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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., Comment: In first version, we falsely claimed that our algorithm is also able to reconstruct a subset that sums to t. In the latest version, we removed this false claim and explained why we cannot do reconstruction

We propose an $\widetilde{O}(n + 1/\eps)$-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 $\widetilde{O}(n + 1/\eps^{5/4})$ 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., Comment: To appear in STOC2024

We investigate polynomial-time approximation schemes for the classic 0-1 knapsack problem. The previous algorithm by Deng, Jin, and Mao (SODA'23) has approximation factor $1 + \eps$ with running time $\widetilde{O}(n + \frac{1}{\eps^{2.2}})$. There is a lower Bound of $(n + \frac{1}{\eps})^{2-o(1)}$ conditioned on the hypothesis that $(\min, +)$ has no truly subquadratic algorithm. We close the gap by proposing an approximation scheme that runs in $\widetilde{O}(n + \frac{1}{\eps^2})$ time.

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 $w_{\max}$ 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 $w_{\max}$ is $O(n + w^3_{\max})$ [Polak, Rohwedder, Wegrzycki '21]. There is a conditional lower bound of $O((n + w_{\max})^{2-o(1)})$ based on $(\min,+)$-convolution hypothesis [Cygan, Mucha, Wegrzycki, Wlodarczyk '17]. We narrow the gap significantly by proposing a $\tilde{O}(n + w^{12/5}_{\max})$-time algorithm. Note that in the regime where $w_{\max} \approx n$, our algorithm runs in $\tilde{O}(n^{12/5})$ time, while all the previous algorithms require $\Omega(n^3)$ time in the worst case. For Bounded Subset Sum, we give two algorithms running in $\tilde{O}(nw_{\max})$ and $\tilde{O}(n + w^{3/2}_{\max})$ time, respectively. These results match the currently best running time for 0-1 Subset Sum. Prior to our work, the best running times (in terms of $n$ and $w_{\max}$) for Bounded Subset Sum is $\tilde{O}(n + w^{5/3}_{\max})$ [Polak, Rohwedder, Wegrzycki '21] and $\tilde{O}(n + \mu_{\max}^{1/2}w_{\max}^{3/2})$ [implied by Bringmann '19 and Bringmann, Wellnitz '21], where $\mu_{\max}$ refers to the maximum multiplicity of item weights., Comment: To appear in SODA2024

Motivated by real-world applications, we study the fair allocation of graphical resources, where the resources are the vertices in a graph. Upon receiving a set of resources, an agent's utility equals the weight of a maximum matching in the induced subgraph. We care about maximin share (MMS) fairness and envy-freeness up to one item (EF1). Regarding MMS fairness, the problem does not admit a finite approximation ratio for heterogeneous agents. For homogeneous agents, we design constant-approximation polynomial-time algorithms, and also note that significant amount of social welfare is sacrificed inevitably in order to ensure (approximate) MMS fairness. We then consider EF1 allocations whose existence is guaranteed. However, the social welfare guarantee of EF1 allocations cannot be better than $1/n$ for the general case, where $n$ is the number of agents.Fortunately, for three special cases, binary-weight, two-agents and homogeneous-agents, we are able to design polynomial-time algorithms that also ensure a constant fractions of the maximum social welfare.

We study a bilevel optimization problem which is a zero-sum Stackelberg game. In this problem, there are two players, a leader and a follower, who pick items from a common set. Both the leader and the follower have their own (multi-dimensional) budgets, respectively. Each item is associated with a profit, which is the same to the leader and the follower, and will consume the leader's (follower's) budget if it is selected by the leader (follower). The leader and the follower will select items in a sequential way: First, the leader selects items within the leader's budget. Then the follower selects items from the remaining items within the follower's budget. The goal of the leader is to minimize the maximum profit that the follower can obtain. Let $s_A$ and $s_B$ be the dimension of the leader's and follower's budget, respectively. A special case of our problem is the bilevel knapsack problem studied by Caprara et al. [SIAM Journal on Optimization, 2014], where $s_A=s_B=1$. We consider the general problem and obtain an $(s_B+\epsilon)$-approximation algorithm when $s_A$ and $s_B$ are both constant. In particular, if $s_B=1$, our algorithm implies a PTAS for the bilevel knapsack problem, which is the first O(1)-approximation algorithm. We also complement our result by showing that there does not exist any $(4/3-\epsilon)$-approximation algorithm even if $s_A=1$ and $s_B=2$. We also consider a variant of our problem with resource augmentation when $s_A$ and $s_B$ are both part of the input. We obtain an O(1)-approximation algorithm with O(1)-resource augmentation, that is, we give an algorithm that returns a solution which exceeds the given leader's budget by O(1) times, and the objective value achieved by the solution is O(1) times the optimal objective value that respects the leader's budget.

