Maximizing Sums of Non-monotone Submodular and Linear Functions: Understanding the Unconstrained Case

Abstract

Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function ℓ. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM).Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function ℓ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function ℓ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).

On Maximizing Sums of Non-monotone Submodular and Linear Functions

“…There are instances of RegularizedUSM with non-positive such that ( ( ), ) is inapproximable for any ( ( ), ) in Table 2. In particular, (0) ≈ 0, matching the result of [BF22, Theorem 1.1], and (1) < 0.478, matching the result of [BF22,Theorem 1.3].…”

“…Observe that for both DeterministicDG and RandomizedDG, increasing improves the dependence of the approximation on but decreases the dependence on . Setting = 1 recovers the guarantees of [BF22]. We also provide examples showing that neither DeterministicDGnor RandomizedDG achieve ( , )-approximations better than Theorems 6.1 and 6.2 in Theorems 6.3 and 6.4, respectively.…”

“…We use blue for approximation algorithms and red for inapproximability results, and the shaded area represents the gap between the best known approximation algorithms and inapproximability results. Observe that Theorem 5.6 unifies the two inapproximability theorems from [BF22]. (0.5, 2 ln 2− )-inapproximability is due to Theorem 5.9.…”

“…Recall that DeterministicDG achieves a 1∕3-approximation for USM, while RandomizedDG achieves a 1∕2approximation for USM in expectation. [BF22] extended these guarantees to RegularizedUSM with non-negative RandomizedDG, [BF22, Theorem 1.5] RandomizedDG (Theorem 6.2)…”

“…Feldman subsequently removed the need for the guessing step and the dependence on ( ) by introducing a distorted objective [Fel18]. Many faster algorithms for the case of monotone have since been developed Following the convention of [BF22], the and axes represent the coefficients of and , respectively. We use blue for approximation algorithms and red for inapproximability results, and the shaded area represents the gap between the best known approximation algorithms and inapproximability results.…”