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Complexity and Approximability of Parameterized MAX-CSPs

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We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types \({\text {AND}}\), \({\text {OR}}\), \({\text {PARITY}}\), and \({\text {MAJORITY}}\), and with various parameters k, and we attempt to fully classify them into the following three cases:

  1. 1.

    The exact optimum can be computed in \(\textsf {FPT}\) time.

  2. 2.

    It is -hard to compute the exact optimum, but there is a randomized \(\textsf {FPT}\) approximation scheme (\(\textsf {FPT\text {-}AS}\)), which computes a \((1{-}\epsilon )\)-approximation in time \(f(k,\epsilon ) \cdot {\text {poly}}(n)\).

  3. 3.

    There is no \(\textsf {FPT\text {-}AS}\) unless .

For the corresponding standard CSPs, we establish \(\textsf {FPT}\) versus -hardness results.

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  1. 1.

    See the next section for a definition of incidence graphs.

  2. 2.

    Akin to neighborhood diversity is the twin-cover number proposed in [11]. On bipartite graphs such as incidence graphs of CSPs, the twin-cover number is essentially the same as the vertex cover number:it differs only on a graph consisting of a single edge, in which the twin-cover number equals 0 while the vertex cover number is 1. Hence, we do not consider the twin-cover number separately as a structural parameter in this paper.

  3. 3.

    We follow here the standard definition of \(\textsf {FPT\text {-}AS}\) given in [23].

  4. 4.

    A CNF formula has bounded modular incidence treewidth if its incidence graph has bounded treewidth after merging variable/clause modules into a single vertex. Here, a variable/clause module is a set of vertices, corresponding to variables/clauses respectively, with same neighborhood outside of the set.In fact, the reduction in [25] constructs a formula whose incidence graph has small feedback vertex set after contracting modules.


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Author information

Correspondence to Holger Dell.

Additional information

Valia Mitsou: Supported by ERC Starting Grant PARAMTIGHT (No. 280152).

Tobias Mömke: This research is supported by Deutsche Forschungsgemeinschaft Grant BL511/10-1.

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Dell, H., Kim, E.J., Lampis, M. et al. Complexity and Approximability of Parameterized MAX-CSPs. Algorithmica 79, 230–250 (2017). https://doi.org/10.1007/s00453-017-0310-8

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  • Constraint satisfaction problems
  • Parameterized complexity
  • Approximation
  • Clique width
  • Neighborhood diversity