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Dynamic Complexity Meets Parameterised Algorithms

Jonas Schmidt, Thomas Schwentick, Nils Vortmeier, Thomas Zeume, Ioannis Kokkinis, Michael Wagner

2020
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Annual Conference for Computer Science Logic
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Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from firstorder logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter k and the extensions allow additional "space" of size f (k) (in the form of an additional structure of
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... his size) or additional time f (k) (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems. ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Logic and databases; Theory of computation → Complexity theory and logic This paper adds the aspect of changing inputs and dynamic maintenance of results to the exploration of the landscape between para-AC 0 and FPT. The study of low-level complexity classes under dynamic aspects was started in [30, 15] in the context of dynamically maintaining the result of database queries. Similarly, as for dynamic algorithms, in this setting a dynamic program can make use of auxiliary relations that can store knowledge about the current input data (database). After a small change of the database (most often: insertion or deletion of a tuple), the program needs to compute the query result for the modified database in very short parallel time. To capture the problems/queries, for which this is possible, Patnaik and Immerman introduced the class DynFO [30]. Here, "FO" stands for first-order logic, which is equivalent to AC 0 , in the presence of arithmetic [7, 24] . In this paper, we study dynamic programs that have additional resources in a "parameterised sense". We explore two such resources, which can be described as parameterised space and parameterised time, respectively. For ease of exposition, we discuss these two resources in the context of AC 0 first. One way to strengthen AC 0 circuit families is to allow circuits of size f (k)poly(|x|). We denote the class thus obtained as para-S-AC 0 (even though it corresponds to the class para-AC 0 ). A second dimension is to let the depth of circuits depend on the parameter. As the depth of circuits corresponds to the (parallel) time the circuits need for a computation, we denote the class of problems captured by such circuits by para-T-AC 0 . Of course, both dimensions can also be combined, yielding the parameterised class para-ST-AC 0 . Surprisingly, several parameterised versions of NP-complete problems can even be solved in para-S-AC 0 . Examples are the vertex cover problem and the hitting set problem parameterised by the size of the vertex cover and the hitting set, respectively [4] . However, classical circuit lower bounds unconditionally imply that this is not possible for all in [3] it was observed that the existence of simple paths of length k (the parameter) cannot be tested in para-S-AC 0 . Likewise, the feedback vertex set problem with the size of the feedback vertex set as parameter cannot be solved in para-ST-AC 0 . When translated from circuits to logical formulas, depth roughly translates into iteration of formulas [24, Theorem 5.22], whereas size translates into the size of an additional structure by which the database is extended before formulas are evaluated. Slightly more formally, para-T-AC 0 corresponds to the class para-T-FO consisting of problems that can be defined by iterating a formula f (k) many times. The class para-S-AC 0 corresponds to the class para-S-FO where formulas are evaluated on structures D extended by an advice structure whose size depends on the parameter only. In the class para-ST-FO both dimensions are combined. The parameterised dynamic classes that we study in this paper are obtained from DynFO just like the above classes are obtained from FO: para-S-DynFO, para-T-DynFO and para-ST-DynFO extend DynFO by an additional structure of parameterised size, f (k) iterations of formulas, or both, respectively. As our first main contribution, we introduce a uniform framework for small dynamic, parameterised complexity classes (Section 3) based on advice structures (corresponding to additional space) or iterations of formulas (corresponding to additional time) and investigate how the resulting classes relate to each other and to other non-dynamic (and even nonparameterised) complexity classes (Section 4). As our second main contribution, we explore how methods for parameterised algorithms can be applied in this framework through case studies for various parameterised problems (Section 5). Due to space limitations, many proofs are omitted and can be found in the full version of this paper.

doi:10.4230/lipics.csl.2020.36
dblp:conf/csl/SchmidtSVZK20
fatcat:u3bofincu5euri7b6w6xu5hgti