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The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal dominating sets of a graph $G=(V,E)$, one usually arrives at the problem to decide for a vertex set $U subseteq V$, if there exists a textit{minimal} dominating set $S$ with $Usubseteq S$. We propose a general, partial-order based formulation of such extension problems and study a number of specific problems which can be expressed in this framework. Possibly contradicting intuition, these problems tend to be NP-hard, even for problems where the underlying optimisation problem can be solved in polynomial time. This raises the question of how fixing a partial solution causes this increase in difficulty. In this regard, we study the parameterised complexity of extension problems with respect to parameters related to the partial solution, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All complexity considerations are also carried out in very restricted scenarios, be it degree restrictions or topological restrictions (planarity) for graph problems or the size of the given partition for the considered extension variant of Bin Packing.
Let $G$ be a graph on $n$ vertices and $mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogous
It is known that the extension complexity of the TSP polytope for the complete graph $K_n$ is exponential in $n$ even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets $mathcal{H}$ of fa
This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called {it minimum normalized cuts}/{it isoperimteric numbers} defined
In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension complexity of fo
Rummikub is a tile-based game in which each player starts with a hand of $14$ tiles. A tile has a value and a suit. The players form sets consisting of tiles with the same suit and consecutive values (runs) or tiles with the same value and different