Reconnect

1
Reconnect 04LP-Based Approximation Algorithms

• Cynthia Phillips
• Sandia National Laboratories

2
Linear Programming (LP) Relaxation-Based
Approximation

• Variables can take rational values (relax
  integrality constraints)
• Efficiently solvable gives lower bound on
  optimal IP solution
• Common technique
• Use structural information from LP solution to
  find feasible IP solution
• Bound quality using LP bound
• Integrality gap (best IP solution)/(best LP
  solution)
• This technique cannot prove anything better than
  integrality gap

3
Integer Program (IP) for capacitated network
design

• A simple IP for capacitated network design
• Where d(C) is the maximum demand di for any pair
  that crosses cut C
• xe 1 if edge e is selected

4
Knapsack Cover (KC) Inequalities
A
C
5
Finding An Approximate Solution

• Let
• Set of edges at least half selected by LP
• Select all these edges
• Increases cost (for A) by factor of 2
• Now much meet demand D(A) D - u(A) with rest of
  edges

6
Finding an Approximate Solutions

• Sort edge by ue
• Consider the three cases

7
Finding an Approximate Solution

• xe q/p rational
• r is least common multiple of denominators so rxe
  integral for all e
• Make 2rxe copies of xe
• (convex multipliers will be 1/r)

8
Approximate solution for knapsack (gap 2)

• 2rxe copies of edge e, sorted by capacity
• Place in r buckets, round robin
• Each bucket will be a solution Si
• No edge in any solution twice

9
All buckets are Feasible
ek4 ek2 e1
lt
ek3 ek1
lt
First Bucket (biggest)
Last Bucket (smallest)
10
All Buckets Feasible

• Suppose
• We have
• So
  for all buckets
• From total capacity
• Contradicts KC inequality

11
Separation

• Only have to satisfy KC inequality for
• Add these cuts if violated till we get an LP
  solution where KC inequality holds for its A.

12
Polynomial Time

• Really only m1 distinct solutions

13
A Scheduling Example

• Given n jobs J1, J2, , Jn
• Job Ji has length pi, weight wi
• Precedence constraints mean Ji
  must finish before Jj starts
• No preemption, one machine
• Cj completion time of job Jj
• Goal minimize
• NP-complete. Well get a 4-approximation

14
Integer Programming Formulation

• Subject to

15
Constraint One Job at a Time
T-pj
t
t-1
t1
t2
tpj-1
...
t-1

• Consider all (job, finish time) pairs that would
  run over (t-1, t

16
Precedence Constraints

• If job Jk finishes by time t pk, then job Jj
  must finish by time t

17
LP relaxation, Fractional Schedule
xjt
pj
18
Fractional Schedule x

• Fractional Completion Time
• Midpoint min t such that

19
Approximation Algorithm

• Solve LP
• Compute midpoints for all jobs
• Order by midpoints

20
Approximate Schedule is feasible

• No preemption
• One job at a time
• Precedence constraints
• Midpoint of Jj lt Midpoint of Jk

21
Proof of Quality Road Map

• Relate Cj to LP values
• Renumber jobs by midpoint
• Well show

22
Upper Bound on Completion Times
t
t-pj

• At time tj fractional schedule has done pj/2
  work.
• Since tk? tj for kltj, schedule has done pk/2
  work on Jk.
• One unit of work/time unit ?
• But by construction

xjt
23
Lower Bound on LP values

• By definition
• So

24
Proof of Quality

• Therefore

25
Comments

• Can create alternative schedules using ? point
  tj?
• LP-based approximation algorithms can give
  feasible solutions in branch and bound
• Other LP-based approximation algorithms for
  scheduling problems are based on
  matching/assignment