arXiv:1705.04195v2 [math.OC] 10 Jun 2017
Improved Bounds for the Greedy Strategy in Optimization Problems with Curvatures Yajing Liua,∗, Edwin K. P. Chonga,b , Ali Pezeshkia,b a Department
of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA b Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
Abstract Consider the problem of choosing a set of actions to optimize an objective function f that is a real-valued polymatroid set function subject to matroid constraints. The greedy strategy provides an approximate solution to the optimization problem; the strategy starts with the empty set and iteratively adds an element to the current solution set with the largest gain in the objective function while satisfying the matroid constraints. The greedy strategy is known to achieve at least a factor of 1/(1 + c(f )) of the maximal objective function value for a general matroid and a factor of (1 − (1 − c(f )/K)K )/c for a uniform matroid with rank K, where c(f ) is the total curvature of the objective function f . (These bounds are both nonincreasing in c(f ), so a smaller curvature implies a better bound.) But in these previous results, the objective function is defined for all sets of actions, and the value of the total curvature c(f ) depends on the value of objective function on sets outside the constraint matroid. If we are given a function defined only on the matroid, the problem still makes sense, but the existing results involving the total curvature do not apply. This is puzzling: If the optimization problem is perfectly well defined, why should the bounds no longer apply? Indeed, the greedy strategy satisfies the bounds 1/(1 + d) and (1 − (1 − d/K)K )/d for a general matroid and a uniform matroid, respectively, ∗ Corresponding
author. Email addresses:
[email protected] (Yajing Liu),
[email protected] (Edwin K. P. Chong),
[email protected] (Ali Pezeshki)
Preprint submitted to Journal of Discrete Applied Mathematics
June 13, 2017
where d = inf c(g), and Ωf is the set of all polymatroid set functions g on 2X g∈Ωf
that agree with f on I. These bounds apply to problems when the objective function is defined only on the matroid. Moreover, when f is defined on the entire power set, d ≤ c(f ), which implies that the bounds are in general improved over the previous ones. We define a curvature b involving only sets in the matroid, and we prove that b(f ) ≤ c(f ) when f is defined on the entire power set. We derive necessary and sufficient conditions for the existence of an extended polymatroid set function g that agrees with f on I such that c(g) = b(f ). This
results in improved bounds 1/(1 + b(f )) and (1 − (1 − b(f )/K)K )/b(f ) for a
general matroid and a uniform matroid, respectively. Moreover, these bounds are not influenced by sets outside the matroid. Finally, we present two examples. We first provide a task scheduling problem to show that a polymatroid set function f defined on the matroid can be extended to a polymatroid set function g defined on the entire power set while satisfying the condition that c(g) = b(f ), which results in a stronger bound. Then we provide an adaptive sensing problem to show that there does not exist any extended polymatroid set function g such that c(g) = b(f ) holds. However, for our extended polymatroid set function g, it turns out that c(g) is very close to b(f ) and much smaller than c(f ), which also results in a stronger bound. Keywords: curvature, greedy, matroid, polymatroid, submodular
1. Introduction 1.1. Background Consider the problem of optimally choosing a set of actions to maximize an objective function. Let X be a finite ground set of all possible actions and f : 2X → R be an objective function defined on the power set 2X of X. The set function f is said to be a polymatroid-set function if it is submodular, nondecreasing, and f (∅) = 0 (definitions of being submodular and nondecreasing are given in Section II) [1]. Let I be a non-empty collection of subsets of the ground set X. The pair (X, I) is called a matroid if I satisfies the hereditary 2
and augmentation properties (definitions of hereditary and augmentation are introduced in Section II). The aim is to find a set in I to maximize the objective function: maximize f (M ) subject to M ∈ I.
(1)
The pair (X, I) is said to be a uniform matroid of rank K (K ≤ |X|) when I = {S ⊆ X : |S| ≤ K}, where | · | denotes cardinality. A uniform matroid is a 5
special matroid, so any result for a matroid constraint also applies to a uniform matroid constraint. Finding the optimal solution to problem (1) in general is NP-hard. The greedy strategy provides a computationally feasible approach to finding an approximate solution to (1). It starts with the empty set, and then iteratively
10
adds to the current solution set one element that results in the largest gain in the objective function, while satisfying the matroid constraints. A detailed definition of the greedy strategy is given in Section II. The performance of the greedy strategy has attracted the attention of many researchers, and some key developments will be reviewed in the following section.
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1.2. Review of Previous Work Nemhauser et al. [2], [3] proved that, when f is a polymatroid set function, the greedy strategy yields a 1/2-approximation1 for a general matroid and a (1− e−1 )-approximation for a uniform matroid. By introducing the total curvature c(f ),2 c(f ) =
f (X) − f (X \ {j}) 1 − , f ({j}) − f (∅) j∈X , f ({j})6=f (∅) max
Conforti and Cornu´ejols [4] showed that, when f is a polymatroid set function, the greedy strategy achieves a 1/(1+c)-approximation for a general matroid and a (1 − (1 − c/K)K )/c-approximation for a uniform matroid, where K is the rank 1 The
term β-approximation means that f (G)/f (O) ≥ β, where G and O denote a greedy
solution and an optimal solution, respectively. 2 When there is no ambiguity, we simply write c to denote c(f ).
