5,066 research outputs found
Breaking the Barrier of 2 for the Storage Allocation Problem
No full textPacking problems are an important class of optimization problems. The probably most well-known problem if this type is knapsack and many generalizations of it have been studied in the literature like Two-dimensional Geometric Knapsack (2DKP) and Unsplittable Flow on a Path (UFP). For the latter two problems, recently the first polynomial time approximation algorithms with better approximation ratios than 2 were presented [Gálvez et al., FOCS 2017][Grandoni et al., STOC 2018]. In this paper we break the barrier of 2 for the Storage Allocation Problem (SAP), a problem which combines properties of 2DKP and UFP. In SAP, we are given a path with capacitated edges and a set of tasks where each task has a start vertex, an end vertex, a size, and a profit. We seek to select the most profitable set of tasks that we can draw as non-overlapping rectangles underneath the capacity profile of the edges where the height of each rectangle equals the size of the corresponding task.
The problem SAP appears naturally in settings of allocating resources like memory, bandwidth, etc. where each request needs a contiguous portion of the resource. The best known polynomial time approximation algorithm for SAP has an approximation ratio of 2+ε [Mömke and Wiese, ICALP 2015] and no better quasi-polynomial time algorithm is known. We present a polynomial time (63/32+ε) < 1.969-approximation algorithm for the important case of uniform edge capacities and a quasi-polynomial time (1.997+ε)-approximation algorithm for non-uniform quasi-polynomially bounded edge capacities. Key to our results are building blocks consisting of stair-blocks, jammed tasks, and boxes that we use to construct profitable solutions and which allow us to compute solutions of these types efficiently. Finally, using our techniques we show that under slight resource augmentation we can obtain even approximation ratios of 3/2+ε in polynomial time and 1+ε in quasi-polynomial time, both for arbitrary edge capacities
Fixed-Parameter Approximation Schemes for Weighted Flowtime
No full textGiven a set of n jobs with integral release dates, processing times and weights, it is a natural and important scheduling problem to compute a schedule that minimizes the sum of the weighted flow times of the jobs. There are strong lower bounds for the possible approximation ratios. In the non-preemptive case, even on a single machine the best known result is a O(sqrt{n})-approximation which is best possible. In the preemptive case on m identical machines there is a O(log min{n/m,P})-approximation (where P denotes the maximum job size) which is also best possible.
We study the problem in the parametrized setting where our parameter k is an upper bound on the maximum (integral) processing time and weight of a job, a standard parameter for scheduling problems. We present a (1+epsilon)-approximation algorithm for the preemptive and the non-preemptive case of minimizing weighted flow time on m machines with a running time of f(k,epsilon,m)* n^{O(1)}, i.e., our combined parameters are k,epsilon, and m. Key to our results is to distinguish time intervals according to whether in the optimal solution the pending jobs have large or small total weight. Depending on this we employ dynamic programming, linear programming, greedy routines, or combinations of the latter to compute the schedule for each respective interval
A (1+epsilon)-Approximation for Unsplittable Flow on a Path in Fixed-Parameter Running Time
No full textUnsplittable Flow on a Path (UFP) is a well-studied problem. It arises in many different settings such as bandwidth allocation, scheduling, and caching. We are given a path with capacities on the edges and a set of tasks, each of them is described by a start and an end vertex and a demand. The goal is to select as many tasks as possible such that the demand of the selected tasks using each edge does not exceed the capacity of this edge. The problem admits a QPTAS and the best known polynomial time result is a (2+epsilon)-approximation. As we prove in this paper, the problem is intractable for fixed-parameter algorithms since it is W[1]-hard. A PTAS seems difficult to construct.
However, we show that if we combine the paradigms of approximation algorithms and fixed-parameter tractability we can break the mentioned boundaries. We show that on instances with |OPT|=k we can compute a (1+epsilon)-approximation in time 2^O(k log k)n^O_epsilon(1) log(u_max) (where u_max is the maximum edge capacity).
