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Knapsack is one of the most fundamental problems in theoretical computer science. In the $(1 - \epsilon)$-approximation setting, although there is a fine-grained lower bound of $(n + 1 / \epsilon) ^ {2 - o(1)}$ based on the $(\min, +)$-convolution hypothesis ([K{\"u}nnemann, Paturi and Stefan Schneider, ICALP 2017] and [Cygan, Mucha, Wegrzycki and Wlodarczyk, 2017]), the best algorithm is randomized and runs in $\tilde O\left(n + (\frac{1}{\epsilon})^{11/5}/2^{\Omega(\sqrt{\log(1/\epsilon)})}\right)$ time [Deng, Jin and Mao, SODA 2023], and it remains an important open problem whether an algorithm with a running time that matches the lower bound (up to a sub-polynomial factor) exists. We answer the question positively by showing a deterministic $(1 - \epsilon)$-approximation scheme for knapsack that runs in $\tilde O(n + (1 / \epsilon) ^ {2})$ time. We first extend a known lemma in a recursive way to reduce the problem to $n \epsilon$-additive approximation for $n$ items with profits in $[1, 2)$. Then we give a simple efficient geometry-based algorithm for the reduced problem.

The All-Pairs Shortest Paths (APSP) problem is one of the fundamental problems in theoretical computer science. It asks to compute the distance matrix of a given $n$-vertex graph. We revisit the classical problem of maintaining the distance matrix under a fully dynamic setting undergoing vertex insertions and deletions with a fast worst-case running time and efficient space usage. Although an algorithm with amortized update-time $\tilde O(n ^ 2)$ has been known for nearly two decades [Demetrescu and Italiano, STOC 2003], the current best algorithm for worst-case running time with efficient space usage runs is due to [Gutenberg and Wulff-Nilsen, SODA 2020], which improves the space usage of the previous algorithm due to [Abraham, Chechik, and Krinninger, SODA 2017] to $\tilde O(n ^ 2)$ but fails to improve their running time of $\tilde O(n ^ {2 + 2 / 3})$. It has been conjectured that no algorithm in $O(n ^ {2.5 - \epsilon})$ worst-case update time exists. For graphs without negative cycles, we meet this conjectured lower bound by introducing a Monte Carlo algorithm running in randomized $\tilde O(n ^ {2.5})$ time while keeping the $\tilde O(n ^ 2)$ space bound from the previous algorithm. Our breakthrough is made possible by the idea of ``hop-dominant shortest paths,'' which are shortest paths with a constraint on hops (number of vertices) that remain shortest after we relax the constraint by a constant factor., Comment: Lemma 3.5 typo fix

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 paper

Exploiting unmanned aerial vehicles (UAVs) to execute tasks is gaining growing popularity recently. To solve the underlying task scheduling problem, the deep reinforcement learning (DRL) based methods demonstrate notable advantage over the conventional heuristics as they rely less on hand-engineered rules. However, their decision space will become prohibitively huge as the problem scales up, thus deteriorating the computation efficiency. To alleviate this issue, we propose a double-level deep reinforcement learning (DL-DRL) approach based on a divide and conquer framework (DCF), where we decompose the task scheduling of multi-UAV into task allocation and route planning. Particularly, we design an encoder-decoder structured policy network in our upper-level DRL model to allocate the tasks to different UAVs, and we exploit another attention based policy network in our lower-level DRL model to construct the route for each UAV, with the objective to maximize the number of executed tasks given the maximum flight distance of the UAV. To effectively train the two models, we design an interactive training strategy (ITS), which includes pre-training, intensive training and alternate training. Experimental results show that our DL-DRL performs favorably against the learning-based and conventional baselines including the OR-Tools, in terms of solution quality and computation efficiency. We also verify the generalization performance of our approach by applying it to larger sizes of up to 1000 tasks. Moreover, we also show via an ablation study that our ITS can help achieve a balance between the performance and training efficiency., Comment: 13 pages, 7 figures

Unbounded SubsetSum is a classical textbook problem: given integers $w_1,w_2,\cdots,w_n\in [1,u],~c,u$, we need to find if there exists $m_1,m_2,\cdots,m_n\in \mathbb{N}$ satisfying $c=\sum_{i=1}^n w_im_i$. In its all-target version, $t\in \mathbb{Z}_+$ is given and answer for all integers $c\in[0,t]$ is required. In this paper, we study three generalizations of this simple problem: All-Target Unbounded Knapsack, All-Target CoinChange and Residue Table. By new combinatorial insights into the structures of solutions, we present a novel two-phase approach for such problems. As a result, we present the first near-linear algorithms for CoinChange and Residue Table, which runs in $\tilde{O}(u+t)$ and $\tilde{O}(u)$ time deterministically. We also show if we can compute $(\min,+)$ convolution for $n$-length arrays in $T(n)$ time, then All-Target Unbounded Knapsack can be solved in $\tilde{O}(T(u)+t)$ time, thus establishing sub-quadratic equivalence between All-Target Unbounded Knapsack and $(\min,+)$ convolution.

The (unweighted) tree edit distance problem for $n$ node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in $O(n ^ 3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007]. The same paper also showed that $O(n ^ 3)$ is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly sub-cubic time under the APSP conjecture [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. In this paper, we break the cubic barrier by showing an $O(n ^ {2.9546})$ time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly sub-cubic time [Bringmann, Grandoni, Saha, and Vassilevska Williams, FOCS 2016]., Comment: Accepted to FOCS'21. Fixed some minor typos

Many NP-Hard problems on general graphs, such as maximum independence set, maximal cliques and graph coloring can be solved efficiently on chordal graphs. In this paper, we explore the problem of non-separating st-paths defined on edges: for a connected undirected graph and two vertices, a non-separating path is a path between the two vertices such that if we remove all the edges on the path, the graph remains connected. We show that on general graphs, checking the existence of non-separating st-paths is NP-Hard, but the same problem can be solved in linear time on chordal graphs. In the case that such path exists, we introduce an algorithm that finds the shortest non-separating st-path on a connected chordal graph of $n$ vertices and $m$ edges with positive edge lengths that runs in $O(n\log{n} + m)$ time., Comment: Fixed one misleading typo as well as some other minor issues in section 6.1.1 Some further polishment for submission to ICALP 2021

We study a natural extension to the well-known convex hull problem by introducing multiplicity: if we are given a set of convex polygons, and we are allowed to partition the set into multiple components and take the convex hull of each individual component, what is the minimum total sum of the perimeters of the convex hulls? We show why this problem is intriguing, and then introduce a novel algorithm with a run-time cubic in the total number of vertices. In the case that the input polygons are disjoint, we show an optimization that achieves a run-time that, in most cases, is cubic in the total number of polygons, within a logarithmic factor., Comment: The result is non-original and inferior compared to https://dl.acm.org/doi/pdf/10.1145/3188745.3188878