3
of the uniform matroid. When K tends to infinity, the bound (1−(1−c/K)K )/c 20
tends to (1 − e−c)/c. For a polymatroid set function, the total curvature c takes values on the interval (0, 1]. In this case, we have 1/(1 + c) ≥ 1/2 and (1 − (1 −
c/K)K )/c ≥ (1 − e−c )/c ≥ (1 − e−1 ), which implies that the bounds 1/(1 + c) and (1−(1−c/K)K )/c are stronger than the bounds 1/2 and (1−e−1 ) in [2] and [3], respectively. Vondr´ ak [5] proved that for a polymatroid set function, the
25
continuous greedy strategy gives a (1 − e−c )/c-approximation for any matroid. Sviridenko et al. [6] proved that, a modified continuous greedy strategy gives a (1−ce−1 )-approximation for any matroid, the first improvement over the greedy (1 − e−c )/c-approximation of Conforti and Cornu´ejols from [4]. Suppose that the objective function f in problem (1) is a polymatroid set
30
function and the cardinality of the maximal set in the matroid (X, I) is K. By the augmentation property of a matroid and the monotoneity of f , any optimal solution can be extended to a set of size K. By the definition of the greedy strategy (see Section II), any greedy solution is of size K. For the greedy strategy, under a general matroid constraint and a uniform matroid constraint,
35
the performance bounds 1/(1 + c) and (1 − (1 − c/K)K )/c from [4] are the best so far, respectively, in terms of the total curvature c. However, the total curvature c, by definition, depends on the function values on sets outside the matroid (X, I). This gives rise to two possible issues when applying existing bounding results involving the total curvature c:
40
1. If we are given a function f defined only on I, then problem (1) still makes sense, but the total curvature is no longer well defined. This means that the existing results involving the total curvature do not apply. But this surely is puzzling: if the optimization problem (1) is perfectly well defined, why should the bound no longer apply?
45
2. Even if the function f is defined on the entire 2X , the fact that the total curvature c involves sets outside the matroid is puzzling. Specifically, if the optimization problem (1) involves only sets in the matroid, why should the bounding results rely on a quantity c that depends on sets outside the 4
matroid? 50
The two reasons above motivate us to investigate more applicable bounds involving only sets in the matroid. 1.3. Contributions In this paper, we prove that any polymatroid set function f defined only on the matroid can be extended to a polymatroid set function g defined on the
55
entire power set of the action set such that g agrees with f on the matroid. Then, it follows that for problem (1), the greedy strategy satisfies the improved bounds 1/(1 + d) and (1 − (1 − d/K)K )/d for a general matroid and a uniform matroid, respectively, where d = inf c(g), and Ωf is the set of all polymatroid g∈Ωf
set functions g on 2X that agree with f on I, i.e., g(A) = f (A) for any A ∈ I. 60
These bounds apply to problems when the objective function is defined only on the matroid. When the objective function is defined on the entire power set, d ≤ c(f ), which implies that the bounds are improved. Then, we define a curvature b involving only sets in the matroid, and we prove that b(f ) ≤ c(f ) when f is defined on the entire power set. We derive necessary and sufficient
65
conditions for the existence of an extended polymatroid set function g that agrees with f on I. This results in improved bounds 1/(1 + b(f )) and (1 −
(1 − b(f )/K)K )/b(f ) for a general matroid and a uniform matroid, respectively.
Moreover, these bounds are not influenced by sets outside the matroid. Finally, we present two examples. We first provide a task scheduling problem to show 70
that a polymatroid set function f defined on the matroid can be extended to a polymatroid set function g defined on the entire power set while satisfying the condition that c(g) = b(f ), which results in a stronger bound. Then, we provide an adaptive sensing problem to show that there does not exist any extended polymatroid set function g such that c(g) = b(f ) holds. However, for
75
our extended polymatroid set function g, it turns out that c(g) is very close to b(f ) and much smaller that c(f ), which also results in a stronger bound.
5
1.4. Organization In Section II, we first introduce definitions of polymatroid set functions, matroids, and curvature. Then, we review performance bounds in terms of the 80
total curvature c from [4]. In Section III, we prove that any polymatroid set function defined only on the matroid can be extended to a polymatroid set function defined on the entire power set. We obtain bounds which apply to problems when the objective function is defined only on the matroid, and these bounds are stronger than those bounds on the entire power set. We define a
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curvature b involving only sets in the matroid and obtain improved bounds in terms of b subject to necessary and sufficient conditions. In Section IV, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.
2. Preliminaries 90
2.1. Polymatroid Set Functions and Curvature The definitions and terminology in this paragraph are standard (see, e.g., [7], [8]), but are included for completeness. Let X be a finite ground set of actions, and I be a non-empty collection of subsets of X. Given a pair (X, I), the collection I is said to be hereditary if it satisfies property i below and has
95
the augmentation property if it satisfies property ii below: i. (Hereditary) For all B ∈ I, any set A ⊆ B is also in I. ii. (Augmentation) For any A, B ∈ I, if |B| > |A|, then there exists j ∈ B \ A such that A ∪ {j} ∈ I. The pair (X, I) is called a matroid if it satisfies both properties i and ii. The
100
pair (X, I) is called a uniform matroid when I = {S ⊆ X : |S| ≤ K} for a given K, called the rank of (X, I).
Let 2X denote the power set of X, and define a set function f : 2X → R.
