Transcript presentation in Powerpoint
Strong LP Formulations & Primal-Dual Approximation Algorithms
David Shmoys (joint work Tim Carnes & Maurice Cheung) June 23, 2011
Introduction
The standard approach to solve combinatorial integer programs in practice – start with a “ simple ” formulation & add valid inequalities Our agenda algorithms : show that same approach can be theoretically justified by approximation An ® approximation algorithm produces a solution of cost within a factor of ® of the optimal solution in polynomial time
Introduction
Primal-dual method a leading approach in the design of approximation algorithms for NP-hard problems Consider several capacitated covering problems - covering knapsack problem - single-machine scheduling problems Give constant approximation algorithms based on strong linear programming (LP) formulations
Approximation Algorithms and LP
Use LP to design approximation algorithms Optimal value for LP gives bound on optimal integer programming (IP) value Want to find feasible IP solution of value within a factor of ® of optimal LP solution Key is to start with “ right ” LP relaxation LP-based approximation algorithm produces additional performance guarantee on each problem instance Empirical success of IP cutting plane methods suggests stronger formulations - needs theory!
Primal-Dual Approximation Algorithms
Do not even need to solve LP!!
Each min LP has a dual max LP of equal optimal value Goal: Construct feasible integer solution S along with feasible solution D to dual of LP relaxation such that cost(S) · ®¢ cost(D) · ®¢ LP-OPT · ®¢ OPT ) ® -approximation algorithm
Adding Valid Inequalities to LP
LP formulation can be too “ weak ” if there is “ big ” integrality gap OPT/LP-OPT is often unbounded Fixed by adding additional inequalities to formulation Restricts set of feasible LP solutions Satisfied by all integer solutions, hence “ valid ” Key technique in practical codes to solve integer programs
Knapsack-Cover Inequalities
Carr, Fleischer, Leung and Phillips (2000) developed valid knapsack-cover inequalities and LP-rounding algorithms for several capacitated covering problems Requires solving LP with ellipsoid method Further complicated since inequalities are not known to be separable GOAL: develop a primal-dual analog!
Highlights of Our Results
For each of the following problems, we have a primal-dual algorithm that achieves a performance guarantee of 2 Min-Cost (Covering) Knapsack Single-Demand Capacitated Facility Location Used valid knapsack-cover inequalities developed by Carr, Fleischer, Leung and Phillips as LP formulation Single-Item Capacitated Lot-Sizing We extend the knapsack-cover inequalities to handle this more general setting Single-Machine Minimum-Weight Late Jobs 1|| w j U j Single-Machine General Minimum-Sum Scheduling 1|| f j
Highlights of Our Results
For each of the following problems, we have a primal-dual algorithm that achieves a performance guarantee of 2 Min-Cost (Covering) Knapsack Single-Demand Capacitated Facility Location Used valid knapsack-cover inequalities developed by Carr, Fleischer, Leung and Phillips as LP formulation Single-Item Capacitated Lot-Sizing We extend the knapsack-cover inequalities to handle this more general setting Single-Machine Minimum-Weight Late Jobs 1|| w j U j Single-Machine General Minimum-Sum Scheduling 1|| f j
Min-Sum 1-Machine Scheduling 1||
f
j Each job j has a cost function f j (C j ) that is non-negative non-decreasing function of its completion time C j Goal: minimize j f j (C j ) What is known? Bansal & Pruhs (FOCS ’10) gave first constant-factor algorithm Main result of Bansal-Pruhs adds release dates, and permits preemption – result is O(loglog(nP))-approximation algorithm OPEN QUESTIONS – Is a constant-factor doable?
10 Open Problems
Better constant factors?
Any constant factor?
Primal-dual when rounding is known?
But – nothing of the type – good constant factor is known, but is a factor of 1+ possible for any >0?
Min-Sum 1-Machine Scheduling 1||
f
j Each job j has a cost function f j (C j ) that is non-negative non-decreasing function of its completion time C j Goal: minimize j f j (C j ) What is known? Bansal & Pruhs (FOCS ’10) gave first constant-factor algorithm Main result of Bansal-Pruhs adds release dates, and permits preemption – result is O(loglog(nP))-approximation algorithm OPEN QUESTIONS – Is a constant-factor doable?
- Can 1+ ² be achieved w/o release dates?
