Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1611.07724

Knapsack Problems: A Parameterized Point of View

Carolin Albrecht·Frank Gurski·Jochen Rethmann·Eda Yilmaz
View original source ↗Read source document ↗Export metadata ↓

Abstract

The knapsack problem (KP) is a very famous NP-hard problem in combinatorial optimization. Also its generalization to multiple dimensions named d-dimensional knapsack problem (d-KP) and to multiple knapsacks named multiple knapsack problem (MKP) are well known problems. Since KP, d-KP, and MKP are integer-valued problems defined on inputs of various informations, we study the fixed-parameter tractability of these problems. The idea behind fixed-parameter tractability is to split the complexity into two parts - one part that depends purely on the size of the input, and one part that depends on some parameter of the problem that tends to be small in practice. Further we consider the closely related question, whether the sizes and the values can be reduced, such that their bit-length is bounded polynomially or even constantly in a given parameter, i.e. the existence of kernelizations is studied. We discuss the following parameters: the number of items, the threshold value for the profit, the sizes, the profits, the number d of dimensions, and the number m of knapsacks. We also consider the connection of parameterized knapsack problems to linear programming, approximation, and pseudo-polynomial algorithms.

Explore related subjects

cs.CCcs.DS

Keep this discovery

BibTeXRIS

Carolin Albrecht, Frank Gurski, Jochen Rethmann, Eda Yilmaz. 2016-11-23. Knapsack Problems: A Parameterized Point of View. https://arxiv.org/abs/1611.07724

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