1 Introduction
Submodularity^{1}^{1}1A function is submodular if for every , , . captures an important diminishingreturns property of discrete functions. Submodular set functions arise from e.g. viral marketing (Kempe et al., 2003), data summarization (Mirzasoleiman and Krause, 2015), and sensor placement (Krause et al., 2008). The optimization of these functions has been studied subject to various types of independence system^{2}^{2}2An independence system on the set is a collection of subsets of such that (i) is nonempty, and (ii) if and , then . constraints, including cardinality (Nemhauser et al., 1978), matroid (Fisher et al., 1978), and the more general independence systems (Calinescu et al., 2011). Formally, the problem (MAXI) considered in this work is the following: given submodular function and independence system on , determine
Even on an independence system where maximal independent sets have the same size, the greedy algorithm may return arbitrarily bad solutions for MAXI. Our results indicate that some exchange property between independent sets must exist if the problem is to be tractable.
Contributions
Our main contributions are summarized as follows.

Let denote the subclass of independences systems where maximal independent sets have the same size. We show that admits no polynomialtime algorithm with approximation ratio better than unless NP = ZPP, even when the submodular function is restricted to be monotone; here, is the size of the ground set, and is arbitrary. On the other hand, under the condition that the system has two disjoint bases, the greedy algorithm does obtain a ratio of . Intuitively, the difficulty of approximation on a system arises from the lack of any exchange property between the independent sets.

Also, we provide a deterministic algorithm TripleGreedy (Alg. 2), which has the ratio on extendible systems in function evaluations, when the objective function is submodular but not necessarily monotone. This is the first approximation algorithm on extendible systems whose runtime is linear up to a logarithmic factor in the size of the ground set and is independent of both and the the maximum size of any independent set. In prior literature, the fastest randomized algorithm is that of Feldman et al. (2017), which achieves expected ratio in evaluations, while the fastest deterministic algorithm is also by Feldman et al. (2017) and achieves ratio in evaluations.
Related work
The maximization of monotone, submodular functions over independence systems has a long history of study; Fisher et al. (1978) proved the approximation ratio of for the greedy algorithm when the independence system is an intersection of matroid constraints, which is a special case of a extendible system. This ratio for the greedy algorithm was extended to extendible systems by Calinescu et al. (2011), as well as to the more general system constraint. A similar ratio for a faster, thresholded greedy algorithm and system constraint was also given by Badanidiyuru and Vondrák (2014).
For the special case when the independence system is a single matroid or cardinality constraint, better approximation guarantess have been obtained: in Calinescu et al. (2011), an optimal approximation is given when is monotone and the independence system is a matroid. For further information, the reader is referred to the survey of Buchbinder and Feldman (2018b) and references therein.
When is nonmonotone and the independence system is a extendible system, Gupta et al. (2010) provided an approximation in function evaluations; this was improved by Mirzasoleiman et al. (2016) to with the same time complexity, and Feldman et al. (2017) improved this to a ratio of in evaluations. Furthermore, Mirzasoleiman et al. (2018) extended these works to a streaming setting. All of these works rely upon an iterated greedy approach, which employs up to iterations of the standard greedy algorithm. In Section 5, we propose a simpler iterated greedy approach for extendible systems, which relies upon only two iterations of the greedy algorithm. We show how to speed up this algorithm to obtain ratio in evaluations.
Organization
The rest of this paper is organized as follows: in Section 2 we define notions used throughout the paper. In Section 3 we prove the hardness result for . Next, we show that the greedy algorithm is indeed the optimal approximation on under a weak assumption in Section 4. Finally, in Section 5 we provide our nearly lineartime for submodular maximization over a extendible system.
2 Preliminaries
Throughout the paper, denotes the ground set of size . In this work, the objective function is a nonnegative function ; typically, the function is given as an oracle that returns, for given set , the value . Our inapproximability result in Section 3 holds in this model, but it also holds when a description of as a polynomialtime computable function is given as input. When is a set and , we occasionally write for .
The members of an independence system are termed independent sets. An independent set is a basis of independence system if for all , .
Definition (Matroid).
An independence system is a matroid if the following property holds: if and , then there exists such that .
