1. Iterative Rounding and Iterative Relaxation
Some of the slides prepared by Lap Chi Lau

2. undirected or directed
Steiner Network
undirected or directed
A graph G = (V,E);
A connectivity requirement r(u,v) for each pair of vertices u,v.
Steiner Network
A subgraph H of G which has r(u,v) disjoint paths between each pair u,v.
edge or vertex

3. Examples of Steiner Network
Spanning tree: r(u,v) = 1 for all pairs of vertices.
Steiner tree: r(u,v) = 1 for all pair of required vertices.
Steiner forest: r(si,ti) = 1 for all source sink pairs.
k-edge-connected subgraph: r(u,v) = k for all pairs of vertices.

4. e.g. minimum Steiner tree, k-edge-connected subgraph
Survivable Network Design
Survivable network design: find a “good” Steiner network
e.g. minimum Steiner tree, k-edge-connected subgraph
Minimum cost Steiner network:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost.
NP-complete…

5. at least r(u,v) edges crossing S
Linear Programming Relaxation
at least r(u,v) edges crossing S S
f(S) := max{ r(u,v) | S separates u and v}. u v
number of constraints is exponential.
can be handled by Ellipsoid algorithm
need a separation oracle 5

6. Approximating Survivable network Design
What does not seem to generalize … ?
primal-dual algorithm for constrained Steiner forest
only logarithmic approx. is known via this approach
one shot rounding as in
vertex cover - rounding up fractions ¸ ½
set cover – randomized rounding
combinatorial algorithms
e.g., Steiner tree (reduction to spanning tree in the
metric completion graph)
New ideas are needed …

7. Basic Solutions (1) A basic solution is the unique solution of
m linearly independent “tight” constraints,
where m is the number of variables in the LP.
A basic solution cannot be represented
as a convex sum of feasible solutions

8. Basic Solutions (2) An edge of 1, pick it. An edge of 0, delete it.
A basic solution is the unique solution of
m linearly independent “tight” constraints,
where m is the number of variables in the LP.
Tight inequalities all come from the connectivity requirements.

9. Petersen Graph: Non-Basic Feasible Solution
use every edge to the extent of 1/3.
15 edges with 0 < x(e) < 1
10 tight constraints (at the vertices)

10. Basic Feasible Solution
black edge – ¼
red edge – ½

11. Tight sets

12. LP Solutions All 1/3 is a feasible solution.
But not a basic feasible solution
This is a basic feasible solution.
Thick edges have value 1/2;
Thin edges have value 1/4.
Theorem [Jain 98] Every basic feasible solution has an edge of value at least 1/2
Corollary. There is a 2-approximation algorithm for survivable network design.

13. Iterative Rounding e Corollary. This is a 2-approximation algorithm
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
Find a basic optimal solution, x, of the LP with function f’.
Add an edge with x(e) ≥ 1/2 into H.
Update f’: for every set S, set
Output H.
By Jain’s Theorem
f’(S)=2
f’(S)=1 e
0.5
The residual problem is feasible.
0.5
Corollary. This is a 2-approximation algorithm
for the minimum cost survivable network problem.
0.5
0.5 13

14. Typical Rounding: Iterative Rounding: Too much Problem Linear
Programming
Solver
Suitable
Rounding
Procedure
Optimal
Fractional
Solution
Integer
Solution
Problem
Instance
Iterative Rounding:
Linear
Programming
Solver
Part
Integer
Good
Part
Optimal
Fractional
Solution
Problem
Instance
Residual
Problem
Too much
Fractional

15. Laminar Basis tree representation any pair of sets in the basis are
either disjoint or contained

16. Weak Supermodularity A weakly supermodular function f satisfies: or
Example: f(S) := max{ r(u,v) | S separates u and v}.
Important: when f is updated by setting x(e)=1 for some edges e, it remains weakly supermodular
Weakly supermodular functions are very useful for obtaining a laminar basis

17. Obtaining a Laminar Basis
Tight constraints:
Uncrossing technique: A basic solution is defined by
a laminar family of tight connectivity constraints.
A[B A B
AÅB

