x extreme ) d linearly independent tight constraints.
Claim: x is extreme ) |E| · n-1
e2 E(S) xe=|S|-1
Standard uncrossing ) linearly independent set of constraints defining x can be chosen to form a laminar family.
Put References,
Put in 1-slide. (Looking Ahead)
No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction.
Put line later.
A[B B A
AÅB
11/12/2018
11/12/2018

16. Extreme Points and Uncrossing
Fact: x extreme ) d linearly independent tight constraints.
Claim: x is extreme ) |E| · n-1
e2 E(S) xe=|S|-1
Standard uncrossing ) linearly independent set of constraints defining x can be chosen to form a laminar family.
Put References,
Put in 1-slide. (Looking Ahead)
No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction.
Put line later.
11/12/2018

17. Extreme Points and Uncrossing
Claim: x is extreme ) |E| · n-1
Number of variables (dimension) = |E|
Number of constraints = |L|
) |E|=|L|
|L| is laminar ) |L|· n-1
) |E|· n-1
Put References,
Put in 1-slide. (Looking Ahead)
No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction.
Put line later.
Theorem [Edmonds]: Spanning tree polyhedron is integral.
11/12/2018

18. Outline Integrality of Spanning Tree B+1 result Extensions
Conclusion and Open Problems
Illustrate the iterative method on MBDST problem.
11/12/2018 18 18

19. Base Problem and Constrained Problem
MBDST problem
Spanning Tree Problem
min c(T) s.t. T is spanning tree degT(v) · Bv 8 v2W
11/12/2018

20. Bounded Degree Spanning Trees
Extend spanning tree polyhedron
min e2 E ce xe
s.t. e2 E(V) xe= |V|-1
e2 E(S) xe· |S|-1 8 S ½ V
e2 (v)xe· Bv v 2 W
xe¸ 0
Here W½V.
Spanning tree
Degree bounds
Outline :
11/12/2018

21. Obtaining B+1 Algorithm
Solve LP to obtain extreme point x.
Remove all edges e s.t. xe=0.
Return E.
Pursued in LNSS.
11/12/2018

22. Obtaining B+1 Algorithm
Relaxation Step
While W 
Solve LP to obtain extreme point x.
Remove all edges e s.t. xe=0.
If 9 v2 W such that degE(v)· Bv+1, then remove the degree constraint of v.
Return E
Lemma:
Solution returned is a tree
Optimal cost
degE(v)· Bv+1
Pursued in LNSS.

23. Main Lemma
Lemma: There exists v2 W such that degE(v)· Bv+1 where E is the support of x.
Proof: Tight constraints =
1. Laminar Family: L
2 . Degree constraints: W
Will show |E|> |L|+|W| ) contradiction
Variables:
Edges E
Collect 1 token for
Each member of L
Each vertex in W
Extra token
Redistribute
1 token for each edge.
|E| total token.
See [Bansal, Khandekar, Nagarajan’08]
11/12/2018

24. Charging Argument
Simpler argument due to Bansal, Khandekar and Nagarajan ‘08.
Redistribution:
Degree constraint for u and v get (1-xuv)/2 tokens each. u v $1
(1-xuv)/2
(1-xuv)/2
11/12/2018

25. Charging Argument
Simpler argument due to Bansal, Khandekar and Nagarajan ‘08.
Redistribution:
Degree constraint for u and v get (1-xuv)/2 tokens each. u v
xuv
Smallest set containing u and v
gets xuv tokens $1
Collection: w
Tokens received : e2 (w) (1-xe)/2
¸ (degE(w)-Bw)/2
¸ 1 (since degree constraint present)
11/12/2018

26. Charging Argument
Simpler argument due to Bansal, Khandekar and Nagarajan ‘08.
Redistribution:
Degree constraint for u and v get (1-xuv)/2 tokens each. u v
xuv
Smallest set containing u and v
gets xuv tokens $1
Collection: S R2 R1
x(E(S))=|S|-1
- x(E(Ri))=|Ri|-1
Tokens to S= Integer
11/12/2018

27. General Methodology Constrained Problem Base Problem Side Constraints
11/12/2018

28. Minimum Bounded Spanning Tree
Degree Constraints
Thm[S, Lau ‘07]: (1,B+1)-approximation
11/12/2018

29. Degree Bounded Steiner Tree
Iterative Rounding [Jain]: ½-edge ) 2-approximation
Steiner Tree
Degree Constraints
Thm [LNSS ‘07]: Obtain (2,2B+3)-approximation
11/12/2018

30. Degree Bounded Steiner Tree
Iterative Rounding [Jain]: ½-edge ) 2-approximation
Steiner Tree
Degree Constraints
Thm [LS ‘08]: Obtain (2,B+3)-approximation
11/12/2018

31. Degree Bounded Arboresence
Degree Constraints
Thm [LNSS ‘07]: Obtain (2,2B+2)-approximation
Thm [BKN ’08]: Obtain (B+4)-approximation.
11/12/2018

32. Bipartite Matching Bipartite Matching Bipartite Matching
Multi-criteria
Makespan Constraints
Thm[S]: 2-approximation for
scheduling unrelated parallel machines.
Thm[FRS]: (1+²)-approximation for
Multi-criteria bipartite matching.
11/12/2018

33. More General Structures
Matroid
Submodular Flow
Degree Constraints
Degree Constraints
Thm[Kiraly, Lau, S ’08]: Additive approximation for degree constrained matroid basis and degree constrained submodular flow problem.
See BKNP ’10 for more general structures.
11/12/2018

34. Multi-Criteria Spanning Tree
lengthi (T) · Li
Thm [Grandoni, Ravi, S. ’09]: Iterative Relaxation gives (1+²)-approximation for fixed ²
11/12/2018

35. General Methodology Base Problem Side Constraints
“Structured” side constraints in “structured” problems are easy to handle
Base Problem
Side Constraints
11/12/2018

36. Applications* Topolgy Control for Future Airborne
Networks, Krishnamurthi et al 2009.
Deploying Mesh Nodes under non-uniform propogation, Robinson et al, 2009
11/12/2018

37. Conclusion Iterative Rounding and Relaxation Open Problems
Obtain new iterative proofs of classical integrality results
Extend these integrality results to multi-criteria optimization and obtain additive approximations
Open Problems
OPT+1 for Bin Packing? OPT + log2 n. Karp Karmarkar ’82.
Discrepancy of Sets? Beck, Fiala ’81, Bansal ’10.
Primal-Dual Algorithms?
11/12/2018

38. Thank You!