Definition (Extendible System).
An independence system is extendible if the following property holds. If , with and if such that , then there exists subset with such that .
Definition (System).
A system is an independence system such that if are bases, then .
We remark that every extendible system is also a system, but that the converse is not true, as the exchange property defining a extendible system may not hold. Furthermore, every matroid is a system, but the converse does not hold. As an example, let , , and . Then is clearly a system but not a matroid.
3 Hardness of Submodular Maximization over Independence Systems
In this section, the main inapproximability result is proven for : maximization of submodular functions over independence systems for which all maximal bases have equal size.
Hardness of is established via an approximationpreserving reduction to the independent set problem (ISG) in a graph, which is to find the maximum size of an edgeindependent set of vertices. Once this reduction is defined, we show that any approximation for yields an approximation for ISG, and our hardness result follows from the hardness of ISG.
Definition (Isg).
The ISG problem is the following: given a finite graph , where , define a set to be edgeindependent iff no pair of vertices in have an edge between them. Then the ISG problem is to determine the maximum size of an edgeindependent set in .
It is easily seen that the set is edgeindependent in is an independence system. In general, may be a system, where ; consider a star graph where all vertices are connected to a center vertex and no other edges exist.
Intuitively, the reduction works by transforming a graph, which is an instance of ISG, into an instance of
through the padding of edgeindependent sets with dummy elements so that maximal independent sets have the same size. A submodular function is then defined that maps the padded independent sets to the size of the original, unpadded, edgeindependent set in the graph. Formally, the reduction is defined as follows.
Definition (Reduction ).
Let be a graph, which is an instance of ISG. Let , where is a set of dummy elements. An independence system is defined on as follows: is in iff. is edgeindependent in and . Define function , by .
We remark that the function is defined on all subsets of , not only members of the independence system. To illustrate the reduction, we provide the following example.
Example 1.
Let be a star graph with five vertices. That is, and . Then the maximal, edgeindependent sets are and . Then maps this graph to the following independence system. The ground set , where is a set of five dummy elements. Then the independence system defined by has bases
That is, consists of all subsets of elements of .
By the following lemma, the reduction takes an instance of ISG to an instance of . Notice that the independence of any subset of may be checked in polynomial time; the same is true for computation of .
Lemma 1.
Let be an instance of ISG, and let . Then

is an independence system; in particular, all maximal bases have equal size.

is monotone and submodular.
Proof.
(i): Clearly, is nonempty, since any singleton vertex is edgeindependent in , and . Furthermore, it is closed under subsets: let , where , , and let . Then , where , . Since any subset of an edgeindependent set of is also edgeindependent, we have that is edgeindependent in , and
Hence . Thus, is an independence system on .
Next, suppose is maximal. Then , for otherwise another dummy element could be added to to produce a larger independent set. Hence is a system.
(ii): Let ; notice that are not necessarily in the independence system . Then , so the function is monotone.
Next, let . If , then
If ,
Hence, in all cases, , so the function is submodular. ∎
Next, we show that is an approximationpreserving reduction.
Lemma 2.
By application of the reduction , any approximation algorithm to yields an approximation to ISG.
Proof.
Let be an instance of ISG, and let . Let . Since membership of a set requires that be edgeindependent in , we have that , where is the maximum size of an edgeindependent set of . Now suppose set satisfies . Then
and by definition of , is edgeindependent in . Therefore, any approximation algorithm for with ratio yields an approximation algorithm for ISG with ratio by the following method: given instance of ISG, transform to an instance of . Apply the approximation to get set such that . Finally, project back to and return the edgeindependent set , which satisfies . ∎
The next theorem follows from Lemma 2 and the results of Hastad (1999) on ISG: namely, for any , there is no polynomialtime algorithm to approximate ISG better than unless NP = ZPP.
Theorem 1.
For any , there is no polynomialtime algorithm that achieves ratio better than on , where is the ground set of the instance of , unless NP = ZPP.
4 The Greedy Ratio on Maxi, when is monotone
When the function is monotone, we further analyze the performance of the greedy algorithm (Alg. 1) on independence systems in this section. When all maximal bases have equal size, we show that the greedy algorithm obtains a ratio that matches our lower bound in the previous section.
