Stochastic Matching in Hypergraphs Amit Chavan, Srijan Kumar and Pan - PowerPoint PPT Presentation

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References Stochastic Matching in Hypergraphs Amit Chavan, Srijan Kumar and Pan Xu May 13, 2014

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References R OADMAP I NTRODUCTION Matching Stochastic Matching B ACKGROUND Stochastic Knapsack Adaptive and Non-adaptive policies Adaptivity Gap S TOCHASTIC k - SET PACKING LP relaxation for optimal adaptive A Solution Policy Related Work

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References M ATCHING 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: http://en.wikipedia.org/wiki/File:Maximum-matching-labels.svg

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References M ATCHING Resource allocation Figure: http://www.phdcomics.com/comics/archive/phd051908s.gif

� � � � I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References � M ATCHING Stable Marriage Problem Example Charlie Charlie Franklin Schroeder Linus Linus Linus Charlie Franklin Franklin Schroeder Lucy Peppermint Marcie Sally Charlie Linus Schroeder Franklin Lucy Marcie Marcie Peppermint Peppermint Sally Lucy Sally Marcie Stable! Stable! Marcie Figure: http://cramton.umd.edu/econ415/deferred-acceptance-algorithm.pdf

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References M ATCHING Latin Square Problem Figure: http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Sudoku-by-L2G-20050714 solution.svg/250px-Sudoku-by- L2G-20050714 solution.svg.png

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References S TOCHASTIC 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].

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References W HY IS IT IMPORTANT ? Motivated by: ◮ Kidney exchange ◮ Online dating

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References W HY IS IT IMPORTANT ? - K IDNEY E XCHANGE Figure: http://www.cartoonstock.com/lowres/animals-transplant-pig-kidney-transplantation-surgeon-dro0315l.jpg

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References W HY IS IT IMPORTANT ? - O NLINE DATING Figure: http://www.cartoonstock.com/newscartoons/cartoonists/bst/lowres/dating-wrong-conversations-arguments-issues- disagree-bstn86l.jpg

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References S TOCHASTIC K NAPSACK P ROBLEM ◮ 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A DAPTIVE AND N ON - 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A DAPTIVITY G AP For an instance I , ADAPT ( I ) ADAPTIVITY-GAP ( I ) = sup NON-ADAPT ( I ) ◮ Studied by Dean, Goemans, and Vondr´ ak [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.

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC k - SET PACKING An instance I consists of

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC k - SET PACKING An instance I consists of ◮ n items/columns

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC 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 )

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References P RELIMINARIES – S TOCHASTIC 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

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References LP RELAXATION FOR OPTIMAL ADAPTIVE (B ANSAL ET AL . [2010])

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References LP RELAXATION FOR OPTIMAL ADAPTIVE (B ANSAL ET AL . [2010]) Let w i = E [ v i ] be the mean profit, for each i ∈ [ n ] .

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References LP RELAXATION FOR OPTIMAL ADAPTIVE (B ANSAL 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.

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References LP RELAXATION FOR OPTIMAL ADAPTIVE (B ANSAL 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. n � maximize w i y i (1) i = 1 n � subject to 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 .

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A S OLUTION P OLICY ◮ Let y ∗ denote an optimal solution to the linear program in 1.

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A S OLUTION P OLICY ◮ Let y ∗ denote an optimal solution to the linear program in 1. ◮ Fix a constant α ≥ 1 (to be specified later).

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A S OLUTION P OLICY ◮ 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.

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A S OLUTION P OLICY ◮ 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 .

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A S OLUTION P OLICY ◮ 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.

I NTRODUCTION B ACKGROUND S TOCHASTIC k - SET PACKING References A PPROXIMATION 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 .