Reconnect ‘04 LP-Based Approximation Algorithms

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. e2 e2 e3 e3 e1 e1 e1 e1 e1 e1 e2 e2 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

9. All buckets are Feasible ek4 ek2 e1 < ek3 ek1 < 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 it’s A.

12. e2 e2 e3 e3 e1 e1 e1 e1 e1 e1 e2 e2 Polynomial Time Really only m+1 distinct solutions

13. A Scheduling Example Given n jobs J1, J2, …, Jn Job Ji has length pi, weight wi Precedence constraints: mean Jimust finish before Jj starts No preemption, one machine Cj= completion time of job Jj Goal: minimize NP-complete. We’ll get a 4-approximation

14. Integer Programming Formulation Subject to

15. Constraint: One Job at a Time Consider all (job, finish time) pairs that would run over (t-1, t] T-pj t-1 t t+1 t+2 t+pj-1 ... t-1

16. Precedence Constraints If job Jk finishes by time t + pk, then job Jjmust 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 < Midpoint of Jk

21. Proof of Quality Road Map • Relate Cj to LP values Renumber jobs by midpoint: We’ll show

22. Upper Bound on Completion Times t t-pj • At time tj* fractional schedule has done pj/2 work. • Since tk* tj* for k<j, 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

26. Appendix

27. … … … General Graphs Let Solutions: one bucket from each multiedge . . .

28. 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.

29. General Graphs - separation • Check smallest bucket for feasibility (meeting all demand pairs) • If cut C violated, add KC inequality: • Don’t know (G) • Run binary search with full algorithm

30. 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.