Data-dependent Evaluations for Budgeted Submodular Maximization

Consistent submodular maximization studies the tradeoff between solution quality and stability when elements arrive over time. For a monotone submodular objective, which models diminishing returns, an algorithm maintains a set of at most $k$ available elements and changes only $O(1)$ elements after each insertion. Dütting et al. [2025] established a tight $2/3$ approximation with unrestricted computation and a polynomial-time $0.51$ approximation. They left open at STOC 2025 whether efficient algorithms can match the offline $1-1/e$ guarantee. We resolve this problem by proving that the supremum approximation achievable with polynomially many value queries and worst-case constant recourse is $$β=2-\sqrt2\approx0.5858<1-1/e.$$ For every $\varepsilon>0$, our randomized algorithm attains $β-\varepsilon$ with $O(\varepsilon^{-2})$ changes per insertion. Any fixed improvement requires exponentially many queries before one critical insertion or linear recourse of $Ω(k)$ changes at that insertion, even with unlimited queries afterwards. This gap quantifies the cost of consistency: the current oracle hides which elements will be needed after an arrival. We also determine the exact curvature-dependent threshold $1-(\sqrt2-1)\vartheta$, attain $1-1/e-\varepsilon$ for weighted coverage with $O(\varepsilon^{-1})$ recourse, and separate the existence of universal future-price certificates from their efficient computation. Our algorithm has a bounded-bit polynomial-time implementation for polynomial-bit rational oracle answers; the lower bound uses only logarithmic-bit rational answers.

Submodular functions -- functions exhibiting diminishing returns -- are central to machine learning. When the objective is monotone and non-negative, the greedy algorithm achieves a tight $63\%$ approximation. But many practical objectives incorporate costs that make them negative on some inputs, and all existing multiplicative guarantees require non-negativity. Prior work handles negativity through additive bounds for the special class of decomposable functions and non-monotonicity through partial-monotonicity parameters, but these address each difficulty in isolation and neither extends the classical structural theory. We extend \emph{curvature} -- a parameter measuring how far a function deviates from linearity -- to all submodular functions, handling both non-monotonicity and negativity through a single classical concept. A greedy algorithm with pruning achieves a curvature-controlled multiplicative ratio for \emph{any} submodular function, including those taking negative values -- the first such guarantee beyond monotonicity and non-negativity. In the non-monotone regime $1 \le c_g < 2.2$, the bound strictly beats the best known uniform ratio of $0.401$ (for non-negative $f$), and it recovers the classical $(1-e^{-c_g})/c_g$ guarantee for monotone functions. A multilinear-extension variant extends the framework to general combinatorial constraints via multilinear relaxation. Experiments on cost-penalized experimental design, coverage, feature selection, and a curvature sweep on Multi-News passage selection support the theory.

We study nonnegative submodular maximization subject to a general matroid when the offline algorithm is given an arbitrary controlled value oracle. Our main result is an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson): without modifying its Poisson intensity, single-element exchange rule, or spiteful drop step, the algorithm retains limiting approximation factors $1/e$ for non-monotone objectives and $1-1/e$ for monotone objectives. More precisely, under every controlled oracle $\widehat f$ satisfying $|\widehat f(S)-f(S)|\le ξ$ for every set $S$, our implementation returns a feasible set with expected value at least $(1/e-\varepsilon)\OPT-O(kξ)$ and $(1-1/e-\varepsilon)\OPT-O(kξ)$, respectively, using $\widetilde O(nk^2\varepsilon^{-2})$ oracle calls. As a consequence, the offline-to-online reduction yields full-bandit CMAB algorithms for general matroid-constrained submodular rewards with exact limiting approximation-regret factors $1/e$ and $1-1/e$ and $\widetilde O(n^{1/5}k^{4/5}T^{4/5})$ regret.