Reconnect

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Reconnect 04LP-Based Approximation Algorithms

• Cynthia Phillips
• Sandia National Laboratories

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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

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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

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Knapsack Cover (KC) Inequalities
A
C
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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

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Finding an Approximate Solutions

• Sort edge by ue
• Consider the three cases

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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)

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Approximate solution for knapsack (gap 2)

• 2rxe copies of edge e, sorted by capacity
• Place in r buckets, round robin
• No edge in any solution twice

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

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

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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.

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Polynomial Time

• Really only m1 distinct solutions

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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

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Integer Programming Formulation

• Subject to

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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

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Precedence Constraints

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

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LP relaxation, Fractional Schedule
xjt
pj
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Fractional Schedule x

• Fractional Completion Time
• Midpoint min t such that

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Approximation Algorithm

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

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Approximate Schedule is feasible

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

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Proof of Quality Road Map

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

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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
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Lower Bound on LP values

• By definition
• So

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Proof of Quality

• Therefore

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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

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Appendix
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General Graphs

• Let
• Solutions one bucket from each multiedge

. . .
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Analysis for General Graphs

• Consider cut C and
• For each edge, maximum uA capacity difference is
  D(Ac).
• Cut from at most ?(G) multiedges (combination of
  buckets), so maximum capacity difference between
  solutions is ?(G) D(Ac).
• By similar arguments, KC cut would be violated if
  any cut were infeasible.

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General Graphs - separation

• Check smallest bucket for feasibility (meeting
  all demand pairs)
• If cut C violated, add KC inequality
• Dont know ?(G)
• Run binary search with full algorithm

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Some Additional Results

• Everything holds if edge e can be chosen b(e)
  times
• Series-parallel graphs have integrality gap
• Capacitated cover bound max nonzeros in any row
• FPTAS for outerplanar graphs with one demand pair
• FPTAS to find approximately most violated KC
  inequality.