18. Proof of Jain’s Theorem
Theorem [Jain]. Every basic solution has an edge with value at least 1/3
There are |L| constraints.
There are |E| variables.
Assume every edge has value 0 < x(e) < 1/3.
Prove that |E| > |L| by a counting argument.
At the beginning we give 2 tokens to each edge, 1 to each endpoint.
At the end we redistribute the tokens so that each member in the
laminar family has at least 2 tokens, and there are still some tokens left.
Then this would imply |E| > |L|

19. Induction Basis
Assume every edge has value 0 < x(e) < 1/3.
1/4
each leaf set has degree ¸ 4
otherwise 9e, x(e) · 1/3
each leaf has 4 tokens
1/4 $ $ $ $
1/4
1/4

20. Inductive Step (1)
Induction Hypothesis: the root has 2 extra tokens (total of 4) +2 +2 +2 +2 +2 +2 +2 +2
Theorem [Jain]. Every basic solution has an edge with value at least 1/3

21. $ $ $ $ $ $ Inductive Step (2) root has two children –
each child can pass up 2 tokens
root has 4 tokens +2 +2 +2 $ $
root has one child and 2 new edges –
child can pass up 2 tokens
root has 4 tokens $ $ $ $

22. Inductive Step (3) root has one child root has no new edges
but \delta(child) = \delta(root)
linear dependence
root has one child and only one new edge e (of the two in picture)
both root and child are tight
f(root) = f(child) + x(e) or
f(root) = f(child) - x(e)
but 0<x(e)<1 contradicting
integrality of f e e

23. Summary of Jain’s Algorithm
Theorem [Jain]. Every basic solution has an edge with value at least 1/3
Theorem [Jain]. Every basic solution has an edge with value at least ½
Proof is more involved …
Open Question: Combinatorial proof ?

24. e.g. minimum spanning tree, minimum Steiner tree
Survivable Network Design
Survivable network design: find a “good” Steiner network
e.g. minimum spanning tree, minimum Steiner tree
Minimum cost Steiner network:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost.
Minimum degree Steiner network:
Find a Steiner network with minimum maximum degree.
e.g. Hamiltonian path, Hamiltonian cycle

25. The Problem Statement
Goal: to find a good Steiner network w.r.t. to both criteria
Minimum cost Steiner network with degree constraints:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost,
so that every vertex has degree at most B.
Without degree bounds, this is the minimum cost Steiner network problem.
Without cost on edges, this is the minimum degree Steiner network problem.

26. Let OPT(B) be the value of an optimal solution to this problem.
Ideal Approximation
Minimum cost Steiner network with degree constraints:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost,
so that every vertex has degree at most B.
Let OPT(B) be the value of an optimal solution to this problem.
Ideally, we would like to return a solution so that:
SOL(B) ≤ c·OPT(B)
However, it cannot be done for any polynomial factor, even for B=2,
since this generalizes the minimum cost Hamiltonian path problem.

27. Let OPT(B) be the value of an optimal solution to this problem.
Bicriteria Approximation Algorithms
Minimum cost Steiner network with degree constraints:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost,
so that every vertex has degree at most B.
Let OPT(B) be the value of an optimal solution to this problem.
A (c,f(B))-approximation algorithm if it returns a solution with
maximum degree f(B)
SOL(f(B)) ≤ c·OPT(B)
e.g. f(B)=2B+1
This implies a c-approximation for minimum cost Steiner network,
and an f(B)-approximation for minimum degree Steiner network.

28. Minimum Bounded Degree Spanning Trees
Theorem [Furer and Raghavachari ’92]
Given k, there is a polynomial time algorithm which does the following:
either the algorithm
finds a spanning tree with maximum degree at most k+1.
shows that there is no spanning tree with maximum degree at most k.
Theorem [Goemans 06]:
Given k, there is a polynomial time algorithm that computes a
spanning tree with cost at most OPT(k) and maximum degree at most k+2.
(1,B+2)
uncrossing!

