Greedy+Singleton for knapsack-constrained monotone submodular maximization has approximation ratio exactly beta* ≈ 0.42944
Year: 2026
Type: preprint
Abstract: Editorial revision; corrected attribution (Wolsey 1982 states the tightness of his bound for the adapted greedy); no theorem changed. Greedy+Singleton (the 'modified greedy') maximizes a monotone submodular function under a knapsack constraint by running cost-benefit greedy, skipping items that do not fit, and returning the better of the greedy set and the best feasible single item. It is a textbook algorithm used for budgeted summarization and related selection problems. Its worst-case ratio was known only to lie in [0.427, 0.42945] (Feldman-Nutov-Shoham; Kulik-Schwartz-Shachnai). We determine it exactly: the approximation ratio is beta* = e z/(e z + 2), where z = -2 W_0(-(1/2) e^{-1/2-1/e}), i.e. beta* = 0.4294397657... (an infimum). The upper bound is a tie-free family of weighted-coverage instances refining the Kulik-Schwartz-Shachnai structure, with a closed-form greedy value. The lower bound strengthens the Feldman-Nutov-Shoham factor-revealing relaxation. It is bounded analytically near the extremal point (with interval-checked side conditions) and certified elsewhere by an exact-integer branch-and-bound. Every computational step of the proof was repeated by a separately written program (including Arb ball arithmetic). The supplement contains all code, certificates (leaf lists), the independent checker and a README mapping every computational claim to a command (MIT license). Version 1.3 changes. No existing theorem, lemma or proposition statement changed, and no number or computation changed (numbers.tex is byte-identical to v1.2). One proposition was added (the warm-up below), which states as a proposition the transfer of Wolsey's bound that earlier versions gave in prose. (1) Restructured for readability: new title stating the result; shorter abstract; introduction with a results box, the key idea and Figure 1 up front, and a practitioner paragraph; a new hand-checkable warm-up (Section 5: Greedy+Singleton gets at least 1-e^{-beta_W} = 0.35780, with a short proof; the lower-bound argument formerly in the proof of the stopping-greedy proposition now lives there); the branch-and-bound machinery and the verification provenance moved to Appendix A; the variants (stopping greedy, CELF rule, cost exponents) form a short Section 7; related work and the history table moved to Section 8.2; the in-paper change log and version line were removed. Small clarifications: hand bounds x* in (1,3/2), 0 < t_1 < a*, beta* > 1/3 used by the upper bound; a trivial case (greedy value already >= OPT) stated in the warm-up proof; floor-division rounding in the certifier stated; the computation-only bound 0.429436684 mentioned in the main text; citation of Jain et al. (JACM 2003) for factor-revealing analysis. (2) Corrected attribution: Wolsey's paper (Math. Oper. Res. 7(3), 1982) has now been read. His Theorem 3 is the lower bound of the stopping-greedy proposition, and on p. 424 he states that the bound is tight for his adapted greedy (our variant (A), comparison with the item that stopped the run), without giving the family. Earlier versions said the tightness was our observation and might be folklore; that is withdrawn. Our contribution there is the explicit tie-free family, the written proof, and variant (B) (comparison with the best feasible singleton). The abstract, introduction, history table and the remark after the stopping-greedy proposition now credit this. The main theorem (exactly beta* for Greedy+Singleton) is unaffected: Wolsey analyses no skipping greedy. Version 1.2 changes (credits to prior work only; no theorem, proof, number or computation changed). (1) The rule that returns the better of cost-benefit greedy and unit-cost greedy was wrongly credited to Krause and Guestrin in version 1.1; it is the rule of CELF (Leskovec, Krause, Guestrin, Faloutsos, VanBriesen, Glance, KDD 2007). Krause and Guestrin (CMU-CALD-05-103, March 2005) extend Greedy+Singleton and its analysis from budgeted maximum coverage (Khuller-Moss-Naor) to general monotone submodular functions. The rule is now credited to CELF throughout (abstract, introduction, Section 7, Proposition 13 'The CELF rule', change log). (2) History table: the Feldman-Nutov-Shoham interval [0.427, 0.462] is dated 2021/23; it first appears in arXiv:2007.04937v2 (February 2021). Tang et al. are dated 2020/21 (first arXiv version); the caption states the dating convention; the Zhang et al. row names the proof concerned (Khuller-Moss-Naor). (3) Khuller, Moss and Naor (1999), now read in the original: their bounds are for Greedy+Singleton (their Algorithm 1); the history table now also lists their instance with ratio about 0.44 (above 0.42945, so consistent with later bounds) and a Lin-Bilmes row. (4) Zhang et al. are credited with the gap in the Khuller-Moss-Naor proof, and Kulik, Schwartz and Shachnai with noting that the Lin-Bilmes proof is also flawed. (5) Proposition 13's remark now cites the skip convention of CELF's LazyForward procedure. (6) The Krause-Guestrin reference now gives the report number (CMU-CALD-05-103, March 2005) and its revised versions (2011/2017); the one-line reason why Wolsey's bound transfers to Greedy+Singleton is given; two qualifiers added (worst-case analysis; the literature checked for the tightness of Wolsey's bound does not include Wolsey's paper). Version 1.1 changes. New Section 7.2 on related variants: (i) the stopping greedy (Wolsey's variant), compared with the last attempted item or with the best feasible singleton, has ratio exactly 1-e^{-beta_W} = 0.35780, where e^{beta_W} = 2-beta_W (we observe that Wolsey's bound is tight; the construction may be folklore) [withdrawn in v1.3: Wolsey states the tightness himself]; (ii) the CELF rule (Leskovec et al., KDD 2007), max(unit-cost greedy, cost-benefit greedy), has ratio exactly beta* (when both greedy runs skip items that do not fit), also with lazy evaluation; (iii) cost-scaled greedy with exponent r != 1 has no constant guarantee even with the singleton check. Practical summary, credit corrections, notation cleanup and minor wording fixes; Theorem 1 unchanged.
Authors: Baliga, Kundan, Xie, Aiden, Byju, Aditya
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225392997