56,515 research outputs found
Optimal Algorithms for Scheduling under Time-of-Use Tariffs
We consider a natural generalization of classical scheduling problems in
which using a time unit for processing a job causes some time-dependent cost
which must be paid in addition to the standard scheduling cost. We study the
scheduling objectives of minimizing the makespan and the sum of (weighted)
completion times. It is not difficult to derive a polynomial-time algorithm for
preemptive scheduling to minimize the makespan on unrelated machines. The
problem of minimizing the total (weighted) completion time is considerably
harder, even on a single machine. We present a polynomial-time algorithm that
computes for any given sequence of jobs an optimal schedule, i.e., the optimal
set of time-slots to be used for scheduling jobs according to the given
sequence. This result is based on dynamic programming using a subtle analysis
of the structure of optimal solutions and a potential function argument. With
this algorithm, we solve the unweighted problem optimally in polynomial time.
For the more general problem, in which jobs may have individual weights, we
develop a polynomial-time approximation scheme (PTAS) based on a dual
scheduling approach introduced for scheduling on a machine of varying speed. As
the weighted problem is strongly NP-hard, our PTAS is the best possible
approximation we can hope for.Comment: 17 pages; A preliminary version of this paper with a subset of
results appeared in the Proceedings of MFCS 201
Approximability of the Subset Sum Reconfiguration Problem
The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in the reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme (PTAS), while the problem is APX-hard if we are given a conflict graph
Efficient Mobile Edge Computing for Mobile Internet of Thing in 5G Networks
We study the off-line efficient mobile edge computing (EMEC) problem for a joint computing to process a task both locally and remotely with the objective of minimizing the finishing time. When computing remotely, the time will include the communication and computing time. We first describe the time model, formulate EMEC, prove NP-completeness of EMEC, and show the lower bound. We then provide an integer linear programming (ILP) based algorithm to achieve the optimal solution and give results for small-scale cases. A fully polynomial-time approximation scheme (FPTAS), named Approximation Partition (AP), is provided through converting ILP to the subset sum problem. Numerical results show that both the total data length and the movement have great impact on the time for mobile edge computing. Numerical results also demonstrate that our AP algorithm obtain the finishing time, which is close to the optimal solution
A QPTAS for the General Scheduling Problem with Identical Release Dates
The General Scheduling Problem (GSP) generalizes scheduling problems with sum of cost objectives such as weighted flow time and weighted tardiness. Given a set of jobs with processing times, release dates, and job dependent cost functions, we seek to find a minimum cost preemptive schedule on a single machine. The best known algorithm for this problem and also for weighted flow time/tardiness is an O(loglog P)-approximation (where P denotes the range of the job processing times), while the best lower bound shows only strong NP-hardness. When release dates are identical there is also a gap: the problem remains strongly NP-hard and the best known approximation algorithm has a ratio of e+epsilon (running in quasi-polynomial time). We reduce the latter gap by giving a QPTAS if the numbers in the input are quasi-polynomially bounded, ruling out the existence of an APX-hardness proof unless NPsubseteq DTIME(2^polylog(n)). Our techniques are based on the QPTAS known for the UFP-Cover problem, a particular case of GSP where we must pick a subset of intervals (jobs) on the real line with associated heights and costs. If an interval is selected, its height will help cover a given demand on any point contained within the interval. We reduce our problem to a generalization of UFP-Cover and use a sophisticated divide-and-conquer procedure with interdependent non-symmetric subproblems.
We also present a pseudo-polynomial time approximation scheme for two variants of UFP-Cover. For the case of agreeable intervals we give an algorithm based on a new dynamic programming approach which might be useful for other problems of this type. The second one is a resource augmentation setting where we are allowed to slightly enlarge each interval
Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling
The Sum of Squares (\sos{}) hierarchy gives an automatized technique to
create a family of increasingly tight convex relaxations for binary programs.
There are several problems for which a constant number of rounds of this
hierarchy give integrality gaps matching the best known approximation
algorithms. For many other problems, however, ad-hoc techniques give better
approximation ratios than \sos{} in the worst case, as shown by corresponding
lower bound instances. Notably, in many cases these instances are invariant
under the action of a large permutation group. This yields the question how
symmetries in a formulation degrade the performance of the relaxation obtained
by the \sos{} hierarchy. In this paper, we study this for the case of the
minimum makespan problem on identical machines. Our first result is to show
that rounds of \sos{} applied over the \emph{configuration linear
program} yields an integrality gap of at least , where is the
number of jobs. Our result is based on tools from representation theory of
symmetric groups. Then, we consider the weaker \emph{assignment linear program}
and add a well chosen set of symmetry breaking inequalities that removes a
subset of the machine permutation symmetries. We show that applying
rounds of the SA hierarchy to this stronger
linear program reduces the integrality gap to , which yields a
linear programming based polynomial time approximation scheme. Our results
suggest that for this classical problem, symmetries were the main barrier
preventing the \sos{}/ SA hierarchies to give relaxations of polynomial
complexity with an integrality gap of~. We leave as an open
question whether this phenomenon occurs for other symmetric problems
Approximation algorithms for the MAXSPACE advertisement problem
In the MAXSPACE problem, given a set of ads A, one wants to schedule a subset
A' of A into K slots B_1, ..., B_K of size L. Each ad A_i in A has a size s_i
and a frequency w_i. A schedule is feasible if the total size of ads in any
slot is at most L, and each ad A_i in A' appears in exactly w_i slots. The goal
is to find a feasible schedule that maximizes the sum of the space occupied by
all slots. We introduce a generalization called MAXSPACE-R in which each ad A_i
also has a release date r_i >= 1, and may only appear in a slot B_j with j >=
r_i. We also introduce a generalization of MAXSPACE-R called MAXSPACE-RD in
which each ad A_i also has a deadline d_i <= K, and may only appear in a slot
B_j with r_i <= j <= d_i. These parameters model situations where a subset of
ads corresponds to a commercial campaign with an announcement date that may
expire after some defined period. We present a 1/9-approximation algorithm for
MAXSPACE-R and a polynomial-time approximation scheme for MAXSPACE-RD when K is
bounded by a constant. This is the best factor one can expect, since MAXSPACE
is NP-hard, even if K = 2
-anonymous Signaling Scheme
We incorporate signaling scheme into Ad Auction setting, to achieve better
welfare and revenue while protect users' privacy. We propose a new
\emph{-anonymous signaling scheme setting}, prove the hardness of the
corresponding welfare/revenue maximization problem, and finally propose the
algorithms to approximate the optimal revenue or welfare
Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack
The 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\u2705]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA\u2709] and it admits a QPTAS [Adamaszek and Wiese, FOCS\u2713; Chuzhoy and Ene, FOCS\u2716]. 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\u2704]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA\u2715]. 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\u27(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\u2717]
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