Primal-Dual for Covering Problems
Early primal-dual algorithms Bar-Yehuda and Even (1981) – weighted vertex cover Chvátal (1979) – weighted set cover Agrawal, Klein and Ravi (1995) Goemans and Williamson (1995) generalized Steiner (cut covering) problems Bertismas & Teo (1998) Jain & Vazirani (1999) uncapacitated facility location problem Inventory problems Levi, Roundy and Shmoys (2006)
Minimum (Covering) Knapsack Problem
Given a set of items F value u i each with a cost c i and a Want to find a subset of items with minimum cost such that the total value is at least D minimize i F c i x i subject to i F u i x i D x i {0,1} for each i F t est ing
Bad Integrality Gap
Consider the min knapsack problem with the following two items c 1 u 1 = 1 = D u 2 c 2 = 0 = D-1 Integer solution must take item 1 and incurs a cost of 1 LP solution can take all of item 2 and just 1/D fraction of item 1, incurring a cost of 1/D Int eger solut ion = LP solut ion 1 1=D = D
Knapsack-Cover Inequalities
Proposed by Carr, Fleischer, Leung and Phillips (2000) Consider a subset A of items in F If we were to take all items in A, then we still need to take enough items to meet leftover demand D A = {1,2,3} i 2 F n A u i x i ¸ D-u(A) u(A) = i 2 A u i 1 2 3 4 5 6 7
Knapsack-Cover Inequalities
i 2 F n A u i x i ¸ D-u(A) This inequality adds nothing new, but we can now restrict the values of the items i 2 F n A u i (A) x i ¸ D-u(A) where u i (A) = min{ u i , D-u(A) } since these inequalities only need to be valid for integer solutions D 1 2 3 4 5 6 7 A = {1,2,3}
Strengthened Min Knapsack LP
When A = ; the knapsack-cover inequality becomes i 2 F n A u i (A) x i ¸ D-u(A) ) i 2 F u i x i ¸ D which is the original min knapsack inequality New strengthened LP is Minimize i 2 F c i x i subject to i 2 F n A u i (A) x i ¸ D- u(A), for each subset A x i
¸
0, for each i
2
F
Dual Linear Program
opt Dual := max A
µ
F (D-u(A))v(A) subject to A µ F : i A u i (A) v(A)
·
c i , for each i
2
F v(A)
¸
0, for each A
µ
F Dual of LP formed by knapsack-cover inequalities
Primal-Dual
D = 5 1.25
2 0.25
2 0.75
Primal-Dual
D = 5 1.25
1 0.25
2 0.75
A = {3} D - u(A) = 3 Increase v(A) Dual Variables: v(;) = 0.25
Primal-Dual
D = 5 1.25
1 0.25
2 0.75
A = {3} D - u(A) = 3 Increase v(A) Dual Variables: v(;) = 0.25
Primal-Dual
D = 5 1 1.75
0 2.92
0.5
v({3}) = 0.5
Primal-Dual
D = 5 1 1.75
0 2.92
0.5
D - u(A) = 1 Increase v(A) Dual Variables: v(;) = 0.25
v({3}) = 0.5
Primal-Dual
D = 5 1 1.25
0 7.25
0 v({3,5}) = 1
Primal-Dual
Primal-Dual Cost = 4.5
c 1 u 1 = 2.5
= 2 Opt. Integer Cost = 4 c 1 u 1 = 2.5
= 2 c 2 u 2 = 2 = 1 c 3 u 3 = 0.5
= 2 c 4 u 4 = 10 = 5 c 5 u 5 = 1.5
= 2 c 2 u 2 = 2 = 1 c 3 u 3 = 0.5
= 2 c 4 u 4 = 10 = 5 c 5 u 5 = 1.5
= 2
Primal-Dual Summary
Start with all variables set to zero and solution A as the empty set Increase variable v(A) until a dual constraint becomes tight for some item i Add item i to solution A and repeat Stop once solution A has large enough value to meet demand D Call final solution S and set x i = 1 for all i 2 S
Analysis
Let l be last item added to solution S If we increased dual variable v(A) then not in A l Thus if v(A) > 0 then A µ ( S \ l ) Since u( S\ l ) < D then u((S\ l )
\
A)
<
D – u(A) was
Analysis (continued)
We have u((S\ l Cost of solution is )\A) < D – u(A) if v(A) > 0 Dual LP
Primal-Dual Theorem
For the min-cost covering knapsack problem, the LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual 2-approximation algorithm.
