Semi-Infinite Relaxations for a Dynamic Knapsack Problem

Transcription

1 Semi-Infinite Relaxations for a Dynamic Knapsack Problem Alejandro Toriello joint with Daniel Blado, Weihong Hu Stewart School of Industrial and Systems Engineering Georgia Institute of Technology MIT April 15, 2016

2 Problem Statement Knapsack with capacity b > 0 and item set N = {1,..., n}. Each item i has 1. deterministic value c i, 2. independent random size A i 0 with known distribution. When attempting to insert i: If i fits collect c i, update capacity. Else process ends. Policy may depend on remaining items and remaining capacity. Goal is to maximize expected value. Problem is at least NP-hard, some versions PSPACE-hard (Vondrák, 05).

3 Outline Pertinent Past Work Approximation and Bound Computational Experiments Asymptotic Analysis Extensions and Conclusions

4 Brief Literature Review Derman/Lieberman/Ross (78): Sizes are exponential r.v. s. Greedy policy w.r.t. ci /E[A i ] is optimal. Dean/Goemans/Vondrák (04,08): Two LP bounds with polynomially many variables. Linear knapsack, polymatroid, both within constant gap. Greedy approximate policies. Gupta/Krishnaswamy/Molinaro/Ravi (11), Ma (14): Integer sizes, LP bounds of pseudo-polynomial size. Randomized policies based on LP optimal solutions. Extensions to models with correlated random item values, preemption, multi-armed bandits. Other work, e.g. Bhalgat/Goel/Kanna (11), Li/Yuan (13), Bansal/Nagarajan (14), Balseiro/Brown (16).

5 Linear Knapsack Bound Dean/Goemans/Vondrák (08) Use x i, probability policy attempts to insert i: max x c i x i i N s.t. i N x i E[A i ] b; 0 x i 1, i N.

6 Linear Knapsack Bound Dean/Goemans/Vondrák (08) Use x i, probability policy attempts to insert i: max c i x i P(A i b) x i N s.t. i N x i E[A i ] b; 0 x i 1, i N.

7 Linear Knapsack Bound Dean/Goemans/Vondrák (08) Use x i, probability policy attempts to insert i: max x c i x i P(A i b) i N s.t. i N x i E[min{b, A i }] b; 0 x i 1, i N. Mean truncated size E[min{b, A i }]: A i above b is irrelevant (insertion will fail).

8 Linear Knapsack Bound Dean/Goemans/Vondrák (08) Use x i, probability policy attempts to insert i: max x c i x i P(A i b) i N s.t. i N x i E[min{b, A i }] 2b; 0 x i 1, i N. Mean truncated size E[min{b, A i }]: A i above b is irrelevant (insertion will fail). Bound intuition: In worst case, policy exactly fills knapsack, then attempts to insert very large item. Worst-case gap is 32/7. Polymatroid bound is extension of same idea, with gap of 4.

9 Dynamic Programming Formulation State: remaining items, remaining capacity (M, s) for M N, s [0, b]. Actions: attempt to insert i M. Bellman recursion is v M(s) = max i M P(A i s)(c i + E[v M\i (s A i) A i s]), v (s) = 0. In doubly infinite LP form: min v N (b) v s.t. v M i (s) P(A i s)(c i + E[v M (s A i ) A i s]), i N, M N \ i, s [0, b] v M : [0, b] R +, M N.

10 Value Function Approximation Any feasible solution to LP yields upper bound. Use affine approximation v M (s) qs + r 0 + r i, i M where q is marginal value of capacity, r i is item i s inherent value, r 0 is value of process continuing ( staying alive ).

11 Value Function Approximation Lemma The best bound given by v M (s) qs + i M 0 r i is the semi-infinite LP min qb + r 0 + r i q,r 0 i N s.t. qe[min{s, A i }] + r 0 P(A i > s) + r i c i P(A i s), i N, s [0, b].

12 Value Function Approximation Lemma The best bound given by v M (s) qs + i M 0 r i is the semi-infinite LP min qb + r 0 + r i q,r 0 i N s.t. qe[min{s, A i }] + r 0 P(A i > s) + r i c i P(A i s), Proof sketch. i N, s [0, b]. v M i (s) P(A i s)e[v M (s A i ) A i s] qs P(A i s)e[q(s A i ) A i s] (focusing on q) = qsp(a i > s) + qp(a i s)e[a i A i s] = qe[min{s, A i }]

13 Multiple-Choice Linear Knapsack Bound Theorem The LP s finite-support dual is solvable and has zero duality gap: c i x i,s P(A i s) max x 0 i N s [0,b] s.t. x i,s E[min{s, A i }] b, (exp. frac. size under b) i N s [0,b] x i,s P(A i > s) 1 (one exp. failure; cf. Ma 14) i N s [0,b] s [0,b] x i,s 1 x has finite support. (insert i once) x i,s : probability policy attempts to insert i when s capacity remains.

14 Multiple-Choice Linear Knapsack Bound Pricing problem { qb + min q,r 0 i N 0 } r i : qe[min{s, A i}] + r 0P(A i > s) + r i c ip(a i s), i N, s [0, b] Pricing/separation: Given q, r, for each i solve { min qe[min{s, Ai }] (c i + r 0 )P(A i s) }. s [0,b] Mean truncated size is concave in s. If CDF is piecewise convex, check only endpoints of convex intervals. Applies to discrete, uniform distributions Polynomially many variables. Other distributions (e.g. exponential, conditional normal) have closed-form solution. Check at most countably many points in general.

