0-1 Knapsack in Nearly Quadratic Time

We study pseudo-polynomial time algorithms for the fundamental \emph{0-1 Knapsack} problem. Recent research interest has focused on its fine-grained complexity with respect to the number of items $n$ and the \emph{maximum item weight} $w_{\max}$. Under $(\min,+)$-convolution hypothesis, 0-1 Knapsack does not have $O((n+w_{\max})^{2-\delta})$ time algorithms (Cygan-Mucha-W\k{e}grzycki-W\l{}odarczyk 2017 and K\"{u}nnemann-Paturi-Schneider 2017). On the upper bound side, currently the fastest algorithm runs in $\tilde O(n + w_{\max}^{12/5})$ time (Chen, Lian, Mao, and Zhang 2023), improving the earlier $O(n + w_{\max}^3)$-time algorithm by Polak, Rohwedder, and W\k{e}grzycki (2021). In this paper, we close this gap between the upper bound and the conditional lower bound (up to subpolynomial factors): - The 0-1 Knapsack problem has a deterministic algorithm in $O(n + w_{\max}^{2}\log^4w_{\max})$ time. Our algorithm combines and extends several recent structural results and algorithmic techniques from the literature on knapsack-type problems: - We generalize the "fine-grained proximity" technique of Chen, Lian, Mao, and Zhang (2023) derived from the additive-combinatorial results of Bringmann and Wellnitz (2021) on dense subset sums. This allows us to bound the support size of the useful partial solutions in the dynamic program. - To exploit the small support size, our main technical component is a vast extension of the "witness propagation" method, originally designed by Deng, Mao, and Zhong (2023) for speeding up dynamic programming in the easier unbounded knapsack settings. To extend this approach to our 0-1 setting, we use a novel pruning method, as well as the two-level color-coding of Bringmann (2017) and the SMAWK algorithm on tall matrices.Comment: This paper supersedes an earlier manuscript arXiv:2307.09454 that contained weaker results. Content from the earlier manuscript is partly incorporated into this paper. The earlier manuscript is now obsolet

An Improved FPTAS for 0-1 Knapsack

The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1+epsilon runs in O~(n + (1/epsilon)^{12/5}) time, where O~ hides polylogarithmic factors. In this paper we present an improved algorithm in O~(n+(1/epsilon)^{9/4}) time, with only a (1/epsilon)^{1/4} gap from the quadratic conditional lower bound based on (min,+)-convolution. Our improvement comes from a multi-level extension of Chan\u27s number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items

B-meson Semi-inclusive Decay to 2−+2^{-+} Charmonium in NRQCD and X(3872)

The semi-inclusive B-meson decay into spin-singlet D-wave $2^{-+}$ charmonium, $B\to \eta_{c2}+X$, is studied in nonrelativistic QCD (NRQCD). Both color-singlet and color-octet contributions are calculated at next-to-leading order (NLO) in the strong coupling constant $\alpha_s$. The non-perturbative long-distance matrix elements are evaluated using operator evolution equations. It is found that the color-singlet $^1D_2$ contribution is tiny, while the color-octet channels make dominant contributions. The estimated branching ratio $B(B\to \eta_{c2}+X)$ is about $0.41\,\times10^{-4}$ in the Naive Dimensional Regularization (NDR) scheme and $1.24\,\times10^{-4}$ in the t'Hooft-Veltman (HV) scheme, with renormalization scale $\mu=m_b=4.8$\,GeV. The scheme-sensitivity of these numerical results is due to cancelation between ${}^1S_0^{[8]}$ and ${}^1P_1^{[8]}$ contributions. The $\mu$-dependence curves of NLO branching ratios in both schemes are also shown, with $\mu$ varying from $\frac{m_b}{2}$ to $2m_b$ and the NRQCD factorization or renormalization scale $\mu_{\Lambda}$ taken to be $2m_c$. Comparison of the estimated branching ratio of $B\to \eta_{c2}+X$ with the observed branching ratio of $B \to X(3872)+K$ may lead to the conclusion that X(3872) is unlikely to be the $2^{-+}$ charmonium state $\eta_{c2}$.Comment: Version published in PRD, references added, 26 pages, 9 figure

Faster Algorithms for All Pairs Non-Decreasing Paths Problem

In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time O~(n^{{3 + omega}/{2}}) = O~(n^{2.686}). Here n is the number of vertices, and omega < 2.373 is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for (max, min)-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for (max, min)-matrix product. The previous best upper bound for APNP on weighted digraphs was O~(n^{1/2(3 + {3 - omega}/{omega + 1} + omega)}) = O~(n^{2.78}) [Duan, Gu, Zhang 2018]. We also show an O~(n^2) time algorithm for APNP in undirected simple graphs which also reaches optimal within logarithmic factors

Approximating Knapsack and Partition via Dense Subset Sums

Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.Comment: To appear in SODA 2023. Corrects minor mistakes in Lemma 3.3 and Lemma 3.5 in the proceedings version of this pape