The challenges of unbounded treewidth in parameterised subgraph counting problems

Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP.

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Tractability in constraint satisfaction problems: a survey

We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT = W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem [25]. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.

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The Constant Inapproximability of the Parameterized Dominating Set Problem

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability.

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Assessing the Computational Complexity of Multi-layer Subgraph Detection

The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph G from some class \(\mathcal K\) of “pattern graphs” can be embedded into a given graph H (that is, is isomorphic to a subgraph of H) is fixed-parameter tractable if \(\mathcal K\) is a class of graphs of bounded tree width and \(W [1]\)-complete otherwise.

The Hardness of Embedding Grids and Walls

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Bingkai Lin

Papers

The Parameterized Complexity ofk-Biclique