In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a (2/3 - ε)-approximation algorithm for any constant ε > 0. This result almost matches the (2/3 + ε) inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a (1/2 - ε)-approximation algorithm for any constant ε > 0. We also prove a hardness bound of (1/2 + ε) for any constant ε > 0.

Existence of long arithmetic progressions in sumsets and subset sums is an important topic in additive combinatorics, and has applications in the design of algorithms for classic combinatorial optimization problems, including Subset Sum and Knapsack. Motivated by these applications, Chen, Mao and Zhang [STOC, 2025] studied arithmetic progressions from a computational perspective: instead of merely knowing the existence of arithmetic progressions, they aim to construct it explicitly and find out how its terms can be represented using integers from the corresponding set. They show that both can be done in near-linear time for long arithmetic progressions in \(kA\), the \(k\)-fold sum of an integer set \(A\), and \(\mathcal{S}(A)\), the set of all subset sums of \(A\), where \(A\) is a set of nonnegative integers and \(|A|\) is relatively large comparing to \(\max(A)\) (the largest element in \(A\)). They left as an open problem whether the same thing can be achieved for long arithmetic progressions in the sumset of different sets, i.e., \(A_1 + A_2 + \cdots + A_k\).

We consider the classic Knapsack problem. Let $t$ and $\mathrm{OPT}$ be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least $\mathrm{OPT}/(1 + \varepsilon)$ and total weight at most $t$, then Knapsack can be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^2)$ time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that $(\min,+)$-convolution cannot be solved in truly subquadratic time [Künnemann, Paturi, and Schneider '17][Cygan, Mucha, Węgrzycki, and Włodarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least $\mathrm{OPT}$ and total weight at most $(1 + \varepsilon)t$. Therefore, it is natural to ask the following question. If one seeks a solution with total profit at least $\mathrm{OPT}/(1+\varepsilon)$ and total weight at most $(1 + \varepsilon)t$, can Knsapck be solved in $\tilde{O}(n + (\frac{1}{\varepsilon})^{2-δ})$ time for some constant $δ> 0$? We answer this open question affirmatively by proposing an $\tilde{O}(n + (\frac{1}{\varepsilon})^{7/4})$-time algorithm.

We investigate pseudo-polynomial time algorithms for Subset Sum. Given a multi-set$X$of$n$positive integers and a target$t$, Subset Sum asks whether some subset of$X$sums to$t$. Bringmann proposes an$\tilde{O}(n+t)$-time algorithm [Bringmann SODA'17], and an open question has naturally arisen: can Subset Sum be solved in$O(n+w)$time? Here$w$is the maximum integer in$X$. We make a progress towards resolving the open question by proposing an$\tilde{O}(n+\sqrt{wt})$-time algorithm.

We investigate pseudopolynomial-time algorithms for Bounded Knapsack and Bounded Subset Sum. Recent years have seen a growing interest in settling their fine-grained complexity with respect to various parameters. For Bounded Knapsack, the number of items n and the maximum item weight wmax are two of the most natural parameters that have been studied extensively in the literature. The previous best running time in terms of n and wmax is [Polak, Rohwedder, Węgrzycki ‘21]. There is a conditional lower bound of (n + wmax)2-o(1) based on (min, +)-convolution hypothesis [Cygan, Mucha, Węgrzycki, Włodarczyk ‘17]. We narrow the gap significantly by proposing an -time algorithm. Our algorithm works for both 0-1 Knapsack and Bounded Knapsack. Note that in the regime where wmax ≈ n, our algorithm runs in Õ(n12/5) time, while all the previous algorithms require Ω(n3) time in the worst case.

We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in O(n + (1/)2) time. Prior to our work, the best running time is O(n + (1/)11/5) [Deng, Jin, and Mao’23]. Our algorithm is the best possible (up to a polylogarithmic factor), as Knapsack has no O((n + 1/)2−δ)-time FPTAS for any constant δ > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.

We propose an O(n + 1/)-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay’22]. Prior to our work, the best known FPTAS for Partition runs in O(n + 1/5/4) time [Deng, Jin and Mao’23, Wu and Chen’22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most $4$. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a $(6 + δ)$-approximation algorithm where $δ$ can be any positive constant, and there is still a gap of roughly $2$. In this paper, we narrow the gap significantly by proposing a $(4+δ)$-approximation algorithm where $δ$ can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is $\mathit{poly}(m,n)\cdot n^{\mathit{poly}(\frac{1}δ)}$ where $n$ is the number of players and $m$ is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to $3 + \frac{21}{26} \approx 3.808$.

The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: roughly 4 for estimation and roughly $6 + 2\sqrt{10}$ for construction. We propose an algorithm that constructs an allocation with value within a factor of $6 + δ$ from the optimum for any constant $δ> 0$. The running time is polynomial in the input size for any constant $δ$ chosen.