Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744877 
Title:  Descriptive complexity of constraint problems  
Author:  Wang, Pengming 
ORCID:
0000000340148248
ISNI:
0000 0004 7230 1451


Awarding Body:  University of Cambridge  
Current Institution:  University of Cambridge  
Date of Award:  2018  
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Abstract:  
Constraint problems are a powerful framework in which many common combinatorial problems can be expressed. Examples include graph colouring problems, Boolean satisfaction, graph cut problems, systems of equations, and many more. One typically distinguishes between constraint satisfaction problems (CSPs), which model strictly decision problems, and socalled valued constraint satisfaction problems (VCSPs), which also include optimisation problems. A key open problem in this field is the longstanding dichotomy conjecture by Feder and Vardi. It claims that CSPs only fall into two categories: Those that are NPcomplete, and those that are solvable in polynomial time. This stands in contrast to Ladner's theorem, which, assuming P$\neq$NP, guarantees the existence of problems that are neither NPcomplete, nor in P, making CSPs an exceptional class of problems. While the FederVardi conjecture is proven to be true in a number of special cases, it is still open in the general setting. (Recent claims affirming the conjecture are not considered here, as they have not been peerreviewed yet.) In this thesis, we approach the complexity of constraint problems from a descriptive complexity perspective. Namely, instead of studying the computational resources necessary to solve certain constraint problems, we consider the expressive power necessary to define these problems in a logic. We obtain several results in this direction. For instance, we show that Schaefer's dichotomy result for the case of CSPs over the Boolean domain can be framed as a definability result: Either a CSP is definable in fixedpoint logic with rank (FPR), or it is NPhard. Furthermore, we show that a dichotomy exists also in the general case. For VCSPs over arbitrary domains, we show that a VCSP is either definable in fixedpoint logic with counting (FPC), or it is not definable in infinitary logic with counting. We show that these definability dichotomies also have algorithmic implications. In particular, using our results on the definability of VCSPs, we prove a dichotomy on the number of levels in the Lasserre hierarchy necessary to obtain an exact solution: For a finitevalued VCSP, either it is solved by the first level of the hierarchy, or one needs $\Omega(n)$ levels. Finally, we explore how other methods from finite model theory can be useful in the context of constraint problems. We consider pebble games for finite variable logics in this context, and expose new connections between CSPs, pebble games, and homomorphism preservation results.


Supervisor:  Dawar, Anuj  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.744877  DOI:  
Keywords:  Complexity theory ; Constraint satisfaction ; Logic ; Computer science ; Optimisation  
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