The set function f is said to be nondecreasing and submodular if it satisfies properties 1 and 2 below, respectively: 6
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1. (Nondecreasing) For any A ⊆ B ⊆ X, f (A) ≤ f (B). 2. (Submodular) For any A ⊆ B ⊆ X and j ∈ X \ B, f (A ∪ {j}) − f (A) ≥ f (B ∪ {j}) − f (B). A set function f : 2X → R is called a polymatroid set function [1] if it is nondecreasing, submodular, and f (∅) = 0, where ∅ denotes the empty set. The
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submodularity in property 2 means that the additional value accruing from an extra action decreases as the size of the input set increases. This property is also called the diminishing-return property in economics. The total curvature of a set function f is defined as [4] f (X) − f (X \ {j}) 1 − c(f ) = max . j∈X ∗ ,f ({j})6=f (∅ f ({j}) − f (∅)
(2)
For convenience, we use c to denote c(f ) when there is no ambiguity. Note that
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0 ≤ c ≤ 1 when f is a polymatroid set function, and c = 0 if and only if f is P additive, i.e., for any set A ⊆ X, f (A) = i∈A f ({i}). When c = 0, it is easy to check that the greedy strategy coincides with the optimal strategy. So in the rest of the paper, when we assume that f is a polymatroid set function, we only consider c ∈ (0, 1]. 2.2. Performance Bounds in Terms of Total Curvature
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In this section, we review two theorems from [4], which bound the performance of the greedy strategy using the total curvature c for general matroid constraints and uniform matroid constraints. We will use these two theorems to derive bounds in Section III. We first define optimal and greedy solutions for (1) as follows: Optimal solution: A set O is called an optimal solution of (1) if O ∈ argmax f (M ), M∈I
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where the right-hand side denotes the collection of arguments that maximize f (·) on I. Note that there may exist more than one optimal solution for problem (1). When (X, I) is a matroid with cardinality K of the maximal set, then any 7
optimal solution can be extended to a set of size K because of the augmentation property of the matroid and the monotoneity of the set function f . Greedy solution: A set G = {g1 , g2 , . . . , gK } is called a greedy solution of (1) if g1 ∈ argmax f ({g}), {g}∈I
and for i = 2, . . . , K, gi ∈ 130
argmax
f ({g1 , g2 , . . . , gi−1 , g}).
g∈X {g1 ,...,gi−1 ,g}∈I
Note that there may exist more than one greedy solution for problem (1). Theorem 1. [4] Let (X, I) be a matroid and f : 2X → R be a polymatroid set function with total curvature c. Then, any greedy solution G satisfies f (G) 1 ≥ , f (O) 1+c where O is any optimal solution to problem (1). When f is a polymatroid set function, we have c ∈ (0, 1], and therefore 1/(1 + c) ∈ [1/2, 1). Theorem 1 applies to any matroid. This means that the bound 1/(1 + c) holds for a uniform matroid too. Theorem 2 below provides a
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tighter bound when (X, I) is a uniform matroid. Theorem 2. [4] Let (X, I) be a uniform matroid of rank K and f : 2X → R be a polymatroid set function with total curvature c. Then, any greedy solution G satisfies f (G) 1 ≥ f (O) c
c K 1 1− 1− ≥ 1 − e−c , K c
where O is any optimal solution to problem (1).
The function (1 − (1 − c/K)K )/c is nonincreasing in K for c ∈ (0, 1] and
(1−(1−c/K)K )/c → (1−e−c)/c when K → ∞; therefore, (1−(1−c/K)K )/c ≥
(1 − e−c )/c when f is a polymatroid set function. Also it is easy to check that 8
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(1 − e−c )/c ≥ 1/(1 + c) for c ∈ (0, 1], which implies that the bound (1 − (1 − c/K)K )/c is stronger than the bound 1/(1 + c) in Theorem 1.
The bounds in Theorems 1 and 2 involve sets not in the matroid, so they do not apply to optimization problems whose objective function is only defined for sets in the matroid. In the following section, we will provide more applicable 145
bounds for general matroid constraints and uniform matriod constraints, which involve only sets in the matroid, and are stronger than the bounds in Theorems 1 and 2, respectively.
3. Main Results 3.1. Function Extension 150
We first prove in the following theorem that any polymatroid set function defined for sets on the matroid (X, I) can be extended to a polymatroid set function defined for any subset of the ground set. Theorem 3. Let (X, I) be a matroid and f : I → R be a polymatroid set
function on I. Then there exists a set function g : 2X → R satisfying the 155
following conditions: 1. g(A) = f (A) for all A ∈ I. 2. g is a polymatroid set function on 2X . Proof. Condition 1 can be satisfied by construction: first set g(A) = f (A)
(3)
for all A ∈ I. To complete the proof of the theorem, we prove the following statement by induction: 160
For any subset Al = {a1 , . . . , al } ∈ / I (1 ≤ l ≤ |X|), there exists g(Al ) satisfying the condition that g is a polymatroid set function for sets of size up to l. First, we prove that the above statement holds for l = 1. For g to be a polymatroid set function for sets of size up to 1, it suffices to have that g(∅) = 0 9
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and g(A1 ) ≥ 0 for any A1 = {a1 } ∈ / I. By the hereditary property of I, we have that ∅ ∈ I. Because f is a polymatroid set function on I and g(A) = f (A) for any A ∈ I, we have that g(∅) = f (∅) = 0. For g(A1 ), A1 ∈ / I, it suffices to pick any nonnegative number. So we conclude that the above statement holds for l = 1.