In selfish bin packing, each item is regarded as a selfish player, who aims to minimize the cost-share by choosing a bin it can fit in. To have a least number of bins used, cost-sharing rules play an important role. The currently best known cost sharing rule has a \emph{price of anarchy} ($PoA$) larger than 1.45, while a general lower bound 4/3 on $PoA$ applies to any cost-sharing rule under which no items have the incentive to move unilaterally to an empty bin. In this paper, we propose a novel and simple rule with a $PoA$ matching the lower bound of $4/3$, thus completely resolving this game. The new rule always admits a Nash equilibrium and its \emph{price of stability} ($PoS$) is one. Furthermore, the well-known bin packing algorithm $BFD$ (Best-Fit Decreasing) is shown to achieve a strong equilibrium, implying that a stable packing with an asymptotic approximation ratio of $11/9$ can be produced in polynomial time. As an extension of the designing framework, we further study a variant of the selfish scheduling game, and design a best coordination mechanism achieving $PoS=1$ and $PoA=4/3$ as well., Comment: Accepted by the 19th Conference on Web and InterNet Economics (WINE 2023)

We consider integer programs whose constraint matrix has a special block structure: $\min\{f(x):H_{com}x=b, l\le x\le u,x\in\mathbb{Z}^{t_B+nt_A}\}$, where the objective function $f$ is separable convex and the constraint matrix $H_ {com}$ is composed of small submatrices $A_i,B,C,D_i$ such that the first row is $(C,D_1,D_2,\ldots,D_n)$, the first column is $(C,B,B,\ldots,B)^{\top}$, the main diagonal is $(C,A_1,A_2,\ldots,A_n)$, and the rest entries are 0. Furthermore, $\text{rank}(B)$=1. We study fixed parameter tractable (FPT) algorithms by taking as parameters the number of rows and columns of small submatrices, together with the largest absolute value over their entries. We call the IP (almost) combinatorial 4-block n-fold IP. It generalizes the generalized n-fold IP and is a special case of the generalized 4-block n-fold IP. The existence of FPT algorithms for the generalized 4-block n-fold IP is a major open problem, which motivates us to study special cases of the generalized 4-block n-fold IP to find structural insights. We show the $\ell_{\infty}$-norm of Graver basis elements of combinatorial 4-block n-fold IP is $\Omega(n)$. There is some FPT-value $\lambda$ such that for any nonzero element $g\in\{x: H_{com} x= 0\}$, $\lambda g$ can always be decomposed into Graver basis elements in the same orthant whose $\ell_{\infty}$-norm is FPT-bounded (while g might not admit such a decomposition). Then we can bound the $\ell_{\infty}$-norm of Graver basis elements by $O_{FPT}(n)$ and develop $O_{FPT}({n^4\hat{L}^2})$-time algorithms ($O_{FPT}$ hides a multiplicative FPT-term, and $\hat{L}$ denotes the logarithm of the largest number occurring in the input). As applications, combinatorial 4-block n-fold IP can be used to model some classical problems, including scheduling with rejection and bicriteria scheduling.

We consider 4-block $n$-fold integer programming, which can be written as $\max\{w\cdot x: H x=b, l\le x\le u, x\in \mathbb{Z}^{N} \}$ where the constraint matrix $H$ is composed of small submatrices $A,B,C,D$ such that the first row of $H$ is $(C,D,D,\cdots,D)$, the first column of $H$ is $(C,B,B,\cdots,B)$, the main diagonal of $H$ is $(C,A,A,\cdots,A)$, and all the other entries are $0$. The special case where $B=C=0$ is known as $n$-fold integer programming. Prior algorithmic results for 4-block $n$-fold integer programming and its special cases usually take $\Delta$, the largest absolute value among entries of $H$ as part of the parameters. In this paper, we explore the possibility of getting rid of $\Delta$ from parameters, i.e., we are looking for algorithms that runs polynomially in $\log\Delta$. We show that, assuming $\text{P}\neq \text{NP}$, this is not possible even if $A=(1,1,\Delta)$ and $B=C=0$. However, this becomes possible if $A=(1,1,\cdots,1)$ or $A\in \mathbb{Z}^{1\times 2}$, or more generally if $A\in\mathbb{Z}^{s_A\times t_A} $ where $t_A=s_A+1$ and the rank of matrix $A$ satisfies that $\text{rank}(A)=s_A$. More precisely, 1. If $A=(1,\ldots,1)\in \mathbb{Z}^{1\times t_A} $, then 4-block $n$-fold IP can be solved in $(t_A+t_B)^{O(t_A+t_B)}\cdot poly(n,\log\Delta)$ time. 2. If $A\in\mathbb{Z}^{s_A\times t_A} $, $t_A=s_A+1$ and $\text{rank}(A)=s_A$, then 4-block $n$-fold IP can be solved in $(t_A+t_B)^{O(t_A+t_B)}\cdot n^{O(t_A)}\cdot poly(\log\Delta)$ time; Specifically, if in addition we have $B=C=0$ (i.e., $n$-fold integer programming), then it can be solved in linear time $n\cdot poly(t_A,\log \Delta)$., Comment: 21 pages

