A distributed greedy algorithm for non-monotone submodular maximization

We consider the problem of maximizing a non-monotone submodular function subject to a hereditary constraint. Our approach is a slight modification of the randomized, distributed greedy algorithm described in Section 4.2, and it builds on the work of Gupta et al. [2010]. Again, we show how to combine the standard Greedy algorithm, together with any algorithm Alg for the non-monotone case in order to obtain a randomized, distributed algorithm for the non-monotone submodular maximization.

4.3.1 Algorithm

Our modified algorithm, NMRandGreeDi (Algorithm 6), works as follows. As in the monotone case, in the first round we distribute the elements of V uniformly at random amongst the m machines. Then, we run the standard greedy algorithm twice to obtain two disjoint solutions S_i1 and S_i2 on each machine. Specifically, each machine first runs Greedy on Vi to obtain a solution Si1, then runs Greedy on Vi\Si1 to obtain a disjoint solution S_i2. In the second round, both of these solutions are sent to a single machine,

which runsAlg onS=S

i(Si1∪Si2)to produce a solutionT. The best solution amongst T and all of the solutions S_i1 andS_i2 is then returned.

4.3.2 Analysis

We devote the rest of this section to the analysis of the algorithm. The space and communication aspects are similar to those required by RandGreeDi, analyzed in Sub- section 4.2.2, and thus omitted here.

Recall the strong greedy property, defined in Section 3.1.1. Given an instance hV,I, fiof non-negative submodular maximization, we assume theGreedyalgorithm has the following property:

For all S∈ I: f(Greedy(V))≥γ·f(Greedy(V)∪S) (GP)

where the values ofγ for different hereditary constraints are given in that section. The analysis is similar to the approach from the previous section. We define V(1/m) as before. We modify the definition of the vectorp in Section 4.2.2 as follows. For each element e∈V, we have

pe=              Pr A∼V(1/m) h e∈Greedy(A∪ {e}) or

e∈Greedy((A∪ {e})\Greedy(A∪ {e}))i if e∈OPT

0 otherwise

We now derive analogues of Lemmas 4.2.1 and 4.2.2.

Lemma 4.3.1. Suppose that Greedy satisfies (GP). For each machine i,

Ef(S_i1) +f(S_i2)≥γ·f−(1OPT−p), and therefore E max f(S_i1), f(S_i2) ≥ γ 2·f −₍₁ OPT−p).

Proof. Consider machineiand let Vi be the set of elements assigned to machineiin the first round. Let

Oi ={e∈OPT :e /∈Greedy(Vi∪ {e}) and

Note that, sinceOPT∈ I and I is hereditary, we haveOi ∈ I. It follows from Lemma 4.1.1 that

S_i1 =Greedy(Vi) =Greedy(Vi∪Oi), (4.5) S_i2 =Greedy(Vi\Si1) =Greedy((Vi\Si1)∪Oi). (4.6)

By combining the equations above with the greedy property (GP), we obtain

f(S_i1)(4=.5)f(Greedy(Vi∪Oi)) (GP) ≥ γ·f(Greedy(Vi∪Oi)∪Oi) (4.5) = γ·f(S_i1∪Oi), (4.7) f(S_i2)(4=.6)f(Greedy((Vi\Si1)∪Oi)) (GP) ≥ γ·f(Greedy((Vi\Si1)∪Oi)∪Oi) (4.6) = γ·f(S_i2∪Oi). (4.8)

Now we observe that

f(S_i1∪Oi) +f(Si2∪Oi)≥f((Si1∪Oi)∩(Si2∪Oi)) +f(Si1∪Si2∪Oi) (f is submodular) =f(Oi) +f(Si1∪Si2∪Oi) (Si1∩Si2 =∅)

≥f(Oi). (f is non-negative)

(4.9)

By combining (4.7), (4.8), and (4.9), we obtain

f(S_i1) +f(S_i2)≥γ·f(Oi). (4.10)

Since the distribution of Vi is the same asV(1/m), for each elemente∈OPT, we have

Pr[e∈Oi] = 1−Pr[e /∈Oi] = 1−pe,

By combining (4.10), (4.11), and Lemma 2.3.1, we obtain

E[f(S_i1) +f(S_i2)]≥γ·E[f(Oi)] (By (4.10)) ≥γ·f−(1OPT−p). (By (4.11) and Lemma 2.3.1)

Lemma 4.3.2. E[f(Alg(S))]≥β·f−(p).