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Assume that the above statement holds for l = k. We prove that the above statement also holds for l = k + 1. By the assumption that the above statement holds for l = k and (3), we have that for any set Ak with |Ak | = k, g(Ak ) is well defined. Consider any set Ak+1 = {a1 , . . . , ak+1 } ∈ / I, and let g(Ak+1 ) = g(Ak ) + dAk+1 (ak+1 ),
(4)
where Ak = {a1 , . . . , ak }. We will prove that there exists dAk+1 (ak+1 ) such that g is a polymatroid set function for sets of size up to k + 1. For g to be a polymatroid set function for sets of size up to k + 1, it requires that g(∅) = 0 (already proved), and that g is nondecreasing and submodular for sets of size up to k + 1. For g to be nondecreasing, it suffices to show that g(C) ≤ g(Ak+1 )
(5)
for any Ak+1 ∈ / I and C ⊂ Ak+1 with |C| = k (for Ak+1 ∈ I, (5) holds based on (3) and that f is a polymatroid set function on I). For g to be submodular, it suffices to show that g(C) − g(C \ {ai }) ≥ g(Ak+1 ) − g(Ak+1 \ {ai })
(6)
for any Ak+1 ∈ / I and ai ∈ C ⊂ Ak+1 with |C| = k (for Ak+1 ∈ I, (6) holds 175
based on (3) and that f is a polymatroid set function on I). By (4), (5) reduces to dAk+1 (ak+1 ) ≥ g(C) − g(Ak ).
(7)
Substituting (4) into (6) yields dAk+1 (ak+1 ) ≤ g(C) − g(Ak ) + g(Ak+1 \ {ai }) − g(C \ {ai }). 10
(8)
From the argument above, to show that g is a polymatroid set function for sets of size up to k + 1, it suffices that dAk+1 (ak+1 ) satisfy (7) and (8). To prove that there exists dAk+1 (ak+1 ) satisfying (7) and (8), we need to prove that for any ai ∈ C ⊂ Ak+1 with |C| = k, g(C) − g(Ak ) ≤ g(C) − g(Ak ) + g(Ak+1 \ {ai }) − g(C \ {ai }).
(9)
Because |Ak+1 | = k + 1, C ⊂ Ak+1 , and |C| = k, we have that |Ak+1 \ {ai }| ≤ k, |C \ {ai }| ≤ k, and C \ {ai } ⊂ Ak+1 \ {ai }. By the assumption that g is a polymatroid set function for sets of size up to k, we have that g(Ak+1 \ {ai }) − g(C \ {ai }) ≥ 0, which implies that (9) holds. 180
Thus, we have that for any Ak+1 ∈ / I with |Ak+1 | = k + 1, there exists dAk+1 (ak+1 ) such that g(Ak+1 ) = g(Ak ) + dAk+1 (ak+1 ) and g is a polymatroid set function for sets of size up to k + 1, which implies that the above statement holds for l = k + 1. This completes the proof.
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Let f : 2X → R be a polymatroid set function. Note that f : 2X → R is
itself an extension of f from I to the entire 2X , and the extended f : 2X → R
is a polymatroid set function on the entire 2X . Therefore, the above theorem
gives rise to the following corollary. Corollary 1. Let (X, I) be a matroid and f : 2X → R a polymatroid set 190
function on 2X . Then c(f ) ≥ inf c(g), where Ωf is the set of all polymatroid g∈Ωf
set functions g on 2X that agree with f on I. 3.2. Improved Bounds
In this section, we will prove that for problem (1), if we set d = inf c(g), the g∈Ωf
greedy strategy yields a 1/(1+d)-approximation and a (1−e−d )/d-approximation 195
under a general matroid and a uniform matroid constraint, respectively. The proofs in this section are straightforward, but are included for completeness. Lemma 1. Let (X, I) be a matroid with maximal set cardinality K and g a set function satisfying the conditions in Theorem 3. Then any greedy solution and 11
optimal solution to the following optimization problem maximize g(M )
(10)
subject to M ∈ I
are also a greedy solution and an optimal solution to problem (1), respectively. Proof. Based on the assumption that (X, I) is a matroid with maximal set cardinality K, there exists a greedy solution of size K for problem (10), denoted 200
as G. Because g(A) = f (A) for any A ∈ I, we have that G is also a greedy solution to problem (1) and g(G) = f (G). Assume that O is an optimal solution to problem (10); then we have that O ∈ I and g(M ) ≤ g(O) for any M ∈ I. Because g(A) = f (A) for any A ∈ I, we have that f (M ) ≤ f (O) for any set M ∈ I, which implies that O is also an optimal solution for problem (1). Theorem 4. Let (X, I) be a matroid with maximal set cardinality K. Then, any greedy solution G to problem (1) satisfies 1 f (G) ≥ , f (O) 1+d
(11)
where O is any optimal solution to problem (1) and d = inf c(g). In particular, g∈Ωf
when (X, I) is a uniform matroid, any greedy solution G to problem (1) satisfies K ! d 1 1 f (G) 1− 1− ≥ (12) 1 − e−d . ≥ f (O) d K d Proof. By Theorems 1 and 2, for any g satisfying the conditions in Theorem 3, we have the following inequalities g(G) 1 ≥ g(O) 1 + c(g) and g(G) 1 ≥ g(O) c(g) 205
K ! c(g) 1 1− 1− ≥ 1 − e−c(g) K c(g)
for problem (10). By Lemma 1, we have that f (G) = g(G) and f (O) = g(O). Thus, (11) and (12) hold for problem (1).
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Remark 1. Because the functions 1/(1 + x), (1 − (1 − x/K)K )/x, and (1 −
e−x )/x are all nonincreasing in x for x ∈ (0, 1] and from Corollary 1 we have
0 < d ≤ c(f ) ≤ 1 when f is defined on the entire power set, we have that
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1/(1 + d) ≥ 1/(1 + c(f )), ((1 − (1 − d/K)K )/d ≥ (1 − (1 − c(f )/K)K )/c(f ),
and (1 − e−d )/d ≥ (1 − e−c(f ) )/c(f ). This implies that our new bounds are, in general, stronger than the previous bounds.
