The object of the thesis is the computational complexity of certain combinatorial optimisation problems called \emph{valued constraint satisfaction problems}, or \emph{VCSPs} for short. The requirements and optimisation criteria of these problems are expressed by sums of \emph{(valued) constraints} (also called \emph{cost functions}). More precisely, the input of a VCSP consists of a finite set of variables, a finite set of cost functions that depend on these variables, and a cost $u$; the task is to find values for the variables such that the sum of the cost functions is at most $u$.
By restricting the set of possible cost functions in the input, a great variety of computational optimisation problems can be modelled as VCSPs. Recently, the computational complexity of all VCSPs for finite sets of cost functions over a finite domain has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain.
We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear (PL) and piecewise linear homogeneous (PLH) cost functions.
The VCSP for a finite set of PLH cost functions can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We
show this by (polynomial-time many-one) reducing the problem to a finite-domain VCSP which can be solved using a linear programming relaxation. We apply this result to show the polynomial-time tractability of VCSPs for {\it submodular} PLH cost functions, for {\it convex} PLH cost functions, and for {\it componentwise increasing} PLH cost functions; in fact, we show that submodular PLH functions and componentwise increasing PLH functions form maximally tractable classes of PLH cost functions.
We define the notion of {\it expressive power} for sets of cost functions over arbitrary domains, and discuss the relation between the expressive power and the set of fractional operations improving the same set of cost functions over an arbitrary countable domain.
Finally, we provide a polynomial-time algorithm solving the restriction of the VCSP for {\it all} PL cost functions to a fixed number of variables.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:71507 |
| Date | 16 July 2020 |
| Creators | Viola, Caterina |
| Contributors | Bodirsky, Manuel, Krokhin, Andrei, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | info:eu-repo/grantAgreement/Deutsche Forschungsgemeinschaft/DFG research training group/1763//Quantitative Logic and Automata/QuantLA, info:eu-repo/grantAgreement/European Research Council/European Union's Horizon 2020 research and innovation programme/681988//CSP-Infinity |
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