Stochastic Matching in Hypergraphs

Transcription

1 Stochastic Matching in Hypergraphs Amit Chavan, Srijan Kumar and Pan Xu May 13, 2014

2 ROADMAP INTRODUCTION Matching Stochastic Matching BACKGROUND Stochastic Knapsack Adaptive and Non-adaptive policies Adaptivity Gap STOCHASTIC k-set PACKING LP relaxation for optimal adaptive A Solution Policy Related Work

3 MATCHING Definition Given a (hyper)graph G(V, E) a matching or independent edge set is a subset of E such that no two of them have a vertex in common. Figure:

4 MATCHING Resource allocation Figure:

5 MATCHING Stable Marriage Problem being single to his/her assignment under µ. A matching is stable if it does not have any blocking pair. Example Schroeder Charlie Charlie Linus Franklin Charlie Franklin Linus Schroeder Linus Franklin Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin Lucy Peppermint Marcie Marcie Sally Marcie Lucy Stable! Peppermint Sally Marcie Figure:

6 MATCHING Latin Square Problem Figure: solution.svg/250px-sudoku-by- L2G solution.svg.png

7 STOCHASTIC MATCHING Setting: Each edge e is present independently with probability p e. Objective: Maximum matching in graph given p e e E. We don t know whether edge is present or not - just the probability. To find, query the edge, and if the edge is present, add it to matching probing of edge. Task: Adaptively query the edge to maximize the expected matching weight. First introduced and studied by Chen, Immorlica, Karlin, Mahdian, and Rudra [2009].

8 WHY IS IT IMPORTANT? Motivated by: Kidney exchange Online dating

9 WHY IS IT IMPORTANT? - KIDNEY EXCHANGE Figure:

10 WHY IS IT IMPORTANT? - ONLINE DATING Figure:

11 STOCHASTIC KNAPSACK PROBLEM Classical Knapsack n items Item i has size s i and profit v i Knapsack capacity W Goal: Compute the max profit feasible subset S Stochastic Knapsack s i are independent random variables with known distribution Goal: Find a policy such that the expected weight of the inserted items is maximized Caveat: Stop when knapsack overflows

12 ADAPTIVE AND NON-ADAPTIVE POLICIES Non-adaptive policy An ordering O = {i 1, i 2,..., i n} of items NON-ADAPT(I) = max O E[val(O)] Optimal O = {1, 2, 3}, E[val(O)] = 1.5 Adaptive policy Function P : 2 [n] [0, 1] [n] Given a set of inserted items J and remaining capacity c, P(J, c) is the next item to insert ADAPT(I) = max P E[val(P(, 1))] Optimal expected value = 1.75

13 ADAPTIVITY GAP For an instance I, ADAPTIVITY-GAP(I) = sup ADAPT(I) NON-ADAPT(I) Studied by Dean, Goemans, and Vondrák [2005]. They show that for d-dimensional knapsack, the gap can be Ω( d). They also give a non-adaptive O(d) approximation to the optimal adaptive.

14 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of

15 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns

16 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns Item i has random profit v i R +, and a random d-dimensional size s i {0, 1} d

17 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns Item i has random profit v i R +, and a random d-dimensional size s i {0, 1} d The probability distributions of different items are independent

18 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns Item i has random profit v i R +, and a random d-dimensional size s i {0, 1} d The probability distributions of different items are independent Each item takes non-zero size in at most k co-ordinates (out of d)

19 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns Item i has random profit v i R +, and a random d-dimensional size s i {0, 1} d The probability distributions of different items are independent Each item takes non-zero size in at most k co-ordinates (out of d) A capacity vector b Z + into which the items must be packed

20 PRELIMINARIES STOCHASTIC k-set PACKING An instance I consists of n items/columns Item i has random profit v i R +, and a random d-dimensional size s i {0, 1} d The probability distributions of different items are independent Each item takes non-zero size in at most k co-ordinates (out of d) A capacity vector b Z + into which the items must be packed Goal: Find an adaptive strategy of choosing items such that the expected profit is maximized

21 LP RELAXATION FOR OPTIMAL ADAPTIVE (BANSAL ET AL. [2010])

22 LP RELAXATION FOR OPTIMAL ADAPTIVE (BANSAL ET AL. [2010]) Let w i = E[v i ] be the mean profit, for each i [n].

23 LP RELAXATION FOR OPTIMAL ADAPTIVE (BANSAL ET AL. [2010]) Let w i = E[v i ] be the mean profit, for each i [n]. Let µ i (j) = E[s i (j)] be the expected size of the i-th item in j-th co-ordinate.

24 LP RELAXATION FOR OPTIMAL ADAPTIVE (BANSAL ET AL. [2010]) Let w i = E[v i ] be the mean profit, for each i [n]. Let µ i (j) = E[s i (j)] be the expected size of the i-th item in j-th co-ordinate. maximize subject to n w i y i (1) i=1 n µ i (j)y i b j, j [d] (2) i=1 y i [0, 1], i [n] (3) y i is the probability that the adaptive algorithm probes item i.

25 A SOLUTION POLICY Let y denote an optimal solution to the linear program in 1.

26 A SOLUTION POLICY Let y denote an optimal solution to the linear program in 1. Fix a constant α 1 (to be specified later).

27 A SOLUTION POLICY Let y denote an optimal solution to the linear program in 1. Fix a constant α 1 (to be specified later). Pick a permutation π : [n] [n] uniformly at random.