Remark 2. The bounds 1/(1 + d) and (1 − e−d )/d apply to problems when the objective function is defined only for sets in the matroid. Now we define a notion of curvature that only involves sets in the matroid. Let h : I → R be a set function. We define the curvature b(h) as follows: h(A) − h(A \ {j}) 1− b(h) = max . (13) h({j}) − h(∅) j∈A∈I,h({j})6=h(∅) 215
For convenience, we use b to denote b(h) when there is no ambiguity. Note that 0 ≤ b ≤ 1 when h is a polymatroid set function on the matroid (X, I), and b = 0 if and only if h is additive for sets in I. When b = 0, the greedy solution to problem (1) coincides with the optimal solution, so we only consider b ∈ (0, 1] in the rest of the paper. For any extension of f : I → R to g : 2X → R , we
220
have that c(g) ≥ b(f ), which will be proved in the following theorem. Theorem 5. Let (X, I) be a matroid. Let f : I → R be a polymatroid set func-
tion. Let g be an extension of f to 2X satisfying the conditions in Theorem 3. Then b(f ) ≤ c(g). Proof. By (13), we have that for any j ∈ A ∈ I, 1−
f (A) − f (A \ {j}) ≤ b(f ). f ({j}) − f (∅)
By submodularity of g and g(A) = f (A) for any j ∈ A ∈ I, we have that f (A) − f (A \ {j}) ≥ g(X) − g(X \ {j}), which implies that for any j ∈ A ∈ I, 1−
f (A) − f (A \ {j}) g(X) − g(X \ {j}) ≤1− . f ({j}) − f (∅) g({j}) − g(∅)
Hence, by (2) and (13), we have b(f ) ≤ c(g). 13
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So far, we have obtained bounds that apply to problems when the objective function is defined only in the matroid. But the bounds still depend on sets not in the matroid. By the definition of the curvature b, we have that b(f ) does not depend on sets outside the matroid. So if there exists an extension of f to g such that c(g) = b(f ) holds, then we can derive bounds that are not influenced
230
by sets outside the matroid. However, it turns out that there does not always exist a g such that c(g) = b(f ); we will give an example in Section 4.2 to show this. In the following theorem, we provide necessary and sufficient conditions for the existence of g such that c(g) = b(f ). Theorem 6. Let (X, I) be a matroid and f : I → R a polymatroid set function.
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Let g be an extension of f satisfying the conditions in Theorem 3. Then c(g) = b(f ) if and only if g(X) − g(X \ {aj }) ≥ (1 − b(f ))g({aj }) for any aj ∈ X and equality holds for some aj ∈ X. Proof. → In this direction, we assume that c(g) = b(f ) and then to prove that g(X) − g(X \ {aj }) ≥ (1 − b(f ))g({aj }) for any aj ∈ X and equality holds for some aj ∈ X. By the definition of the total curvature c of g and c(g) = b(f ), we have for any aj ∈ X, g(X) − g(X \ {aj }) ≥ (1 − b(f ))g({aj }), and equality holds for some aj ∈ X.
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← Now we assume that g(X) − g(X \ {aj }) ≥ (1 − b(f ))g({aj }) for any aj ∈ X and equality holds for some aj ∈ X, then try to prove that c(g) = b(f ). By the assumptions, we have 1−
g(X) − g(X \ {aj }) ≤ b(f ) g({aj }) − g(∅)
for any aj ∈ X, and equality holds for some aj ∈ X. By the definition of the total curvature c of g, we have c(g) =
g(X) − g(X \ {aj }) 1 − = b(f ). aj ∈X , g({aj })6=g(∅) g({aj }) − g(∅) max
14
This completes the proof. Combining Theorems 4 and 6, we have the following corollary. Corollary 2. Let (X, I) be a matroid with maximal set cardinality K. Let g be an extension of f satisfying the conditions in Theorem 3 and g(X) − g(X \ {aj }) ≥ (1 − b(f ))g({aj }) for any aj ∈ X. Then, any greedy solution G to problem (1) satisfies f (G) 1 ≥ , f (O) 1 + b(f )
(14)
where O is any optimal solution to problem (1). In particular, when (X, I) is a uniform matroid, any greedy solution G to problem (1) satisfies K ! b(f ) 1 1 f (G) 1− 1− ≥ 1 − e−b(f ) . ≥ f (O) b(f ) K b(f )
(15)
K
The bounds 1/(1 + b(f )) and (1−(1 − b(f )/K) )/b(f ) are not influenced by sets outside the matroid, so they apply to problems when the objective function 245
is only defined on the matroid. When f is defined on the entire power set, from Theorem 5, we have b(f ) ≤ c(f ), which implies that the bounds are stronger than those from [4].
