263 research outputs found
Ising formulations of many NP problems
We provide Ising formulations for many NP-complete and NP-hard problems,
including all of Karp's 21 NP-complete problems. This collects and extends
mappings to the Ising model from partitioning, covering and satisfiability. In
each case, the required number of spins is at most cubic in the size of the
problem. This work may be useful in designing adiabatic quantum optimization
algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more
references; v3: substantial revision and extension, to-be-published versio
On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling
Vertex colouring is a well-known problem in combinatorial optimisation, whose
alternative integer programming formulations have recently attracted
considerable attention. This paper briefly surveys seven known formulations of
vertex colouring and introduces a formulation of vertex colouring using a
suitable clique partition of the graph. This formulation is applicable in
timetabling applications, where such a clique partition of the conflict graph
is given implicitly. In contrast with some alternatives, the presented
formulation can also be easily extended to accommodate complex performance
indicators (``soft constraints'') imposed in a number of real-life course
timetabling applications. Its performance depends on the quality of the clique
partition, but encouraging empirical results for the Udine Course Timetabling
problem are reported
Cost-optimal constrained correlation clustering via weighted partial Maximum Satisfiability
Peer reviewe
Incomplete MaxSAT Solving by Linear Programming Relaxation and Rounding
NP-hard optimization problems can be found in various real-world settings such as scheduling, planning and data analysis.
Coming up with algorithms that can efficiently solve these problems can save various rescources.
Instead of developing problem domain specific algorithms we can encode a problem instance as an instance of maximum satisfiability (MaxSAT), which is an optimization extension of Boolean satisfiability (SAT).
We can then solve instances resulting from this encoding using MaxSAT specific algorithms.
This way we can solve instances in various different problem domains by focusing on developing algorithms to solve MaxSAT instances.
Computing an optimal solution and proving optimality of the found solution can be time-consuming in real-world settings.
Finding an optimal solution for problems in these settings is often not feasible.
Instead we are only interested in finding a good quality solution fast.
Incomplete solvers trade guaranteed optimality for better scalability.
In this thesis, we study an incomplete solution approach for solving MaxSAT based on linear programming relaxation and rounding.
Linear programming (LP) relaxation and rounding has been used for obtaining approximation algorithms on various NP-hard optimization problems.
As such we are interested in investigating the effectiveness of this approach on MaxSAT.
We describe multiple rounding heuristics that are empirically evaluated on random, crafted and industrial MaxSAT instances from yearly MaxSAT Evaluations.
We compare rounding approaches against each other and to state-of-the-art incomplete solvers SATLike and Loandra.
The LP relaxation based rounding approaches are not competitive in general against either SATLike or Loandra
However, for some problem domains our approach manages to be competitive against SATLike and Loandra
LinSets.zip: Compressing Linear Set Diagrams
Linear diagrams are used to visualize set systems by depicting set
memberships as horizontal line segments in a matrix, where each set is
represented as a row and each element as a column. Each such line segment of a
set is shown in a contiguous horizontal range of cells of the matrix indicating
that the corresponding elements in the columns belong to the set. As each set
occupies its own row in the matrix, the total height of the resulting
visualization is as large as the number of sets in the instance. Such a linear
diagram can be visually sparse and intersecting sets containing the same
element might be represented by distant rows. To alleviate such undesirable
effects, we present LinSets.zip, a new approach that achieves a more
space-efficient representation of linear diagrams. First, we minimize the total
number of gaps in the horizontal segments by reordering columns, a criterion
that has been shown to increase readability in linear diagrams. The main
difference of LinSets.zip to linear diagrams is that multiple non-intersecting
sets can be positioned in the same row of the matrix. Furthermore, we present
several different rendering variations for a matrix-based representation that
utilize the proposed row compression. We implemented the different steps of our
approach in a visualization pipeline using integer-linear programming, and
suitable heuristics aiming at sufficiently fast computations in practice. We
conducted both a quantitative evaluation and a small-scale user experiment to
compare the effects of compressing linear diagrams.Comment: To be presented at PacificVis 202
Integer Programming in Parameterized Complexity: Three Miniatures
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra\u27s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation.
To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k) poly(n). We focus on:
- Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.
- Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.
- Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds.
Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension
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