Price of dependence: stochastic submodular maximization with dependent items
In this paper, we study the stochastic submodular maximization problem with dependent items subject to downward-closed and prefix-closed constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are...
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| Published in: | Journal of combinatorial optimization Vol. 39; no. 2; pp. 305 - 314 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
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Springer US
01.02.2020
Springer Nature B.V |
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| ISSN: | 1382-6905, 1573-2886 |
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| Abstract | In this paper, we study the stochastic submodular maximization problem with dependent items subject to downward-closed and prefix-closed constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are able to observe its state. We assume a monotone and submodular utility function over items and states, and our objective is to select a group of items adaptively so as to maximize the expected utility. Previous studies on stochastic submodular maximization often assume that items’ states are independent, however, this assumption may not hold in general. This motivates us to study the stochastic submodular maximization problem with dependent items. We first introduce the concept of
degree of independence
to capture the degree to which one item’s state is dependent on others’. Then we propose a non-adaptive policy that approximates the optimal adaptive policy within a factor of
α
1
-
e
-
κ
2
+
κ
18
m
2
-
κ
+
2
3
m
κ
where the value of
α
is depending on the type of constraints, e.g.,
α
=
1
for matroid constraint,
κ
is the degree of independence, e.g.,
κ
=
1
for independent items, and
m
is the number of items. We also analyze the adaptivity gap, i.e., the ratio of the values of best adaptive policy and best non-adaptive policy, of our problem with prefix-closed constraints. |
|---|---|
| AbstractList | In this paper, we study the stochastic submodular maximization problem with dependent items subject to downward-closed and prefix-closed constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are able to observe its state. We assume a monotone and submodular utility function over items and states, and our objective is to select a group of items adaptively so as to maximize the expected utility. Previous studies on stochastic submodular maximization often assume that items’ states are independent, however, this assumption may not hold in general. This motivates us to study the stochastic submodular maximization problem with dependent items. We first introduce the concept of degree of independence to capture the degree to which one item’s state is dependent on others’. Then we propose a non-adaptive policy that approximates the optimal adaptive policy within a factor of α1-e-κ2+κ18m2-κ+23mκ where the value of α is depending on the type of constraints, e.g., α=1 for matroid constraint, κ is the degree of independence, e.g., κ=1 for independent items, and m is the number of items. We also analyze the adaptivity gap, i.e., the ratio of the values of best adaptive policy and best non-adaptive policy, of our problem with prefix-closed constraints. In this paper, we study the stochastic submodular maximization problem with dependent items subject to downward-closed and prefix-closed constraints. The input of our problem is a finite set of items, and each item is in a particular state from a set of possible states. After picking an item, we are able to observe its state. We assume a monotone and submodular utility function over items and states, and our objective is to select a group of items adaptively so as to maximize the expected utility. Previous studies on stochastic submodular maximization often assume that items’ states are independent, however, this assumption may not hold in general. This motivates us to study the stochastic submodular maximization problem with dependent items. We first introduce the concept of degree of independence to capture the degree to which one item’s state is dependent on others’. Then we propose a non-adaptive policy that approximates the optimal adaptive policy within a factor of α 1 - e - κ 2 + κ 18 m 2 - κ + 2 3 m κ where the value of α is depending on the type of constraints, e.g., α = 1 for matroid constraint, κ is the degree of independence, e.g., κ = 1 for independent items, and m is the number of items. We also analyze the adaptivity gap, i.e., the ratio of the values of best adaptive policy and best non-adaptive policy, of our problem with prefix-closed constraints. |
