Computing knapsack solutions with cardinality robustness

Kakimura, Naonori; Makino, Kazuhisa; Seimi, Kento

doi:10.1007/s13160-012-0075-z

Cited by 12 publications

(8 citation statements)

References 25 publications

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“…A variant of the knapsack problem with a similar notion of robustness was proposed by Kakimura et al [22]. In this problem a knapsack solution needs to be computed, such that, for every k, the value of the k most valuable items in the knapsack compares well with the optimum solution using k items, for every k. Kakimura et al [22] restrict themselves to polynomial time algorithms and show that under this restriction a bounded competitive ratio is possible only if the rank quotient of the knapsack system is bounded. In contrast, our results show that if we do not restrict the running time and if we only require our solution to contain a good packing with k items for every k, then we can be (1 + ϕ)-competitive using our generic algorithm, even for generalizations like Multi-Dimensional Knapsack.…”

Section: Theorem 2 For Every Cardinality Constrained Problem With a Mmentioning

confidence: 99%

General bounds for incremental maximization

Bernstein¹,

Disser

Groß

2020

Math. Program.

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We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $$k\in {\mathbb {N}}$$ k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

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Section: Theorem 2 For Every Cardinality Constrained Problem With a Mmentioning

confidence: 99%

General bounds for incremental maximization

Bernstein¹,

Disser

Groß

2020

Math. Program.

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Section: Corollary 3 If There Is a Polynomial Time α-Approximation Al...mentioning

confidence: 99%

“…A variant of the knapsack problem with a similar notion of robustness was proposed by Kakimura et al [19]. In this problem a knapsack solution needs to be computed, such that, for every k, the value of the k most valuable items in the knapsack compares well with the optimum solution using k items, for every k. Kakimura et al [19] restrict themselves to polynomial time algorithms and show that under this restriction a bounded competitive ratio is possible only if the rank quotient of the knapsack system is bounded. In contrast, our results show that if we do not restrict the running time and if we only require our solution to contain a good packing with k items for every k, then we can be (1 + ϕ)-competitive using our generic algorithm, even for generalizations like Multi-Dimensional Knapsack.…”

Section: Corollary 3 If There Is a Polynomial Time α-Approximation Al...mentioning

confidence: 99%

General Bounds for Incremental Maximization

Bernstein¹,

Disser²,

Groß³

2017

Preprint

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We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below 2.18 in general.In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) 2.313 for the class of problems that satisfy this relaxed submodularity condition.Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.

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“…A related knapsack variant with an uncertain cardinality bound has been studied recently in [14]. It has been shown that any knapsack instance I admits a ν(I)-robust solution, where ν(I) is the rank of the knapsack system corresponding to I [11], and it is N P-hard to decide if a given instance admits an improved or even optimal robustness factor [14]. On the positive side, there is an FPTAS that finds solutions with nearly optimal robustness factor [14].…”

Section: Related Workmentioning

confidence: 99%

“…As our work shows, it may still be possible to get better solutions by focusing on instancessensitive guarantees. It is worth noting that this approach has been pursued, albeit not in a systematic way, in other contexts such as low-distortion embeddings of metrics into geometric spaces, e.g., in [6], subset selection problems [14], and earliest arrival network flows [9]. We hope that our work stimulates further research on instance-sensitive worst-case guarantees for other combinatorial optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Instance-sensitive robustness guarantees for sequencing with unknown packing and covering constraints

Megow

Mestre

2013

Proceedings of the 4th Conference on Innovations in Theoretical Computer Science

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Sequencing problems with an unknown covering or packing constraint appear in various applications, e.g., in real-time computing environments with uncertain run-time availability. A sequence is called α-robust when, for any possible constraint, the maximal or minimal prefix of the sequence that satisfies the constraint is at most a factor α from an optimal packing or covering. It is known that the covering problem always admits a 4-robust solution, and there are instances for which this factor is tight. For the packing variant no such constant robustness factor is possible in general.In this work we address the fact that many problem instances may allow for a much better robustness guarantee than the pathological worst case instances. We aim for more meaningful, instance-sensitive performance guarantees. We present an algorithm that constructs for each instance a solution with a robustness factor arbitrarily close to optimal. This implies nearly optimal solutions for previously studied problems such as the universal knapsack problem and for universal scheduling on an unreliable machine. The crucial ingredient and main result is a nearly exact feasibility test for dual-value sequencing with a given target function. We show that deciding exact feasibility is strongly N P-hard, and thus, our test is best possible, unless P=N P.We hope that the idea of instance-sensitive performance guarantees inspires to revisit other optimization problems and design algorithm tailored to perform well for each individual instance.

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Computing knapsack solutions with cardinality robustness

Cited by 12 publications

References 25 publications

General bounds for incremental maximization

General bounds for incremental maximization

General Bounds for Incremental Maximization

Instance-sensitive robustness guarantees for sequencing with unknown packing and covering constraints

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