4. Examples In this section, we first provide an example to extend f : I → R to a
250
polymatroid set function g : 2X → R that agrees with f on I and satisfies the condition that c(g) = b(f ). Then we provide an example to show that there does not exist any extension of f to g such that c(g) = b(f ) holds. However, in this example, it turns out that for our extension g, c(g) is very close to b(f ) and much smaller that c(f ). These two examples both show our bounds are
255
stronger than the previous bounds from [4]. 4.1. Task Scheduling As a canonical example of problem (1), we will consider the task assignment problem that was posed in [9], and was further analyzed in [10]–[12]. In this 15
problem, there are n subtasks and a set X of N agents aj (j = 1, . . . , N ). At 260
each stage, a subtask i is assigned to an agent aj , who successfully accomplishes the task with probability pi (aj ). Let Xi (a1 , a2 , . . . , ak ) denote the Bernoulli random variable that describes whether or not subtask i has been accomplished
265
after performing the sequence of actions a1 , a2 , . . . , ak over k stages. Then Pn 1 i=1 Xi (a1 , a2 , . . . , ak ) is the fraction of subtasks accomplished after k stages n by employing agents a1 , a2 , . . . , ak . The objective function f for this problem is
the expected value of this fraction, which can be written as k n Y X 1 1 − (1 − pi (aj )) . f ({a1 , . . . , ak }) = n i=1 j=1
Assume that pi (a) > 0 for any a ∈ X; then it is easy to check that f is non-decreasing. Therefore, when I = {S ⊆ X : |S| ≤ K}, the solution to this problem should be of size K. Also, it is easy to check that the function f has 270
the submodular property. For convenience, we only consider the special case n = 1; our analysis can be generalized to any n ≥ 2. In this case, we have f ({a1 , . . . , ak }) = 1 −
k Y
j=1
(1 − p(aj )),
(16)
where p(·) = p1 (·). Let X = {a1 , a2 , a3 , a4 }, p({a1 }) = 0.4, p({a2 }) = 0.6, p({a3 }) = 0.8, and p({a4 }) = 0.9. Then, f (A) is defined as in (16) for any A = {ai , . . . , ak } ⊆ X. Consider K = 2, then I = {S ⊆ X : |S| ≤ 2}. 275
We now construct a set function g : 2X → R satisfying the conditions that g
is a polymatroid set function on 2X , g(A) = f (A) for any A ∈ I, and c(g) = b(f ) given that f : I → R is a polymatroid set function. Then, we prove that the greedy strategy for the task assignment problem achieves a better bound than the previous result.
16
280
Define g as
g(A) =
f (A)
|A| ≤ 2
(17)
g(B) + dA (A \ B) |A| > 2,
where B ⊆ A with |B| = |A| − 1 and dA (A \ B) is defined as below to satisfy
the conditions that g is a polymatroid set function on 2X and c(g) = b(f ). By the definition of the curvature b of f , we have f (A) − f (A \ {j}) 1− b(f ) = max f ({j}) − f (∅) j∈A⊆X,|A|=2, f ({j})6=0
=
f ({ai , aj }) − f ({ai }) 1− f ({aj }) {ai ,aj }⊆X max
= max {p({ai })} = 0.9. ai ∈X
First, we will define d{a1 ,a2 ,a3 } ({a3 }). By inequality (8), it suffices to have that d{a1 ,a2 ,a3 } ({a3 }) ≤ min{f ({a1 , a3 }) − f ({a1 }), f ({a2 , a3 }) − f ({a2 }), f ({a1 , a3 }) − f ({a1 , a2 }) + f ({a2 , a3 }) − f ({a3 })}. which is d{a1 ,a2 ,a3 } ({a3 }) ≤ min{0.48, 0.32, 0.24} = 0.24.
(18)
By Theorem 6, if we want to achieve c(g) = b(f ), it suffices to have that d{a1 ,a2 ,a3 } ({a3 }) ≥ max{(1 − b)f ({a3 }), f ({a2 , a3 }) − f ({a1 , a2 }) + (1 − b)f ({a1 }), f ({a1 , a3 }) − f ({a1 , a2 }) + (1 − b)f ({a2 })}, which is d{a1 ,a2 ,a3 } ({a3 }) ≥ max{0.08, 0.2, 0.18} = 0.2.
(19)
Setting d{a1 ,a2 ,a3 } ({a3 }) = 0.2 satisfies (18) and (19). This gives g({a1 , a2 , a3 }) = g({a1 , a2 }) + d{a1 ,a2 ,a3 } ({a3 }) = 0.96. 17
Similarly, we set g({a1 , a2 , a4 }) = f ({a1 , a2 }) + d{a1 ,a2 ,a4 } ({a4 }) = 1, g({a1 , a3 , a4 }) = f ({a1 , a3 }) + d{a1 ,a3 ,a4 } ({a4 }) = 1.02, g({a2 , a3 , a4 }) = f ({a2 , a3 }) + d{a2 ,a3 ,a4 } ({a4 }) = 1.04. We now define d{a1 ,a2 ,a3 ,a4 } ({a4 }). By inequality (8), it suffices to have that d{a1 ,a2 ,a3 ,a4 } ({a4 }) ≤ min{g({a1 , a2 , a4 }) − f ({a1 , a2 }), g({a1 , a3 , a4 }) − f ({a1 , a3 }), g({a2 , a3 , a4 }) − f ({a2 , a3 }), g({a1 , a2 , a4 }) − g({a1 , a2 , a3 }) + g({a2 , a3 , a4 }) − f ({a2 , a4 }), g({a1 , a3 , a4 }) − g({a1 , a2 , a3 }) + g({a2 , a3 , a4 }) − f ({a3 , a4 }), g({a1 , a2 , a4 }) − g({a1 , a2 , a3 }) + g({a1 , a3 , a4 }) − f ({a1 , a4 })}, which is d{a1 ,a2 ,a3 ,a4 } ({a4 }) ≤ min{0.24, 0.14, 0.12, 0.12, 0.12, 0.12} = 0.12.