28 A SOLUTION POLICY Let y denote an optimal solution to the linear program in 1. Fix a constant α 1 (to be specified later). Pick a permutation π : [n] [n] uniformly at random. Inspect items/columns in the order of π.

29 A SOLUTION POLICY Let y denote an optimal solution to the linear program in 1. Fix a constant α 1 (to be specified later). Pick a permutation π : [n] [n] uniformly at random. Inspect items/columns in the order of π. Probe item c with probability y c/α if and only if it is safe to do so.

30 APPROXIMATION RATIO For any column c [n], let {I c,l } k l=1 denote the indicator random variable that the l-th constraint in the support of c is tight when c is considered in π.

31 APPROXIMATION RATIO For any column c [n], let {I c,l } k l=1 denote the indicator random variable that the l-th constraint in the support of c is tight when c is considered in π. Pr[c is safe when considered] = Pr[ k l=1 I c,l ] 1 k l=1 Pr[I c,l]

32 APPROXIMATION RATIO For any column c [n], let {I c,l } k l=1 denote the indicator random variable that the l-th constraint in the support of c is tight when c is considered in π. Pr[c is safe when considered] = Pr[ k l=1 I c,l ] 1 k l=1 Pr[I c,l] Lemma Pr[I c,l ] 1 2α

33 2k-APPROXIMATION Let j [d] be the l-th constraint in the support of c.

34 2k-APPROXIMATION Let j [d] be the l-th constraint in the support of c. Let U j c denote the usage of constraint j, when c is considered.

35 2k-APPROXIMATION Let j [d] be the l-th constraint in the support of c. Let U j c denote the usage of constraint j, when c is considered. E[U j c] = = n Pr[column a appears before c AND a is probed]µ a(j) a=1 n a=1 n a=1 b j 2α Pr[column a appears before c] ya α µa(j) y a 2α µa(j)

36 2k-APPROXIMATION Let j [d] be the l-th constraint in the support of c. Let U j c denote the usage of constraint j, when c is considered. E[U j c] = = n Pr[column a appears before c AND a is probed]µ a(j) a=1 n a=1 n a=1 b j 2α Pr[column a appears before c] ya α µa(j) y a 2α µa(j) Since I c,l = {U j c b j }, by Markov s inequality, Pr[I c,l ] E[Uj c ] b j 1 2α.

37 2k-APPROXIMATION By union bound, the probability that a particular column c is safe when considered under π is at least 1 k 2α.

38 2k-APPROXIMATION By union bound, the probability that a particular column c is safe when considered under π is at least 1 k. 2α The probability of probing c is atleast yc (1 k ). α 2α

39 2k-APPROXIMATION By union bound, the probability that a particular column c is safe when considered under π is at least 1 k. 2α The probability of probing c is atleast yc (1 k ). α 2α By linearity of expectation, the expected profit is at least 1 (1 k ) n α 2α c=1 wcyc.

40 2k-APPROXIMATION By union bound, the probability that a particular column c is safe when considered under π is at least 1 k. 2α The probability of probing c is atleast yc (1 k ). α 2α By linearity of expectation, the expected profit is at least 1 (1 k ) n α 2α c=1 wcyc. Setting α = k implies an approximation ratio of 2k.

41 RELATED WORK Stochastic Knapsack Dean, Goemans, and Vondrák [2008] A non-adaptive 4 approximation to the optimal adaptive A adaptive (3 + ε) approximation to the optimal adaptive

42 RELATED WORK Stochastic Knapsack Dean, Goemans, and Vondrák [2008] A non-adaptive 4 approximation to the optimal adaptive A adaptive (3 + ε) approximation to the optimal adaptive Stochastic k-set Packing Bansal et al. [2010] 2k approximation in the general case k + 1 approximation in the monotone outcome case

43 RELATED WORK Stochastic Knapsack Dean, Goemans, and Vondrák [2008] A non-adaptive 4 approximation to the optimal adaptive A adaptive (3 + ε) approximation to the optimal adaptive Stochastic k-set Packing Bansal et al. [2010] 2k approximation in the general case k + 1 approximation in the monotone outcome case Stochastic Matching in graphs (with patience constraints) Bansal et al. [2010] 3 approximation for bipartite graphs 4 approximation for general graphs Matching in k-uniform hypergraph Chan and Lau [2012] (k 1 + 1/k) approximation in the deterministic case The standard LP has an integrality gap of (k 1 + 1/k) Füredi, Kahn, and Seymour [1993]

44 REFERENCES Nikhil Bansal, Anupam Gupta, Jian Li, Julián Mestre, Viswanath Nagarajan, and Atri Rudra. When lp is the cure for your matching woes: Improved bounds for stochastic matchings. In Algorithms ESA 2010, pages Springer, Yuk Hei Chan and Lap Chi Lau. On linear and semidefinite programming relaxations for hypergraph matching. Mathematical programming, 135(1-2): , Ning Chen, Nicole Immorlica, Anna R Karlin, Mohammad Mahdian, and Atri Rudra. Approximating matches made in heaven. Automata, Languages and Programming, pages , Brian C Dean, Michel X Goemans, and Jan Vondrák. Adaptivity and approximation for stochastic packing problems. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages Society for Industrial and Applied Mathematics, Brian C Dean, Michel X Goemans, and Jan Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. Mathematics of Operations Research, 33(4): , Zoltán Füredi, Jeff Kahn, and Paul D. Seymour. On the fractional matching polytope of a hypergraph. Combinatorica, 13(2): , 1993.