The ridesharing problem is that given a set of trips, each trip consists of an individual, a vehicle of the individual and some requirements, select a subset of trips and use the vehicles of selected trips to deliver all individuals to their destinations satisfying the requirements. Requirements of trips are specified by parameters including source, destination, vehicle capacity, preferred paths of a driver, detour distance and number of stops a driver is willing to make, and time constraints. We analyze the relations between the time complexity and parameters for two optimization problems: minimizing the number of selected vehicles and minimizing total travel distance of the vehicles. We consider the following conditions: (1) all trips have the same source or same destination, (2) no detour is allowed, (3) each participant has one preferred path, (4) no limit on the number of stops, and (5) all trips have the same departure and same arrival time. It is known that both minimization problems are NP-hard if one of Conditions (1), (2) and (3) is not satisfied. We prove that both problems are NP-hard and further show that it is NP-hard to approximate both problems within a constant factor if Conditions (4) or (5) is not satisfied. We give $\frac{K+2}{2}$-approximation algorithms for minimizing the number of selected vehicles when condition (4) is not satisfied, where $K$ is the largest capacity of all vehicles., Comment: 39 pages, 6 figures

In this paper, we consider the bandwidth-delay-hop constrained routing problem in large-scaled software defined networks. A number of demands, each of which specifies a source vertex and a sink vertex, are required to route in a given network. We are asked to select a subset of demands, and assign a routing path for each selected demand without violating the hop and delay constraints, while assuring that the bandwidth occupied in each edge is not beyond its capacity. The goal is to maximize the throughput (the total bandwidth) of the selected demands. We develop an efficient heuristic algorithm for the problem, which consists of three main steps, namely, computing feasible paths for each demand, sorting the demands with some priority rules, selecting a path for each demand. The algorithm is tested with networks of actual sizes and topologies, generated by Huawei Technologies Company. The experiments show that the proposed approach outperforms existing algorithms both in throughput and in running time. In the experiments, our algorithm achieves more than 90\% of the total bandwidth of the given demands within 10 seconds. Moreover, a large part of our algorithm can run in parallel which largely speeds up the process when using multi-core processors., Comment: 15 pages

We consider the classical scheduling problem on parallel identical machines to minimize the makespan, and achieve the following results under the Exponential Time Hypothesis (ETH) 1. The scheduling problem on a constant number $m$ of identical machines, which is denoted as $Pm||C_{max}$, is known to admit a fully polynomial time approximation scheme (FPTAS) of running time $O(n) + (1/\epsilon)^{O(m)}$ (indeed, the algorithm works for an even more general problem where machines are unrelated). We prove this algorithm is essentially the best possible in the sense that a $(1/\epsilon)^{O(m^{1-\delta})}+n^{O(1)}$ time FPTAS for any $\delta>0$ implies that ETH fails. 2. The scheduling problem on an arbitrary number of identical machines, which is denoted as $P||C_{max}$, is known to admit a polynomial time approximation scheme (PTAS) of running time $2^{O(1/\epsilon^2\log^3(1/\epsilon))}+n^{O(1)}$. We prove this algorithm is nearly optimal in the sense that a $2^{O((1/\epsilon)^{1-\delta})}+n^{O(1)}$ time PTAS for any $\delta>0$ implies that ETH fails, leaving a small room for improvement. To obtain these results we will provide two new reductions from 3SAT, one for $Pm||C_{max}$ and another for $P||C_{max}$. Indeed, the new reductions explore the structure of scheduling problems and can also lead to other interesting results. For example, using the framework of our reduction for $P||C_{max}$, Chen et al. (arXiv:1306.3727) is able to prove the APX-hardness of the scheduling problem in which the matrix of job processing times $P=(p_{ij})_{m\times n}$ is of rank 3, solving the open problem mentioned by Bhaskara et al. (SODA 2013)., Comment: 33 pages, to appear in SODA 2014