Knapsack-Cover Inequalities Everywhere
Bansal, Buchbinder, Naor (2008) weighted paging) Randomized competitive algorithms for generalized caching (and Bansal, Gupta, & Krishnaswamy (2010) 485 approximation algorithm for min-sum set cover Bansal & Pruhs (2010) O(log log nP)-approximation algorithm for general 1-machine preemptive scheduling + O(1) with identical deadlines
Minimum-Weight Late Jobs on 1 Machine
Each job j has processing time p j , deadline d j , weight w j Choose a subset L of jobs of minimum-weight to be late scheduled to complete by deadline This problem is (weakly) NP-hard can be solved in O( n j p j ) time [ Lawler & Moore ], (1+ ² )-approximation in O(n 3 /
²
) time [ Sahni ] - not If there also are release dates that constrain when a job may start, no approximation result is possible - focus on max-weight set of jobs scheduled on time [ Bar-Noy, Bar-Yehuda, Freund, Naor, & Schieber ] - allow preemption [ Bansal & Pruhs ]
What if all deadlines are the same?
Total processing time is j p j !
P WLOG assume schedule runs through [0,P] Deadline D done after D
)
at least P-D units of processing are So just select set of total processing at least P-D of minimum total weight
,
minimum-cost covering knapsack problem
Same Idea for General Deadlines
Total processing time is j p j P WLOG assume schedule runs through [0,P] Assume d 1
·
d 2
·
…
·
d n Deadline d i P(i)-d i where
)
among all jobs with deadlines
·
d i units of processing are done after d i , S(i) = { j : d j d i } and P(i) = j S(i) p j Minimize subject to j w j y S(i) y j j ¸ p j y j ¸ P(i)-d i , i=1,…,n 0 , j=1,…,n
Strengthened LP – Knapsack Covers
Minimize subject to w j j y j S(L,i) p j (L,i) y j D(L,i), for each L,i where S(L,i) = { j: d j D(L,i) = max{ j d i p j (L,i) = min{ p j , j S(L,i) L} p j , D(L,i) } - d i , 0} Dual: Maximize D(L,i) v(L,i) subject to (L,i): j 2 S(L,i) p j (L,i) v(L,i) v(L,i) ¸ · w j 0 for each j for each L,i
Primal-Dual Summary
Start with all dual variables set to 0 and solution A as the empty set Increase variable v( A,i ) with largest D(A,i) until a dual constraint becomes tight for some item i Add item i to solution A and repeat Stop once solution A is sufficient so remaining jobs N-A can be scheduled on time Examine each item j in A in reverse order and delete j if reduced late set is still feasible Call final solution L * and set y j = 1 for all j 2 L *
Highlights of the Analysis
Lemma . Suppose current iteration increases v(L,i), and let L(i) be jobs put in final late set L * afterwards. Then 9 job k L(i) so that L * -{k} is not feasible.
Note: in previous case, since all deadlines were equal, the last job l added satisfies this property.
Here, the reverse delete process is set exactly to ensure that the Lemma holds.
Highlights of the Analysis
Lemma . Suppose current iteration increases v(L,i), and let L(i) be jobs put in final late set L * afterwards. Then 9 job k L(i) so that L * -{k} is not feasible.
Lemma .
j: j i, j L(i) – {k} p j (L,i) < D(L,i) if v(L,i)>0.
Highlights of the Analysis
Lemma . Suppose current iteration increases v(L,i), and let L(i) be jobs put in final late set L * afterwards. Then 9 job k L(i) so that L * -{k} is not feasible.
Lemma .
j: j i, j L(i) – {k} p j (L,i) < D(L,i) if v(L,i)>0.
Fact.
p k (L,i) D(L,i) (by definition of p k (L,i) )
Highlights of the Analysis
Lemma . Suppose current iteration increases v(L,i), and let L(i) be jobs put in final late set L * afterwards. Then 9 job k L(i) so that L * -{k} is not feasible.
Lemma .
j: j i, j L(i) – {k} p j (L,i) < D(L,i) if v(L,i)>0 .
Fact.
p k (L,i) D(L,i) (by definition of p k (L,i) ) Corollary.
j: j i, j L(i) p j (L,i) < 2D(L,i) if v(L,i)>0.
Previous Analysis (flashback)
We have u((S\ l Cost of solution is )\A) < D – u(A) if v(A) > 0 Dual LP
Highlights of the Analysis
Lemma . Suppose current iteration increases v(L,i), and let L(i) be jobs put in final late set L * afterwards. Then 9 job k L(i) so that L * -{k} is not feasible.