| Author | Tang, Shaojie |
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| References | Hellerstein, Kletenik, Lin (CR11) 2015 Asadpour, Nazerzadeh (CR3) 2015; 62 CR5 Golovin, Krause (CR9) 2011; 42 Ageev, Sviridenko (CR2) 2004; 8 CR8 Chekuri, Vondrák, Zenklusen (CR7) 2014; 43 Adamczyk, Sviridenko, Ward (CR1) 2016; 41 CR13 CR12 Calinescu, Chekuri, Pál, Vondrák (CR6) 2011; 40 CR10 Asadpour, Nazerzadeh, Saberi (CR4) 2008 D Golovin (470_CR9) 2011; 42 G Calinescu (470_CR6) 2011; 40 Arash Asadpour (470_CR4) 2008 C Chekuri (470_CR7) 2014; 43 M Adamczyk (470_CR1) 2016; 41 470_CR12 470_CR13 470_CR10 AA Ageev (470_CR2) 2004; 8 470_CR8 A Asadpour (470_CR3) 2015; 62 470_CR5 Lisa Hellerstein (470_CR11) 2015 |
| References_xml | – volume: 43 start-page: 1831 issue: 6 year: 2014 end-page: 1879 ident: CR7 article-title: Submodular function maximization via the multilinear relaxation and contention resolution schemes publication-title: SIAM J Comput doi: 10.1137/110839655 – volume: 8 start-page: 307 issue: 3 year: 2004 end-page: 328 ident: CR2 article-title: Pipage rounding: a new method of constructing algorithms with proven performance guarantee publication-title: J Comb Optim doi: 10.1023/B:JOCO.0000038913.96607.c2 – ident: CR12 – ident: CR13 – ident: CR10 – start-page: 235 year: 2015 end-page: 248 ident: CR11 article-title: Discrete Stochastic Submodular Maximization: Adaptive vs. Non-adaptive vs. Offline publication-title: Lecture Notes in Computer Science – volume: 62 start-page: 2374 issue: 8 year: 2015 end-page: 2391 ident: CR3 article-title: Maximizing stochastic monotone submodular functions publication-title: Manag Sci doi: 10.1287/mnsc.2015.2254 – volume: 40 start-page: 1740 issue: 6 year: 2011 end-page: 1766 ident: CR6 article-title: Maximizing a monotone submodular function subject to a matroid constraint publication-title: SIAM J Comput doi: 10.1137/080733991 – start-page: 477 year: 2008 end-page: 489 ident: CR4 article-title: Stochastic Submodular Maximization publication-title: Lecture Notes in Computer Science – ident: CR5 – ident: CR8 – volume: 42 start-page: 427 year: 2011 end-page: 486 ident: CR9 article-title: Adaptive submodularity: theory and applications in active learning and stochastic optimization publication-title: J Artif Intell Res – volume: 41 start-page: 1022 issue: 3 year: 2016 end-page: 1038 ident: CR1 article-title: Submodular stochastic probing on matroids publication-title: Math Oper Res doi: 10.1287/moor.2015.0766 – ident: 470_CR8 – volume: 43 start-page: 1831 issue: 6 year: 2014 ident: 470_CR7 publication-title: SIAM J Comput doi: 10.1137/110839655 – start-page: 235 volume-title: Lecture Notes in Computer Science year: 2015 ident: 470_CR11 – volume: 42 start-page: 427 year: 2011 ident: 470_CR9 publication-title: J Artif Intell Res – ident: 470_CR10 doi: 10.1137/1.9781611974782.111 – volume: 41 start-page: 1022 issue: 3 year: 2016 ident: 470_CR1 publication-title: Math Oper Res doi: 10.1287/moor.2015.0766 – volume: 8 start-page: 307 issue: 3 year: 2004 ident: 470_CR2 publication-title: J Comb Optim doi: 10.1023/B:JOCO.0000038913.96607.c2 – volume: 62 start-page: 2374 issue: 8 year: 2015 ident: 470_CR3 publication-title: Manag Sci doi: 10.1287/mnsc.2015.2254 – start-page: 477 volume-title: Lecture Notes in Computer Science year: 2008 ident: 470_CR4 – ident: 470_CR5 – ident: 470_CR13 doi: 10.24963/ijcai.2017/546 – volume: 40 start-page: 1740 issue: 6 year: 2011 ident: 470_CR6 publication-title: SIAM J Comput doi: 10.1137/080733991 – ident: 470_CR12 doi: 10.1145/3084041.3084043 |
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| SubjectTerms | Combinatorics Convex and Discrete Geometry Dependent variables Expected utility Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations Research/Decision Theory Optimization Theory of Computation |
| Title | Price of dependence: stochastic submodular maximization with dependent items |
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