We begin with a performance ratio for the greedy algorithm on an arbitrary independence system in terms of the size of the largest independent set.
Proposition 1.
Let be an independence system, and let . Let be the solution returned by the greedy algorithm, and let be the optimal solution to MAXI. Then .
Proof.
Let be the ground set of , and let , and observe that . Now let ; then by submodularity, . It follows that . ∎
The next corollary, combined with the hardness result from the previous section, shows that if the independence system has two disjoint bases, the greedy algorithm is the optimal approximation on systems where bases have equal size.
Corollary 1.
Let be a system where maximal bases have equal size, with at least two disjoint bases. Then the greedy algorithm is a approximation algorithm to on .
Proof.
Let be bases of , such that . Since is a system, for some , ; hence . Hence, , so the result follows from Prop. 1. ∎
5 The TripleGreedy Algorithm
In this section, the TripleGreedy (TG, Algorithm 2) is presented. The algorithm TG is the first nearly lineartime algorithm to approximately maximize a submodular function with respect to a extendible system.
We start with an abstract subproblem required by TG.
Definition (MaxUnion).
Given and independence system , determine , such that for any , . Even if no such exists, by an approximation to MAXUNION, it is meant an algorithm that finds , such that for any , .
Notice that in the requirement of MAXUNION may not be a member of the independence system.
The TG algorithm employs two subroutines, one to approximate the MAXUNION problem and one for the unconstrained maximization problem; the unconstrained maximization problem is to determine . Since a total of three calls to these subroutines are required, and since variants of greedy algorithms may be used for each subroutine, Alg. 2 is termed TripleGreedy. First, TG determines a set approximating MAXUNION with the function ; second, TG determines a set is found approximating MAXUNION with the restriction of to . Third, a set is found, approximating the maximum value of restricted to . Finally, the set in maximizing is returned.
We remark that TG functions similarly to the algorithm for maximizing submodular functions with respect to cardinality constraint developed in Gupta et al. (2010); in place of MAXUNION, Gupta et al. (2010) simply uses the greedy algorithm. By abstracting out this subproblem, we see that 1) a performance ratio may be proved in a much more general setting than cardinality constraint, namely for extendible systems, and 2) the faster thresholding approach developed by Badanidiyuru and Vondrák (2014) (THRESHOLD) for monotone submodular maximization can be used for MAXUNION, which results in nearly linear runtime.
If is submodular, then the approximation ratio of TG depends on the ratios of the algorithms used for MAXUNION and UNCONSTRAINEDMAX.
Theorem 2.
Let be submodular, let be an independence system, and let , and let TG . Then
where and are the ratios of the algorithms used for UNCONSTRAINEDMAX, and MAXUNION, respectively.
Proof.
Let have their values at termination of TG . Suppose a approximation algorithm is used for UNCONSTRAINEDMAX. Then any set satisfies . Suppose an approximation algorithm is used for MAXUNION; so and .
where the second and third inequalities follow from the submodularity of and the fact that is nonnegative and . ∎
Next, we establish that THRESHOLD approximates MAXUNION on extendible systems; the proof is provided in Appendix A.
Lemma 3.
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Appendix A Appendix
Proof of Lemma 3.
Let be returned by THRESHOLD. Let , . The set will be partitioned into at most subsets , each of size at most , as follows. Let , . Suppose have been obtained, such that , which is initially satisfied at . By the definition of extendible system, there exists , with , such that . Then let and let ; clearly . If , stop; otherwise, continue inductively until . Let be the index at which this procedure terminates. If , let and redefine for all .
Claim 1.
For each , , for all .
Proof.
Since , and , the claim follows by definition of independence system. ∎
Claim 2.
Proof.
where the last inequality is by the stopping condition of THRESHOLD and the fact that , so for all . The other inequalities follow from submodularity and the definition of . ∎
Then
where the first inequality is by Claim 2, the first two equalities are by telescoping and the definition of , the second and third inequalities are by submodularity. The fourth inequality holds by the following argument: when was added to , it holds that the threshold has its initial value , in which case for any , or all were not added during the previous threshold . Hence by submodularity. Since , the lemma follows. ∎
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