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Iterative Rounding and
Iterative Relaxation
Some of the slides prepared by Lap Chi
Lau
Steiner Network
• 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.
undirected or directed
edge or vertex
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.
Survivable Network Design
Survivable network design: find a “good” Steiner network
Minimum cost Steiner network:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost.
e.g. minimum Steiner tree, k-edge-connected subgraph
NP-complete…
Linear Programming Relaxation
S
u v f(S) := max{ r(u,v) | S separates u
and v}.
at least r(u,v) edges crossing
S
• number of constraints is
exponential.
• can be handled by
Ellipsoid algorithm
• need a separation oracle
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 …
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 representedas a convex sum of feasible solutions
Basic Solutions (2)
An edge of 0, delete it.An edge of 1, pick it.
Tight inequalities all come from the connectivity requirements.
A basic solution is the unique solution of
m linearly independent “tight” constraints,
where m is the number of variables in the LP.
Petersen Graph: Non-Basic Feasible Solution
use every edge to the extent of 1/3.
e uvc 1 r, 1.
15 edges with 0 < x(e) < 110 tight constraints (at the vertices)
Basic Feasible Solution
black edge – ¼red edge – ½
Tight sets
LP Solutions
All 1/3 is a feasible solution.
But not a basic feasible solution
Thick edges have value 1/2;
Thin edges have value 1/4.
This is a basic feasible solution.
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.
Iterative Rounding
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
o Find a basic optimal solution, x, of the LP with function f’.
o Add an edge with x(e) ≥ 1/2 into H.
o Update f’: for every set S, set
Output H.
By Jain’s Theorem
0.5
0.5 0.5
0.5
ef’(S)=2f’(S)=1
Corollary. This is a 2-approximation algorithm
for the minimum cost survivable network problem.
The residual problem is feasible.
LinearProgrammingSolver
SuitableRoundingProcedure
ProblemInstance
OptimalFractional
Solution
Integer
Solution
LinearProgrammingSolver
ProblemInstance
OptimalFractional
Solution
Part
IntegerGoodPart
Too muchFractional
Residual
Problem
Typical Rounding:
Iterative Rounding:
Laminar Basis
any pair of sets in the basis are either disjoint or contained
tree representation
Weak Supermodularity
A weakly supermodular function f satisfies:
Example: f(S) := max{ r(u,v) | S separates u
and v}.
or
Weakly supermodular functions are very useful
for obtaining a laminar basis
Important: when f is updated by setting x(e)=1
for some edges e, it remains weakly
supermodular
Obtaining a Laminar Basis
Uncrossing technique: A basic solution is defined by
a laminar family of tight connectivity constraints.
A B
A[B
AÅB
Tight constraints:
Proof of Jain’s Theorem
•There are |L| constraints.
•There are |E| variables.
Theorem [Jain]. Every basic solution has an edge with value at least 1/3
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|
Induction Basis
Assume every edge has value 0 < x(e) < 1/3.
1/4
1/41/4
1/4each leaf set has degree ¸ 4 otherwise 9e, x(e) · 1/3each leaf has 4 tokens
Inductive Step (1)
+2 +2 +2
+2 +2+2 +2
+2
Theorem [Jain]. Every basic solution has an edge with value at least 1/3
Induction Hypothesis: the root has 2 extra tokens (total of 4)
Inductive Step (2)
+2 +2
root has two children –each child can pass up 2 tokensroot has 4 tokens
root has one child and 2 new edges –child can pass up 2 tokensroot has 4 tokens
+2
Inductive Step (3)
root has one childroot has no new edgesbut \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 contradictingintegrality of f
e
e
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 ?
Survivable Network Design
Survivable network design: find a “good” Steiner network
Minimum cost Steiner network:
Given a cost c(e) on each edge,
find a Steiner network with minimum total cost.
e.g. minimum spanning tree, minimum Steiner tree
Minimum degree Steiner network:
Find a Steiner network with minimum maximum
degree.
e.g. Hamiltonian path, Hamiltonian cycle
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.
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.
Bicriteria Approximation Algorithms
This implies a c-approximation for minimum cost Steiner network,
and an f(B)-approximation for minimum degree Steiner network.
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
SOL(f(B)) ≤ c·OPT(B)maximum degree f(B) e.g. f(B)=2B+1
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
(i) finds a spanning tree with maximum degree at most k+1.
(ii) 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.
uncrossing!
(1,B+2)
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] B+1 [FR] (O(logn),O(logn)B)
Steiner forest 2 [AKR] ? ?
k-ec subgraph 2 [KV] O(log n)·B [FMZ] ?
Steiner network 2 [Jain] ? ?
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.
(2,2B+3)
(2,2B+3)
(2,2B+3)
(2,2B+3)
2B+3
2B+3
2B+3
Linear Programming Relaxation
S
u v f(S) := max{ r(u,v) | S separates u
and v}.
At least r(u,v) edges crossing
S
Nonuniform degree bounds
First Try
Observation: Half edges are good for degree bounds as well.
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
o Find a basic optimal solution, x, of the LP with function f’.
o Add an edge with x(e) ≥ 1/2 into H.
o Update f’: for every set S, set
o Update degree bounds:
Output H.
set Bv:=Bv-0.5 if e is incident on v.
The residual problem is feasible.
0.5
0.50.5
0.5eBv=
2Bv=1.
5
Problem: A half edge may not
exist!
The Difference
But integrality is important in the counting argument.
Uncrossing would just work fine.
fractional values
Bv=0.5
0.25 0.25
New Idea
Idea: Relax the problem by removing the
degree constraint for v if v is of “low” degree.
Intuition: Removing a constraint decreases the number of linearly
independent tight constraints, and makes the counting argument work.
Effect: Only violates the degree bound by
an additive constant.
Bv=0.5
0.25 0.25
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.
Counting
+2 +2 +3
+2 +2
By linear independence
+2 +2
+2
Theorem [LNSS]. Every basic solution has an edge with value at least 1/3
if every degree constraint has at least 5 edges.
Induction Hypothesis: The root has 2 extra tokens.
Iterative Relaxation
Initialization: H = , f’ = f.
While f’ ≠ 0 do:
o Find a basic optimal solution, x, of the LP with function f’.
o (Rounding) Add an edge with x(e) ≥ 1/2 into H.
o (Relaxing) Remove the degree constraint of v if v has degree ≤ 4
o Update the connectivity requirement f’
o Update degree bounds: set Bv:=Bv-1/2 if e is incident at v.
Output H.
An additive constant +3
A multiplicative factor 2
Theorem: There is a (2,2B+3)-approximation algorithm for
the minimum bounded degree Steiner network problem.
Additive Approximation
Theorem [Lau,Singh,STOC 08]: There is a (2,B+O(rmax))-approximation algorithm for the minimum bounded degree Steiner network problem.
Theorem [Lau,Singh,STOC 08]: There is a (2,B+3)-approximation algorithm
for the minimum bounded degree Steiner forest problem.
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
Directed Connectivity
(const, const)-approximation for certain directed
Connectivity problems
[LNSS STOC 07] [Bansal,Khandekar,Nagarajan,STOC 08]
Exact Formulations
•Spanning Tree
•Arborescence
•Matroid intersection
•Perfect matching in general graphs
•Rooted k-out-connected subgraphs
•Submodular flows
This method can be applied to prove LP formulations are exact.
Limitation: not simple to prove the dual is integral.
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]
Open Problems
• TSP, ATSP?
• Other applications?
• Combinatorial algorithms?
• Connection to existing approaches?