(20)
By Theorem 6, if we want to achieve c(g) = b(f ), it suffices to have that d{a1 ,a2 ,a3 ,a4 } ({a3 }) ≥ max{(1 − b)f ({a4 }), g({a2 , a3 , a4 }) − g({a1 , a2 , a3 }) + (1 − b)f ({a1 }), g({a1 , a3 , a4 }) − g({a1 , a2 , a3 }) + (1 − b)f ({a2 }), g({a1 , a2 , a4 }) − g({a1 , a2 , a3 }) + (1 − b)f ({a3 })}, which is d{a1 ,a2 ,a3 ,a4 } ({a4 }) ≥ max{0.09, 0.12, 0.12, 0.12} = 0.12. 285
(21)
Setting d{a1 ,a2 ,a3 ,a4 } ({a4 }) = 0.12 satisfies (20) and (21). This gives us g({a1 , a2 , a3 , a4 }) = g({a1 , a2 , a3 }) + d{a1 ,a2 ,a3 ,a4 } ({a4 }) = 1.08. We have now defined g({a1 , a2 , a3 }), g({a1 , a2 , a4 }), g({a1 , a3 , a4 }), g({a2 ,
a3 , a4 }), and g({a1 , a2 , a3 , a4 }). Then, the total curvature c of g : 2X → R is g({a1 , a2 , a3 , a4 }) − g({a1 , a2 , a3 , a4 } \ {ai }) = 0.9 = b(f ). c(g) = max 1 − ai ∈X g({ai }) − g(∅) 18
In contrast, the total curvature c of f is f ({a1 , a2 , a3 , a4 }) − f ({a1 , a2 , a3 , a4 } \ {ai }) c(f ) = max 1 − ai ∈X f ({ai }) − f (∅) =
{1 − (1 − p({ai })) (1 − p({aj }))(1 − p({ak }))}
max
{ai ,aj ,ak }⊂X
= 0.992 > c(g). By Corollary 2, we have that the greedy strategy for the task scheduling problem satisfies the bound (1 − (1 − b(f )/2)2 )/b(f ) = 0.775, which is better than the
previous bound (1 − (1 − c(f )/2)2 )/c(f ) = 0.752. 290
4.2. Adaptive Sensing For our second example, we consider the adaptive sensing design problem posed in [10]–[12]. Consider a signal of interest x ∈ IR2 with normal prior distribution N (0, I), where I is the 2 × 2 identity matrix; our analysis easily √ √ generalizes to dimensions larger than 2. Let A = {Diag( α, 1 − α) : α ∈ {α1 , . . . , αN }}, where α ∈ [0.5, 1] for 1 ≤ i ≤ N . At each stage i, we make a measurement yi of the form yi = ai x + wi , where ai ∈ A and wi represents i.i.d. Gaussian measurement noise with mean zero and covariance I, independent of x. The objective function f for this problem is the information gain, which can be written as [13] f ({a1 , . . . , ak }) = H0 − Hk , Here, H0 =
N 2 log(2πe)
(22)
is the entropy of the prior distribution of x and Hk is
the entropy of the posterior distribution of x given {yi }ki=1 ; that is, Hk = −1 where Pk = Pk−1 + aTk ak
N 1 log det(Pk ) + log(2πe), 2 2
−1
is the posterior covariance of x given {yi }ki=1 .
∗ The objective is to choose a set of measurement matrices {a∗i }K i=1 , ai ∈ A, to 295
maximize the information gain f ({a1 , . . . , aK }) = H0 − HK . It is easy to check 19
that f is nondecreasing, submodular, and f (∅) = 0; i.e., f is a polymatroid set function. Let X = {a1 , a2 , a3 }, α1 = 0.5, α2 = 0.6, and α3 = 0.8. Then, f (A) is defined as in (22) for any A = {ai , . . . , ak } ⊆ X. Consider K = 2, then 300
I = {S ⊆ X : |S| ≤ 2}.
We now construct a set function g : 2X → R satisfying the conditions that
g is a polymatroid set function on 2X , g(A) = f (A) for any A ∈ I given that f : I → R is a polymatroid set function. Then, we show that there does not exist a polymatroid set function g such that c(g) = b(f ). However, for the 305
polymatroid set function g we construct, it turns out that c(g) is very close to b(f ) and much smaller than c(f ). Define g as g(X) = g({a1 , a2 }) + dX ({a3 }).
(23)
By the definition of the curvature b of f , we have f (A) − f (A \ {j}) 1− b(f ) = max f ({j}) − f (∅) j∈A⊆X,|A|=2, f ({j})6=0
f ({ai , aj }) − f ({ai }) 1− = max {ai ,aj }⊆X f ({aj }) = 0.3001. Now we define dX ({a3 }). By inequality (8), it suffices to have that dX ({a3 }) ≤ min{f ({a1 , a3 }) − f ({a1 }), f ({a2 , a3 }) − f ({a2 }), f ({a1 , a3 }) − f ({a1 , a2 }) + f ({a2 , a3 }) − f ({a3 })}. which is dX ({a3 }) ≤ min{log
√ √ √ √ 1.7378, log 1.7143, log 1.6799} = log 1.6799. (24)
By Theorem 6, if we want to achieve c(g) = b(f ), it suffices to have that dX ({a3 }) ≥ max{(1 − b(f ))f ({a3 }), f ({a2 , a3 }) − f ({a1 , a2 }) + (1 − b(f ))f ({a1 }), f ({a1 , a3 }) − f ({a1 , a2 }) + (1 − b(f ))f ({a2 })}, 20
which is dX ({a3 }) ≥ max{log
√ √ √ √ 1.7143, log 1.6977, log 1.7232} = log 1.7232. (25)
Comparing (24) and (25), we have that there does not exist dX ({a3 }) such that g is a polymatroid set function and c(g) = b(f )(i.e., the necessary and sufficient condition in Theorem 6 fails). Now we just define dX ({a3 }) to satisfy (24), √ which makes g a polymatroid set function on 2X . Let dX ({a3 }) = log 1.6799, √ then we have g(X) = log 6.7028. Then the total curvature of g is g({a1 , a2 , a3 }) − g({a1 , a2 , a3 } \ {ai }) c(g) = max 1 − ai ∈X g({ai }) − g(∅) = 0.3317. In contrast, the total curvature c of f is f ({a1 , a2 , a3 }) − f ({a1 , a2 , a3 } \ {ai }) 1 − c(f ) = max ai ∈X f ({ai }) − f (∅) = 0.4509 > c(g). By Theorem 4, we have that the greedy strategy for the adaptive sensing problem satisfies the bound (1 − (1 − c(g)/2)2 )/c(g) = 0.9172, which is stronger than
the previous bound (1 − (1 − c(f )/2)2 )/c(f ) = 0.8873. 310