We consider the classical minimum makespan scheduling problem, where the processing time of job $j$ on machine $i$ is $p_{ij}$, and the matrix $P=(p_{ij})_{m\times n}$ is of a low rank. It is proved in (Bhaskara et al., SODA 2013) that rank 7 scheduling is NP-hard to approximate to a factor of $3/2-\epsilon$, and rank 4 scheduling is APX-hard (NP-hard to approximate within a factor of $1.03-\epsilon$). We improve this result by showing that rank 4 scheduling is already NP-hard to approximate within a factor of $3/2-\epsilon$, and meanwhile rank 3 scheduling is APX-hard., Comment: 14 pages

We consider the classical online scheduling problem P||C_{max} in which jobs are released over list and provide a nearly optimal online algorithm. More precisely, an online algorithm whose competitive ratio is at most (1+\epsilon) times that of an optimal online algorithm could be achieved in polynomial time, where m, the number of machines, is a part of the input. It substantially improves upon the previous results by almost closing the gap between the currently best known lower bound of 1.88 (Rudin, Ph.D thesis, 2001) and the best known upper bound of 1.92 (Fleischer, Wahl, Journal of Scheduling, 2000). It has been known by folklore that an online problem could be viewed as a game between an adversary and the online player. Our approach extensively explores such a structure and builds up a completely new framework to show that, for the online over list scheduling problem, given any \epsilon>0, there exists a uniform threshold K which is polynomial in m such that if the competitive ratio of an online algorithm is \rho<=2, then there exists a list of at most K jobs to enforce the online algorithm to achieve a competitive ratio of at least \rho-O(\epsilon). Our approach is substantially different from that of Gunther et al. (Gunther et al., SODA 2013), in which an approximation scheme for online over time scheduling problems is given, where the number of machines is fixed. Our method could also be extended to several related online over list scheduling models., Comment: 21 pages

The 2D Online Bin Packing is a fundamental problem in Computer Science and the determination of its asymptotic competitive ratio has attracted great research attention. In a long series of papers, the lower bound of this ratio has been improved from 1.808, 1.856 to 1.907 and its upper bound reduced from 3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In this paper, we rewrite the upper bound record to 2.5545. Our idea for the improvement is as follows. In SODA 2002 \cite{SS03}, Seiden and van Stee proposed an elegant algorithm called $H \otimes B$, comprised of the {\em Harmonic algorithm} $H$ and the {\em Improved Harmonic algorithm} $B$, for the two-dimensional online bin packing problem and proved that the algorithm has an asymptotic competitive ratio of at most 2.66013. Since the best known online algorithm for one-dimensional bin packing is the {\em Super Harmonic algorithm} \cite{S02}, a natural question to ask is: could a better upper bound be achieved by using the Super Harmonic algorithm instead of the Improved Harmonic algorithm? However, as mentioned in \cite{SS03}, the previous analysis framework does not work. In this paper, we give a positive answer for the above question. A new upper bound of 2.5545 is obtained for 2-dimensional online bin packing. The main idea is to develop new weighting functions for the Super Harmonic algorithm and propose new techniques to bound the total weight in a rectangular bin.

In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We first observe there is a Polynomial Time Approximation Scheme (PTAS) for the problem of packing weighted squares into rectangular bins with large resources, then apply the PTAS to the problem of packing squares with profits into a rectangular bin and get a $\frac65+\epsilon$ approximation algorithm.

In this paper, we study the 3D strip packing problem in which we are given a list of 3-dimensional boxes and required to pack all of them into a 3-dimensional strip with length 1 and width 1 and unlimited height to minimize the height used. Our results are below: i) we give an approximation algorithm with asymptotic worst-case ratio 1.69103, which improves the previous best bound of $2+\epsilon$ by Jansen and Solis-Oba of SODA 2006; ii) we also present an asymptotic PTAS for the case in which all items have {\em square} bases., Comment: Submitted to SODA 2007

In this paper we establish a general algorithmic framework between bin packing and strip packing, with which we achieve the same asymptotic bounds by applying bin packing algorithms to strip packing. More precisely we obtain the following results: (1) Any offline bin packing algorithm can be applied to strip packing maintaining the same asymptotic worst-case ratio. Thus using FFD (MFFD) as a subroutine, we get a practical (simple and fast) algorithm for strip packing with an upper bound 11/9 (71/60). A simple AFPTAS for strip packing immediately follows. (2) A class of Harmonic-based algorithms for bin packing can be applied to online strip packing maintaining the same asymptotic competitive ratio. It implies online strip packing admits an upper bound of 1.58889 on the asymptotic competitive ratio, which is very close to the lower bound 1.5401 and significantly improves the previously best bound of 1.6910 and affirmatively answers an open question posed by Csirik et. al., Comment: 12 pages, 3 figures