arXiv subjects

Danny Hermelin

Publications and source records attributed to Danny Hermelin.

At least 37 records · Page 2Linked to original sources

arXiv2021

Equitable Scheduling on a Single Machine

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 $k \le m$ days. This corresponds to an equitable schedule where each client is guaranteed a minimal level of service throughout the period of $m$ days. We provide a thorough analysis of the computational complexity of three main variants of this problem, identifying both efficient algorithms and worst-case intractability results.

cs.DM↗

arXiv2021

A comparison between D-wave and a classical approximation algorithm and a heuristic for computing the ground state of an Ising spin glass

Finding the ground state of an Ising-spin glass on general graphs belongs to the class of NP-hard problems, widely believed to have no efficient polynomial-time algorithms for solving them. An approach developed in computer science for dealing with such problems is to devise approximation algorithms that run in polynomial time, and provide solutions with provable guarantees on their quality in terms of the optimal unknown solution. Recently, several algorithms for the Ising-spin glass problem on a graph that provide different approximation guarantees were introduced albeit without implementation. Also recently, D-wave company constructed a physical realization of an adiabatic quantum computer, and enabled researchers to access it. D-wave is particularly suited for computing an approximation for the ground state of an Ising spin glass on its chimera graph -- a graph with bounded degree. In this work, we compare the performance of a recently developed approximation algorithm for solving the Ising spin glass problem on graphs of bounded degree against the D-wave computer. We also compared a heuristic tailored specifically to handle the fixed D-wave chimera graph. D-wave computer was able to find better approximations to all the random instances we studied. Furthermore the convergence times of D-wave were also significantly better. These results indicate the merit of D-wave computer under certain specific instances. More broadly, our method is relevant to other performance comparison studies. We suggest that it is important to compare the performance of quantum computers not only against exact classical algorithms with exponential run-time scaling, but also to approximation algorithms with polynomial run-time scaling and a provable guarantee on performance.

cond-mat.dis-nn↗

arXiv2021

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

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. One of the most intriguing open problems 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 $T^{1-\varepsilon}\cdot 2^{o(n)}$ for any $\varepsilon>0$, unless the Strong Exponential Time Hypothesis (SETH) fails. This is one of the strongest known connections between any two of the core problems of fine-grained complexity. 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 $\tilde{O}(L+m)$, in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

cs.DS↗

arXiv2020

Approximating Sparse Quadratic Programs

Given a matrix $A \in \mathbb{R}^{n\times n}$, we consider the problem of maximizing $x^TAx$ subject to the constraint $x \in \{-1,1\}^n$. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an $Ω(1/\lg n)$ approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an $Ω(1)$ approximation when $A$ corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is $\tilde{O}(n^{1.5}\cdot \min\{N,n^{1.5}\})$, where $N$ is the number of nonzero entries in $A$ and $\tilde{O}$ ignores polylogarithmic factors. In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where $A$ is sparse (i.e., has $O(n)$ nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that: - MaxQP admits a $(1/2Δ)$-approximation in $O(n \lg n)$ time, where $Δ$ is the maximum degree of the corresponding graph. - UnitMaxQP, where $A \in \{-1,0,1\}^{n\times n}$, admits a $(1/2d)$-approximation in $O(n)$ time when the corresponding graph is $d$-degenerate, and a $(1/3δ)$-approximation in $O(n^{1.5})$ time when the corresponding graph has $δn$ edges. - MaxQP admits a $(1-\varepsilon)$-approximation in $O(n)$ time when the corresponding graph and each of its minors have bounded local treewidth. - UnitMaxQP admits a $(1-\varepsilon)$-approximation in $O(n^2)$ time when the corresponding graph is $H$-minor free.

cs.DS↗

arXiv2020

Efficient fully dynamic elimination forests with applications to detecting long paths and cycles

We present a data structure that in a dynamic graph of treedepth at most $d$, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time $2^{{\cal O}(d^2)}$, which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time $f(d)$ for some non-elementary (i.e. tower-exponential) function $f$. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth $d$ from doubly-exponential in $d$ to $d^{{\cal O}(d)}$. As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter $k$ and a dynamic graph $G$, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: - Does $G$ contain a simple path on $k$ vertices? - Does $G$ contain a simple cycle on at least $k$ vertices? In the first case, the data structure achieves amortized update time $2^{{\cal O}(k^2)}$. In the second case, the amortized update time is $2^{{\cal O}(k^4)} + {\cal O}(k \log n)$. In both cases we assume access to a dictionary on the edges of $G$.

cs.DS↗

arXiv2020

Scheduling Lower Bounds via AND Subset Sum

Given $N$ instances $(X_1,t_1),\ldots,(X_N,t_N)$ 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 $X_i$ has a subset that sums up to the target integer $t_i$. We prove that this problem cannot be solved in time $\tilde{O}((N \cdot t_{max})^{1-ε})$, for $t_{max}=\max_i t_i$ and any $ε> 0$, assuming the $\forall \exists$ Strong Exponential Time Hypothesis ($\forall \exists$-SETH). We then use this result to exclude $\tilde{O}(n+P_{max} \cdot n^{1-ε})$-time algorithms for several scheduling problems on $n$ jobs with maximum processing time $P_{max}$, based on $\forall \exists$-SETH. These include classical problems such as $1||\sum w_jU_j$, the problem of minimizing the total weight of tardy jobs on a single machine, and $P_2||\sum U_j$, the problem of minimizing the number of tardy jobs on two identical parallel machines.