Proof. Recall that S_i1 =Greedy(Vi),Si2 =Greedy(Vi\Si1), andS = S

i(Si1∪Si2). Since OPT∈ I and I is hereditary,S∩OPT∈ I. Since Algis aβ-approximation, we have

f(Alg(S))≥β·f(S∩OPT). (4.12)

Consider an element e∈OPT. For each machinei, we have

Pr[e∈S|eis assigned to machine i]

= Pr[e∈Greedy(Vi) or e∈Greedy(Vi\Greedy(Vi))|e∈Vi]

= Pr

A∼V(1/m)[e

∈Greedy(A) or e∈Greedy(A\Greedy(A))|e∈A]

= Pr

B∼V(1/m)[e∈Greedy(B∪ {e}) or e∈Greedy((B∪ {e})\Greedy(B∪ {e}))] =pe.

The first equality above follows from the fact that eis included in S iff eis included in either S_i1 or S_i2. The second equality follows from the fact that the distribution of Vi is the same as V(1/m). The third equality follows from the fact that the distribution of A ∼ V(1/m) conditioned on e∈ A is identical to the distribution of B∪ {e} where B ∼ V(1/m). Therefore

Pr[e∈S∩OPT] =pe,

E[1S∩OPT] =p. (4.13)

By combining (4.12), (4.13), and Lemma 2.3.1, we obtain

We can now combine Lemmas 4.3.2 and 4.3.1 to obtain our main result for non- monotone submodular maximization.

Theorem 4.3.3. Let f : 2V →_R+ be a submodular function, let I ⊆2V be a hereditary set system, and letAlgis aβ-approximation algorithm formaxS∈If(S). SupposeGreedy

satisfies (GP) with factor γ. The algorithm NMRandGreeDi is (in expectation) a _2ββγ_+γ- approximation algorithm for the same problem.

Proof. LetS_i1 =Greedy(Vi),Si2 =Greedy(Vi\Si1), andS = S

i(Si1∪Si2)be the set of ele- ments on the last machine, andT =Alg(S)be the solution produced on the last machine. Then, the output D produced by RandGreeDi satisfiesf(D)≥maximax{f(Si1), f(S2i)} and f(D)≥f(T). Thus, from Lemmas 4.3.1 and 4.3.2 we have:

E[f(D)]≥ γ

2 ·f −₍₁

OPT−p), (4.14)

E[f(D)]≥β·f−(p). (4.15)

By combining (4.14) and (4.15), we obtain

(2β+γ)E[f(D)]≥βγ[f−(p) +f−(1OPT−p)] ≥βγ·f−(1OPT)

=βγ·f(OPT).

In the second inequality, we have used the fact thatf− is convex andf−(c·x)≥cf−(x) for any constantc∈[0,1].

Finally, we remark that one can use the following approach on the last machine, and use a lemma from Gupta et al. [2010]. As in the first round, we run Greedytwice to obtain two solutions T1 =Greedy(S) andT2 =Greedy(S\T1). Additionally, we select a subset T3 ⊆T1 using an unconstrained submodular maximization algorithm onT1, such as the Double Greedy algorithm of [Buchbinder et al., 2012], which is a 1₂-approximation. The final solution T is the best solution among T1, T2, T3. If Greedy satisfies property GP, then it follows from the analysis of [Gupta et al., 2010] that the resulting solution T satisfiesf(T)≥ ₂₍₁₊γ_γ)·f(OPT) (shown in the following). This gives us the following corollary of Theorem 4.3.3:

Corollary 4.3.4. Let f : 2V → _R+ be a submodular function, and let I ⊆ 2V be

a hereditary set system. Suppose Greedy satisfies (GP) with factor γ. The resulting algorithm described above (which usesGreedytwice andDoubleGreedy) is (in expectation) a ₄₊₂γ_γ-approximation formaxS∈If(S).

Proof. By (GP) and the approximation guarantee of the Double Greedy algorithm, we have: f(T)≥f(T1)≥γ·f(T1∪OPT) (4.16) f(T)≥f(T2)≥γ·f(T2∪(OPT\T1)) (4.17) f(T)≥f(T3)≥ 1 2f(T1∩OPT). (4.18)

Additionally, from [Gupta et al., 2010, Lemma 2], we have:

f(T1∪OPT) +f(T2∪(OPT\T1)) +f(T1∩OPT)≥f(OPT)

By combining the inequalities above, we obtain:

(1 +γ)f(T)≥ γ

2(f(T1∪OPT) +f(T2∪(OPT\T1)) +f(T1∩OPT))≥ γ

2f(OPT) and hence f(T)≥ ₂₍₁₊γ_γ)·f(OPT)as claimed. Setting β = _2(γγ₊₁₎ in Theorem 4.3.3, we obtain an approximation ratio of ₄₊₂γ_γ.

4.4 An improved distributed greedy algorithm for non-monotone sub-