15 Multiple-Choice Linear Knapsack Bound Pricing problem: Exponential distribution example { min qe[min{s, Ai }] (c i + r 0 )P(A i s) } s [0,b] Suppose A i exp(λ): P(A i s) = 1 e λs E[min{s, A i }] = P(A i s)/λ. Thus qe[min{s, A i }] (c i + r 0 )P(A i s) = (q/λ c i r 0 )P(A i s) minimized at s {0, b}.

16 Multiple-Choice Linear Knapsack Bound So if sizes are exponentially distributed, the bound is max c i x i,b P(A i b) x i N s.t. i N x i,b E[min{b, A i }] b 0 x i,b 1, i N. This is DGV linear knapsack with capacity cut in half. Applies to other size distributions, e.g. conditional normal, uniform, geometric. Theorem The MCLK bound dominates the DGV knapsack bound on any instance. Conjecture: MCLK also dominates DGV polymatroid.

17 Computational Experiments Generated instances from deterministic knapsack instances. 8 small, n [5, 24]: people.sc.fsu.edu/ jburkardt 10 large, n = 100: pisinger/codes.html (uncorrelated) For a deterministic size a i, generated: Exponential (1/a i ) Uniform [0, 2a i ] and [a i /2, 3a i /2] Conditional normal (a i, a i /3) Bound comparison: average of deterministic knapsack over 400 simulations ( perfect information relaxation ). Not reporting: DGV polymatroid bound not competitive. Benchmark: Adaptive greedy policy w.r.t. version studied in DGV). c i P(A i s) E[min{s,A i }] (basic

18 Computational Experiments Geometric mean of gap Small Large PIR MCLK PIR MCLK Exponential 48% 5% 22% 0.5% Uniform 1 41% 12% 12% 1% Uniform 2 26% 12% 4% 0.6% Normal 30% 12% 5% 0.5% Greedy benchmark is optimal (Derman/Lieberman/Ross 78). MCLK gives consistently better bound across instance types. Tighter for most small, all large instances. All gaps improve as number of items increases. Especially stark advantage for exponential instances.

19 Asymptotic Analysis Motivation All gaps improve as number of items increases. Check MCLK versus simple greedy policy, which chooses items in order c 1 E[A 1 ] c n E[A n ]. This policy is optimal in at least one case, exponential distributions (Derman/Lieberman/Ross 78). Expect an averaging effect as n grows.

20 Asymptotic Analysis Take infinite collection of items sorted in greedy order: c 1 /E[A 1 ] c i /E[A i ] Main technical condition: ( E[Ai A i > s] s ) < sup i,s Implies uniformly bounded moments. Satisfied by several distributions, e.g. bounded support, exponential, normal,..., but not e.g. power law. Theorem Under these conditions, See also Balseiro/Brown (16). Greedy(b) lim b MCLK(b) 1.

21 Asymptotic Analysis Proof sketch. Take b k := i k E[A i]. Two main ingredients: 1. MCLK(b k ) c 0 + i k c ip(a i b k ) Requires constructing dual feasible MCLK solution for each k. Constant c 0 comes from technical condition. 2. Greedy(b k ) i k c ip(a i b k ) 1 as k. More standard arguments from probability, mostly Markov s inequality. Working on proof extension to weaken condition, e.g. using different growth regimes.

22 Extensions Correlated value: Much of analysis applies, but must use conditional value E[C i A i s] (GKMR 11, Ma 14). If items have integer support: Use non-parametric pseudo-polynomial approximation v M (s) s r i + w σ. i M σ=0 Yields Ma bound (14). Can use to show Ma bound dominates GKMR bound (strengthen Ma s result). Policies: MCLK and pseudo-polynomial bounds can be used for policy design. E.g. from value function approximation, rounding, ad hoc methods.

23 Ongoing Work MCLK and other bounds still leave gap in small-to-medium instances. Improve approximation with quadratic variables: v M (s) qs + r 0 + r i r ij. i M i,j M Submodular in M, reflecting decreasing marginal value of items. LP must now separate over sets M. Separation problem is IP with totally unimodular constraint matrix, preserving polynomial solvability.

24 Preliminary Computations - Quad LP Small 20-Item MCLK Quad Ma MCLK Quad Ma D1 27.7% 26.5% 16.5% 10.6% 10.0% 10.2% D2 15.7% 15.3% 10.4% 6.3% 6.2% 6.2% D3 13.8% 13.7% 8.4% 5.6% 5.5% 5.2% D4 20.4% 16.7% 8.7% 7.2% 5.7% 5.0% Quad LP significantly faster than Ma bound (poly vs. pseudo-poly). Competitive, particularly in medium instances. Room left for improvement with all bounds.

25 Conclusions MCLK bound has theoretical guarantees and good empirical performance on various item size distributions. Value function approximation is systematic way to generate bounds for dynamic problems. Big picture questions: 1. Exact algorithms: cutting planes, branching? 2. Extend to general stochastic and dynamic IP (Vondrák 05). atoriello3