5. Conclusion We considered the optimization problem (1), which is to choose a set of actions to optimize an objective function f that is a polymatroid set function, while satisfying the matroid constraints. We reviewed some previous results on performance bounds of the greedy strategy. The best prior performance
315
bounds in terms of the total curcature c(f ) for a general matroid and a uniform matroid with rank K are 1/(1 + c(f )) and (1 − (1 − c(f )/K)K )/c(f ) from [4], where the total curvature c(f ) depends on the values of the objective function on sets outside the matroid. In this paper, we proved that any polymatroid set function f defined only on the matroid can be extended to a polymatroid 21
320
set function g defined on the entire power set of the action set that agrees with f on the matroid. Indeed, for problem (1), the greedy strategy satisfies the bounds 1/(1 + d) and (1 − (1 − d/K)K )/d for a general matroid and a uniform matroid, respectively, where d = inf c(g), and Ωf is the set of all g∈Ωf
polymatroid set functions g on 2X that agree with f on I. These bounds 325
apply to problems when the objective function is defined only on the matroid. When the objective function is defined on the entire power set, d ≤ c(f ), which implies that our bounds are stronger than the previous ones. We defined a curvature b involving only sets in the matroid, and we proved that b(f ) ≤ c(f ) when f is defined on the entire power set. We derived necessary and
330
sufficient conditions for the existence of an extended polymatroid set function g that agrees with f on I such that c(g) = b(f ). This results in improved
bounds 1/(1 + b(f )) and (1 − (1 − b(f )/K)K )/b(f ) for a general matroid and a uniform matroid, respectively. Moreover, these bounds are not influenced by
sets outside the matroid. Finally, we presented two examples. We first provided 335
a task scheduling problem to show that a polymatroid set function f defined on the matroid can be extended to a polymatroid set function g defined on the entire power set while satisfying the condition that c(g) = b(f ), which results in a stronger bound. Then we provided an adaptive sensing problem to show that there does not exist any extended polymatroid set function g such that
340
c(g) = b(f ) holds. However, for our extended polymatroid set function g, it turns out that c(g) is very close to b(f ) and much smaller than c(f ), which also results in a stronger bound.
6. Acknowledgments This work is supported in part by NSF under award CCF-1422658, and 345
by the Colorado State University Information Science and Technology Center (ISTeC).
22
References References [1] E. Boros, K. Elbassioni, V. Gurvich, L. Khachiyan, An inequal350
ity for polymatroid functions and its applications, Discrete Appl. Math. 131 (2) (2003) 255–281. [2] M.L. Fisher, G.L. Nemhauser, L.A. Wolsey, An analysis of approximations for maximizing submodular set functions-II, Math. Programming Study 8 (1978) 73–87.
355
[3] G.L. Nemhauser, L.A. Wolsey, M.L. Fisher, An analysis of approximations for maximizing submodular set functions-I, Math. Programming 14 (1) (1978) 265–294. [4] M. Conforti, G. Cornu´ejols, Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the
360
Rado-Edmonds theorem, Discrete Appl. Math. 7 (3) (1984) 251–274. [5] J. Vondr´ ak, Submodularity and curvature: The optimal algorithm, RIMS Kokyuroku Bessatsu B23 (2010) 253–266. [6] M. Sviridenko, J. Vondr´ ak, J. Ward, Optimal approximation for submodular and supermodular optimization with bounded curvature, in: Proceed-
365
ings of the 26th Annual ACM-SIAM Symp. Discrete Algorithms, 2015, pp. 1134–1148. [7] J. Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Optimization 2570 (2003) 11–26. [8] W.T. Tutte,
370
Bureau
of
Lecture on matroids, Standards-B.
J. Research of the National
Mathematics
69B (1 and 2) (1965) 1–47.
23
and
Mathematical
Physics
[9] M. Streeter, D. Golovin, An online algorithm for maximizing submodular functions, in: Proceedings of NIPS: Advances in Neural Information Processing Systems 21, 2008, pp. 67–74. 375
[10] Z. Zhang, E.K.P. Chong, A. Pezeshki, W. Moran, String submodular functions with curvature constraints, IEEE Trans. Autom. Control 61 (3) (2016) 601–616. [11] Y. Liu, E.K.P. Chong, and A. Pezeshki, Bounding the greedy strategy in finite-horizon string optimization, in: Proceedings of the 54th IEEE
380
Conference on Decision and Control, 2015, pp. 3900–3905. [12] Y. Liu, E.K.P. Chong, and A. Pezeshki, Performance bounds for the k-batch greedy strategy in optimization problems with curvature, in: Proceedings of 2016 American Control Conference, 2016, pp. 7177–7182. [13] E. Liu, E.K.P. Chong, and L.L. Scharf, Greedy adaptive linear compression
385
in signal-plus-noise models, IEEE Trans. Inf. Theory 60 (4) (2014), 2269– 2280.
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