29. Results on Minimum Degree Survivable Networks
Minimum cost
Minimum degree
Bicriteria
Spanning tree 1
B+1 [FR]
(1,B+2) [G]
Steiner tree
1.55 [RZ]
(O(logn),O(logn)B)
Steiner forest
2 [AKR] ?
k-ec subgraph
2 [KV]
O(log n)·B [FMZ]
Steiner network
2 [Jain]
(2,2B+3)
2B+3
(2,2B+3)
2B+3
(2,2B+3)
2B+3
(2,2B+3)
More results at the end add citation
Theorem [Lau,Naor,Salavatipour,Singh, STOC 07]:
There is a (2,2B+3)-approximation algorithm for the
minimum bounded degree Steiner network problem.
Corollary: There is a constant factor approximation algorithm
for the Minimum Degree Steiner Network problem.

30. At least r(u,v) edges crossing S
Linear Programming Relaxation
At least r(u,v) edges crossing S S
f(S) := max{ r(u,v) | S separates u and v}. u v
Max flow min cut
Nonuniform degree bounds

31. Problem: A half edge may not exist!
First Try
Observation: Half edges are good for degree bounds as well.
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
Find a basic optimal solution, x, of the LP with function f’.
Add an edge with x(e) ≥ 1/2 into H.
Update f’: for every set S, set
Update degree bounds:
Output H.
Problem: A half edge may not exist!
set Bv:=Bv-0.5 if e is incident on v.
Bv=1.5
Bv=2 e
0.5
0.5
The residual problem is feasible.
0.5
0.5

32. The Difference fractional values Uncrossing would just work fine. 0.25
But integrality is important in the counting argument.
Bv=0.5

33. New Idea Idea: Relax the problem by removing the
degree constraint for v if v is of “low” degree.
0.25
0.25
Bv=0.5
Effect: Only violates the degree bound by
an additive constant.
Lemma [LNSS]: If every vertex is of degree 5 when its degree
constraint is present, then there is a half edge in a basic solution.
Intuition: Removing a constraint decreases the number of linearly
independent tight constraints, and makes the counting argument work.

34. By linear independence
Counting
Induction Hypothesis: The root has 2 extra tokens. +2 +2 +2 +2 +2 +2 +2 +3
By linear independence
Theorem [LNSS]. Every basic solution has an edge with value at least 1/3
if every degree constraint has at least 5 edges.

35. A multiplicative factor 2
Iterative Relaxation
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
Find a basic optimal solution, x, of the LP with function f’.
(Rounding) Add an edge with x(e) ≥ 1/2 into H.
(Relaxing) Remove the degree constraint of v if v has degree ≤ 4
Update the connectivity requirement f’
Update degree bounds: set Bv:=Bv-1/2 if e is incident at v.
Output H.
A multiplicative factor 2
An additive constant +3
Theorem: There is a (2,2B+3)-approximation algorithm for
the minimum bounded degree Steiner network problem.

36. Additive Approximation
Theorem [Lau,Singh,STOC 08]: There is a (2,B+3)-approximation algorithm
for the minimum bounded degree Steiner forest problem.
Theorem [Lau,Singh,STOC 08]: There is a (2,B+O(rmax))-approximation algorithm for the minimum bounded degree Steiner network problem.

37. Minimum Bounded Degree Spanning Trees
Theorem [Singh,Lau,STOC 07]
There is an (1,B+1)-approximation algorithm for the
minimum bounded degree spanning tree problem.
Improves on the (1,B+2)-approximation of Goemans 2006

38. Directed Connectivity
(const, const)-approximation for certain directed
Connectivity problems
[LNSS STOC 07] [Bansal,Khandekar,Nagarajan,STOC 08]

39. Exact Formulations
This method can be applied to prove LP formulations are exact.
Spanning Tree
Arborescence
Matroid intersection
Perfect matching in general graphs
Rooted k-out-connected subgraphs
Submodular flows
Limitation: not simple to prove the dual is integral.

40. Approximation Algorithms
Some NP-hard problems are variants of basic problems.
General assignment (bipartite matching)
Multicriteria spanning trees
Partial vertex cover
Prize collecting Steiner trees
Degree bounded matroids [Király,L,Singh,08]
Degree bounded submodular flows [Király,L,Singh,08]

41. Open Problems TSP, ATSP? Other applications? Combinatorial algorithms?
Connection to existing approaches?