To obtain this algorithm we develop new insights for UFP and enrich a recent dynamic programming framework for the problem. Our results yield a PTAS for (unweighted) UFP instances where |OPT| is at most O(log n/log log n) and they imply that the problem does not admit an EPTAS, unless W[1]=FPT
De Origine Et Ivre Collectarvm Provincialivm Ad Elocandas Filias Illvstres : Dissertatio Inavgvralis
No full textRostock, Univ., Jur. Diss., 1790Qvam Illvstris Ivreconsvltorvm Ordinis Avctoritate Pro Svmmis In Vtroqve Ivre Honotibvs Capessendis Die XXIII. Febrvarii MDCCXC. Pvblicae Ervditorvm Disqvisitioni Svbiicit Andreas Bernhard Carl Wiese RostochiensisAutopsie nach Ex. der ULB Sachsen-AnhaltVorlageform des Erscheinungsvermerks: Rostochii, Litteris Adlerianis
Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack
No full textThe area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability:
- In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k.
- In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR.
For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time.
For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17]
On the Two-Dimensional Knapsack Problem for Convex Polygons
No full textWe study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O(1)-approximation algorithm for the general case and a polynomial time O(1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Also, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation, i.e., we allow to increase the size of the knapsack by a factor of 1+δ for some δ > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles
Better Approximations for General Caching and UFP-Cover Under Resource Augmentation
No full textIn the Unsplittable Flow on a Path Cover (UFP-cover) problem we are given a path with a demand for each edge and a set of tasks where each task is defined by a subpath, a size and a cost. The goal is to select a subset of the tasks of minimum cost that together cover the demand of each edge. This problem models various resource allocation settings and also the general caching problem. The best known polynomial time approximation ratio for it is 4 [Bar-Noy et al., STOC 2000]. In this paper, we study the resource augmentation setting in which we need to cover only a slightly smaller demand on each edge than the compared optimal solution. If the cost of each task equals its size (which represents the natural bit-model in the related general caching problem) we provide a polynomial time algorithm that computes a solution of optimal cost. We extend this result to general caching and to the packing version of Unsplittable Flow on a Path in their respective natural resource augmentation settings. For the case that the cost of each task equals its "area", i.e., the product of its size and its path length, we present a polynomial time (1+ε)-approximation for UFP-cover. If additionally the edge capacities are in a constant range we compute even a solution of optimal cost and also obtain a PTAS without resource augmentation
Packing Cars into Narrow Roads: PTASs for Limited Supply Highway
No full textIn the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers’s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA'11, SICOMP'16].
In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway.
As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem
On Minimizing the Makespan When Some Jobs Cannot Be Assigned on the Same Machine
No full textWe study the classical scheduling problem of assigning jobs to machines in order to minimize the makespan. It is well-studied and admits an EPTAS on identical machines and a (2-1/m)-approximation algorithm on unrelated machines. In this paper we study a variation in which the input jobs are partitioned into bags and no two jobs from the same bag are allowed to be assigned on the same machine. Such a constraint can easily arise, e.g., due to system stability and redundancy considerations. Unfortunately, as we demonstrate in this paper, the techniques of the above results break down in the presence of these additional constraints.
Our first result is a PTAS for the case of identical machines. It enhances the methods from the known (E)PTASs by a finer classification of the input jobs and careful argumentations why a good schedule exists after enumerating over the large jobs. For unrelated machines, we prove that there can be no (log n)^{1/4-epsilon}-approximation algorithm for the problem for any epsilon > 0, assuming that NP nsubseteq ZPTIME(2^{(log n)^{O(1)}}). This holds even in the restricted assignment setting. However, we identify a special case of the latter in which we can do better: if the same set of machines we give an 8-approximation algorithm. It is based on rounding the LP-relaxation of the problem in phases and adjusting the residual fractional solution after each phase to order to respect the bag constraints
A Simpler QPTAS for Scheduling Jobs with Precedence Constraints
No full textWe study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a (2-1/m)-approximation. Interestingly, it is open whether the problem is NP-hard if m = 3 which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that m is a constant. In a break-through paper, Levey and Rothvoss presented a (1+ε)-approximation with a running time of n^{(log n)^{O((m²/ε²)log log n)}} [STOC 2016, SICOMP 2019] and this running time was improved to quasi-polynomial by Garg [ICALP 2018] and to even n^O_{m,ε}(log³log n) by Li [SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules.
In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly
- …