cs.DS↗

arXiv2020

Faster Minimization of Tardy Processing Time on a Single Machine

This paper is concerned with the $1||\sum p_jU_j$ 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 a very 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 fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs 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 $1||\sum p_jU_j$, each improving on Lawler and Moore's algorithm in a different scenario. 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, while for the first algorithm we define a new "skewed" version of $(\max,\min)$-convolution which is interesting in its own right.

cs.DS↗

arXiv2019

Single Machine Batch Scheduling to Minimize the Weighted Number of Tardy Jobs

The $1|B,r_j|\sum w_jU_j$ scheduling problem takes as input a batch setup time $Δ$ and a set of $n$ jobs, each having a processing time, a release date, a weight, and a due date; the task is to find a sequence of batches that minimizes the weighted number of tardy jobs. This problem was introduced by Hochbaum and Landy in 1994; as a wide generalization of {\sc Knapsack}, it is $\mathsf{NP}$-hard. In this work we provide a multivariate complexity analysis of the $1|B,r_j|\sum w_jU_j$ problem with respect to several natural parameters. That is, we establish a thorough classification into fixed-parameter tractable and $\mathsf{W}[1]$-hard problems, for parameter combinations of (i) $\#p$ = distinct number of processing times, (ii) $\#w$ = number of distinct weights, (iii) $\#d$ = number of distinct due dates, (iv) $\#r$ = number of distinct release dates, and (v) $b$ = batch sizes. Thereby, we significantly extend the work of Hermelin et al. (2018) who analyzed the parameterized complexity of the non-batch variant of this problem without release dates. As one of our key results, we prove that $1|B,r_j|\sum w_jU_j$ is $\mathsf{W}[1]$-hard parameterized by the number of distinct processing times and distinct due dates. To the best of our knowledge, these are the first parameterized intractability results for scheduling problems with few distinct processing times. Further, we show that $1|B,r_j|\sum w_jU_j$ is fixed-parameter tractable with respect to parameter $\#p+\#d+\#r$ and with respect to parameter $\#w+\#d$ if there is just a single release date. Both results hold even if the number of jobs per batch is limited by some integer $b$.

cs.DS↗

arXiv2019

On Computing Centroids According to the $p$-Norms of Hamming Distance Vectors

In this paper we consider the $p$-Norm Hamming Centroid problem which asks to determine whether some given binary strings have a centroid with a bound on the $p$-norm of its Hamming distances to the strings. Specifically, given a set of strings $S$ and a real $k$, we consider the problem of determining whether there exists a string $s^*$ with $\big(\sum_{s \in S}d^p(s^*,s)\big)^{1/p} \leq k$, where $d(,)$ denotes the Hamming distance metric. This problem has important applications in data clustering, and is a generalization of the well-known polynomial-time solvable \textsc{Consensus String} $(p=1)$ problem, as well as the NP-hard \textsc{Closest String} $(p=\infty)$ problem. Our main result shows that the problem is NP-hard for all fixed rational $p > 1$, closing the gap for all rational values of $p$ between $1$ and $\infty$. Under standard complexity assumptions the reduction also implies that the problem has no $2^{o(n+m)}$-time or $2^{o(k^{\frac{p}{(p+1)}})}$-time algorithm, where $m$ denotes the number of input strings and $n$ denotes the length of each string, for any fixed $p > 1$. Both running time lower bounds are tight. In particular, we provide a $2^{k^{\frac{p}{(p+1)}+\varepsilon}}$-time algorithm for each fixed $\varepsilon > 0$. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-$2$ approximation algorithm for the problem.

cs.CC↗

arXiv2019

Domination When the Stars Are Out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K_{1,4} as an induced subgraph (K_{1,4}-free graphs). We show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that Connected Dominating Set has no polynomial kernel on claw-free graphs and is not fixed-parameter tractable on K_{1,4}-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K_{1,L}-free graphs and show that the problem is fixed-parameter tractable if and only if L <= 3.

cs.DS↗

arXiv2018

The Clever Shopper Problem

We investigate a variant of the so-called "Internet Shopping Problem" introduced by Blazewicz et al. (2010), where a customer wants to buy a list of products at the lowest possible total cost from shops which offer discounts when purchases exceed a certain threshold. Although the problem is NP-hard, we provide exact algorithms for several cases, e.g. when each shop sells only two items, and an FPT algorithm for the number of items, or for the number of shops when all prices are equal. We complement each result with hardness proofs in order to draw a tight boundary between tractable and intractable cases. Finally, we give an approximation algorithm and hardness results for the problem of maximising the sum of discounts.

cs.DS↗

arXiv2018

How hard is it to satisfy (almost) all roommates?

