This paper focuses on kernelization algorithms for the fundamental Knapsack problem. A kernelization algorithm (or kernel) is a polynomial-time reduction from a problem onto itself, where the output size is bounded by a function of some problem-specific parameter. Such algorithms provide a theoretical model for data reduction and preprocessing and are central in the area of parameterized complexity. In this way, a kernel for Knapsack for some parameter $k$ reduces any instance of Knapsack to an equivalent instance of size at most $f(k)$ in polynomial time, for some computable function $f(\cdot)$. When $f(k)=k^{O(1)}$ then we call such a reduction a polynomial kernel. Our study focuses on two natural parameters for Knapsack: The number of different item weights $w_{\#}$, and the number of different item profits $p_{\#}$. Our main technical contribution is a proof showing that Knapsack does not admit a polynomial kernel for any of these two parameters under standard complexity-theoretic assumptions. Our proof discovers an elaborate application of the standard kernelization lower bound framework, and develops along the way novel ideas that should be useful for other problems as well. We complement our lower bounds by showing the Knapsack admits a polynomial kernel for the combined parameter $w_{\#}+p_{\#}$.

In this paper we consider the fundamental scheduling problem of minimizing the weighted number of tardy jobs on a single machine. We present a simple pseudo polynomial-time algorithm for this problem that improves upon the classical Lawler and Moore algorithm from the late 60’s under certain scenarios and parameter settings. Our algorithm uses (max,+)-convolutions as its main tool, exploiting recent improved algorithms for computing such convolutions, and obtains several different running times depending on the specific improvement used. We also provide a related lower bound for the problem under a variant of the well-known Strong Exponential Time Hypothesis (SETH). History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by the Israel Science Foundation [Grant 1070/20].

This paper is concerned with the \(1|| \sum p_j U_j\) problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also an important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version in \(O(P \cdot n)\) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for problem, each improving on Lawler and Moore’s algorithm in a different scenario. Our first algorithm runs in \({\tilde{O}}(P^{7/4})\) time, and outperforms Lawler and Moore’s algorithm in instances where \(n={\tilde{\omega }}(P^{3/4})\) . Our second algorithm runs in \({\tilde{O}}(\min \{P \cdot D_{\#}, P + D\})\) time, where \(D_{\#}\) is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when \(n={\tilde{\omega }}(D_{\#})\) or \(n={\tilde{\omega }}(D/P)\) . Further, it extends the known \({\tilde{O}}(P)\) algorithm for the single due date special case of \(1||\sum p_jU_j\) in a natural way. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new “skewed” version of \((\max ,\min )\) -Convolution which is interesting in its own right.

Given N instances (X1,t1),…,(XN,tN) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers Xi has a subset that sums up to the target integer ti. We prove that this problem cannot be solved in time O˜((N⋅tmax)1−ε), for tmax=maxi⁡ti and any ε>0, assuming the ∀∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude O˜(n+pmax⋅n1−ε)-time algorithms for several scheduling problems on n jobs with maximum processing time pmax, assuming ∀∃-SETH. These include classical problems such as 1||∑wjUj, the problem of minimizing the total weight of tardy jobs on a single machine, and P2||∑Uj, the problem of minimizing the number of tardy jobs on two identical parallel machines.

Subset Sumand k -SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. An important open problem in this area is to base the hardness of one of these problems on the other. Our main result is a tight reduction from k -SAT to Subset Sum on dense instances, proving that Bellman’s 1962 pseudo-polynomial O * ( T )-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T 1-ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails. As a corollary, we prove a “Direct-OR” theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time ( N T ) 1-o(1) . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s,t -Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L , we show that the O ( Lm ) pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to Õ( L + m ), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

We introduce a natural but seemingly yet unstudied generalization of the problem of scheduling jobs on a single machine so as to minimize the number of tardy jobs. Our generalization lies in simultaneously considering several instances of the problem at once. In particular, we have n clients over a period of m days, where each client has a single job with its own processing time and deadline per day. Our goal is to provide a schedule for each of the m days, so that each client is guaranteed to have their job meet its deadline in at least k

Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. An important open problem in this area is to base the hardness of one of these problems on the other. Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial O*(T)-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T1–ε · 2o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails. As a corollary, we prove a “Direct-OR” theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (NT)1–o(1). As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-PATH problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to Õ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters. We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.