Lemma .
j: j i, j L(i) – {k} p j (L,i) < D(L,i) if v(L,i)>0 .
Fact.
p k (L,i) D(L,i) (by definition of p k (L,i) ) Corollary.
j: j i, j L(i) p j (L,i) < 2D(L,i) if v(L,i)>0.
Highlights of the Analysis
Corollary j: j i, j L(i) p j (L,i) < 2D(L,i) if v(L,i)>0.
Same trick here: j L* w j = j L* (L,i): j 2 S(L,i) p j (L,i) v(L,i) = (L,i) v(L,i) j: j i, j L(i) p j (L,i) · (L,i) 2D(L,i) v(L,i) · 2 OPT
Primal-Dual Theorem
For the 1-machine min-weight late jobs scheduling problem with a common deadline, the LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual 2 approximation algorithm.
General 1-Machine Min-Cost Scheduling
Each job j has its own nondecreasing cost function f j (C j ) – where C j denotes completion time of job j Assume that all processing times are integer Goal: construct schedule to minimize total cost incurred LP variables – x jt =1 means job j has C j =t Knapsack cover constraint: for each t and L, require that total processing time of jobs finishing at time t or later is sufficiently large
Primal-Dual Theorem(s)
For 1-machine min-cost scheduling, LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual pseudo-polynomial 2-approximation algorithm.
For 1-machine min-cost scheduling, LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual (2+ ² )-approximation algorithm.
“Weak” LP Relaxation
Total processing time is j p j P WLOG assume schedule runs through [0,P] x jt = 1 means job j completes at time
t
Minimize f j (t) x jt subject to t j {1,…n} s 1,…,P {t,…,P} p j x jt x js x jt = 1 , j=1,…,n D(t) t=1,…,P 0 j=1,…,n; t=1,…,P where D(t) = P-t+1 .
Strong LP Relaxation
Total processing time is j p j P WLOG assume schedule runs through [0,P] x jt = 1 means job j completes at time
t
Minimize f j (t) x jt subject to t {1,…,P} j L s {t,…,P} p j x jt (L,t) x js x jt = 1 , for all j ¸ ¸ D(L,t) 0 , for all L,t for all j,t where D(t) = P-t+1, D(L,t) = max{0, D(t) j p j (L,t) = min{p j , D(L,t)} .
L p j }, and
Primal and Dual LP
Minimize f j (t) x jt subject to t {1,…,P} L s=t,…,P p j x jt = 1 , for all j j (L,t) x js x jt ¸ D(L,t) ¸ 0 for all L,t , D(t) = P-t+1 D(L,t) = max{0, j L for all j,t p j –t+1} p j (L,t) = min{p j , D(L,t)} Maximize L t subject to L: j D(L,t) v(L,t) L t=1,…,s p j (L,t) v(L,t) v(L,t) f j (s) for all j,s 0 for all L,t
Primal-Dual Summary
Start w/ all dual variables set to 0 and each A t = ; Increase variable v( A t ,t ) with largest D(A t ,t) until a dual constraint becomes tight for some item i (break ties by selecting latest time) Add item i to solution A s for all s · t and repeat Stop once solution A is sufficient so remaining jobs N-A satisfy all demand constraints Focus on pairs (j,t) where t is latest job j is in A t perform a reverse delete Set d j =t for job j by remaining pairs (j,t) Schedule in Earliest Due Date order and
Primal-Dual Theorem
For 1-machine min-cost scheduling, LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual pseudo-polynomial 2-approximation algorithm.
Removing the “Pseudo” with a (1+
²
) Loss
This requires only standard techniques
For each job j, partition the potential job completion times {1,…,P} into blocks so that within block the cost for j increases by · 1+ ² Consider finest partition based on all n jobs Now consider variables x jt that assign job j to finish in block t of this partition.
All other details remain basically the same.
Fringe Benefit: more general models, such as possible periods of machine non-availability
Primal-Dual Theorem
For 1-machine min-cost scheduling, LP relaxation with knapsack-cover inequalities can be used to derive a (simple) primal-dual (2+ ² )-approximation algorithm.
Some Open Problems
Give a constant approximation algorithm for 1-machine min-sum scheduling with release dates allowing preemption Give a (1+ ² )-approximation algorithm for 1-machine min-sum scheduling, for arbitrarily small ² > 0 Give an LP-based constant approximation algorithm for capacitated facility location Use “ configuration LP ” to find an approximation algorithm for bin-packing problem that uses at most ONE bin more than optimal
Thank you!