The classic Stable Roommates problem (which is the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost γ, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable Roommates, which seeks to find a matching with a minimum number β of blocking pairs. Our main result is that Egal Stable Roommates parameterized by γ is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by β is W[1]-hard, even if the length of each preference list is at most five.

cs.CC↗

arXiv2017

Fractals for Kernelization Lower Bounds

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim. 2011], we show that, unless NP $\subseteq$ coNP / poly, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most $k$ edges such that the resulting graph has no $s$-$t$ path of length shorter than $\ell$) parameterized by the combination of $k$ and $\ell$ has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems. Along the way, we show that LBEC remains NP-hard on planar graphs, a result which we believe is interesting in its own right.

cs.CC↗

arXiv2017

New Algorithms for Minimizing the Weighted Number of Tardy Jobs On a Single Machine

In this paper we study the classical single machine scheduling problem where the objective is to minimize the total weight of tardy jobs. Our analysis focuses on the case where one or more of three natural parameters is either constant or is taken as a parameter in the sense of parameterized complexity. These three parameters are the number of different due dates, processing times, and weights in our set of input jobs. We show that the problem belongs to the class of fixed parameter tractable (FPT)problems when combining any two of these three parameters. We also show that the problem is polynomial-time solvable when the latter two parameters are constant, complementing Karp's result who showed that the problem is NP-hard already for a single due date.

cs.DS↗

arXiv2017

Scheduling Two Agents on a Single Machine: A Parameterized Analysis of NP-hard Problems

Scheduling theory is an old and well-established area in combinatorial optimization, whereas the much younger area of parameterized complexity has only recently gained the attention of the community. Our aim is to bring these two areas closer together by studying the parameterized complexity of a class of single-machine two-agent scheduling problems. Our analysis focuses on the case where the number of jobs belonging to the second agent is considerably smaller than the number of jobs belonging to the first agent, and thus can be considered as a fixed parameter k. We study a variety of combinations of scheduling criteria for the two agents, and for each such combination we pinpoint its parameterized complexity with respect to the parameter k. The scheduling criteria that we analyze include the total weighted completion time, the total weighted number of tardy jobs, and the total weighted number of just-in-time jobs. Our analysis draws a borderline between tractable and intractable variants of these problems.

cs.DS↗

arXiv2017

On The Parameterized Tractability of the Just-In-Time Flow-Shop Scheduling Problem

Since its development in the early 90's, parameterized complexity has been widely used to analyze the tractability of many NP-hard combinatorial optimization problems with respect to various types of problem parameters. While the generic nature of the framework allows the analysis of any combinatorial problem, the main focus along the years was on analyzing graph problems. In this paper we diverge from this trend by studying the parameterized complexity of Just-In-Time (JIT) flow-shop scheduling problems. Our analysis focuses on the case where the number of due dates is considerably smaller than the number of jobs, and can thus be considered as a parameter. We prove that the two-machine problem is W[1]-hard with respect to this parameter, even if all processing times on the second machine are of unit length, while the problem is in XP even for a parameterized number of machines. We then move on to study the tractability of the problem when combining the different number of due dates with either the number of different weights or the number of different processing times on the first machine. We prove that in both cases the problem is fixed-parameter tractable for the two machine case, and is W[1]-hard for three or more machines.

cs.DS↗

arXiv2017

Diminishable Parameterized Problems and Strict Polynomial Kernelization

Kernelization---a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems---plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of NP $\nsubseteq$ coNP$/$poly. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [Theory Comput. Syst. 2011] and developing a general and remarkably simple framework, we show that a variety of FPT problems does not admit strict polynomial kernels under the weaker assumption of P $\neq$ NP. In particular, we show that various (multicolored) graph problems and Turing machine computation problems do not admit strict polynomial kernels unless P $=$ NP. To this end, a key concept we use are diminishable problems; these are parameterized problems that allow to decrease the parameter of the input instance by at least one in polynomial time, thereby outputting an equivalent problem instance. Finally, we study a relaxation of the notion of strict kernels and reveal its limitations.

cs.CC↗

arXiv2017

Coordination of Mobile Mules via Facility Location Strategies

In this paper, we study the problem of wireless sensor network (WSN) maintenance using mobile entities called mules. The mules are deployed in the area of the WSN in such a way that would minimize the time it takes them to reach a failed sensor and fix it. The mules must constantly optimize their collective deployment to account for occupied mules. The objective is to define the optimal deployment and task allocation strategy for the mules, so that the sensors' downtime and the mules' traveling distance are minimized. Our solutions are inspired by research in the field of computational geometry and the design of our algorithms is based on state of the art approximation algorithms for the classical problem of facility location. Our empirical results demonstrate how cooperation enhances the team's performance, and indicate that a combination of k-Median based deployment with closest-available task allocation provides the best results in terms of minimizing the sensors' downtime but is inefficient in terms of the mules' travel distance. A k-Centroid based deployment produces good results in both criteria.

cs.NI↗