On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

Elberfeld, Michael; Stockhusen, Christoph; Tantau, Till

doi:10.1007/s00453-014-9944-y

Cited by 45 publications

(104 citation statements)

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“…Parameterized circuit complexity. We explain the definitional layer of parameterized circuit complexity basing our notation on Bannach et al [4], though prior foundational work on parameterized circuit complexity was done by Elberfeld et al [20]. An AC-circuit C is a directed acyclic graph with node set consisting of input, conjunction (AND), disjunction (OR), and negation (NOT) gates.…”

Section: Preliminariesmentioning

confidence: 99%

“…Curiously, even though the complexity-theoretical foundations of parameterized complexity are expressed using circuit complexity, the question of what are the appropriate analogues of standard circuit complexity classes in parameterized complexity was not systematically studied up to very recently, when Elberfeld et al [20] and Bannach et al [4] introduced an appropriate definitional layer and gave several foundational results. Slightly informally, a parameterized problem is in the class para-AC i (where i > 0) if it can be solved by an (appropriately uniform) family of AC-circuits (C n,k ) n,k∈N , where the circuit C n,k solves the problem on inputs of size n and parameter value k, such that each C n,k has size f (k) · n c and depth f (k) + c · log i n, for a computable function f and universal constant c. The classes para-NC i are defined similarly using NC-circuits.…”

Section: Introductionmentioning

confidence: 99%

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Parameterized circuit complexity of model-checking on sparse structures

Pilipczuk

Siebertz

Toruńczyk

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We prove that for every class C of graphs with effectively bounded expansion, given a first-order sentence ϕ and an n-element structure A whose Gaifman graph belongs to C , the question whether ϕ holds in A can be decided by a family of AC-circuits of size f (ϕ) · n c and depth f (ϕ) + c log n, where f is a computable function and c is a universal constant. This places the model-checking problem for classes of bounded expansion in the parameterized circuit complexity class para-AC 1 . On the route to our result we prove that the basic decomposition toolbox for classes of bounded expansion, including orderings with bounded weak coloring numbers and low treedepth decompositions, can be computed in para-AC 1 .

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameterized circuit complexity of model-checking on sparse structures

Pilipczuk

Siebertz

Toruńczyk

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…In [11] Elberfeld et al introduced the parameterized class para-AC 0 as the AC 0 analog of the class FPT: A problem is in para-AC 0 if it can be computed by dlogtime-uniform AC 0 -circuits after Following the framework proposed in [12], we first compare two possible definitions of para-AC 0 depending on different ways to obtain parameterized classes from classical ones. We have already mentioned the first one, in which an arbitrary precomputation can be performed on the parameter before a standard computation according to the corresponding classical class.…”

Section: Introductionmentioning

confidence: 99%

Some lower bounds in parameterized AC0

Chen

Flum

2019

Information and Computation

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We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical AC 0 . Among others, we derive such a lower bound for all fptapproximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong AC 0 version of the planted clique conjecture: AC 0 -circuits asymptotically almost surely can not distinguish between a random graph and this graph with a randomly planted clique of any size ≤ n ξ (where 0 ≤ ξ < 1).an (arbitrarily complex) precomputation [12] on the parameter. Later in [3] it was shown that para-AC 0 contains the parameterized vertex cover problem (p-VERTEX-COVER), one of the archetypal fixed-parameter tractable problems. For various other problems the authors of [3] also proved their membership in para-AC 0 . Concerning nonmembership, a result in [6] shows that the parameterized st-connectivity problem (p-STCONN), i.e., the problem of deciding whether there is a path of length at most k between vertices s and t in a graph G, parameterized by k, is not in para-AC 0 . It is worth noting that st-connectivity is solvable in polynomial time, and hence p-STCONN ∈ FPT.The class AC 0 is one of the best understood classical complexity classes. Already in [1,14] it was shown that PARITY, the problem of deciding whether a binary string contains an even number of 1's, is not in AC 0 . Since PARITY has a very low complexity, for many other problems, including VERTEX-COVER and CLIQUE, the AC 0 -lower bound can be easily derived by reductions from PARITY. Similarly, as p-CLIQUE / ∈ para-AC 0 , it is not very hard to see, using some appropriate weak parameterized reductions, that many other parameterized problems, including the dominating set problem, are not in para-AC 0 .It is well known that the class AC 0 is intimately connected to first-order logic (FO). In fact, the problems decidable by dlogtime-uniform AC 0 -circuits are precisely those definable in FO(<, +, ×), that is, in first-order logic for ordered structures with built-in predicates of addition and multiplication. Now we can also study various parameterized classes based on fragments of FO(<, +, ×). Let us emphasize that this is not merely an academic exercise. Logic and parameterized complexity are surprisingly intertwined with each other, which, among others, is witnessed by various algorithmic meta-theorems (see e.g. [16]). Moreover, the problem whether there is a logic for PTIME, a central problem of descriptive complexity, turned out (see [9] for a thorough discussion) to be related to the complexity of the parameterized halting problem p-HALT Instance: n ∈ N in unary and a nondeterministic Turing machine (NTM) M. Parameter: |M|, the size of he machine M.Question: Does M accept the empty input tape in at most n steps?In fact, already in [20] it was shown that PTIME has a logic if p-HALT has an algorithm with running ...

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“…Following [10], it is not hard to see that p-HALT ∈ para-AC 0 implies that there is a logic capturing (+, ×)-invariant FO. Recall that para-AC 0 ⊆ FPT [13], and there is good evidence that p-HALT / ∈ FPT [9], so the conjecture below seems highly plausible.…”

mentioning

confidence: 97%

“…Parameterized AC 0 , or para-AC 0 , can be viewed as an analog of AC 0 in the parameterized world. There is some recent interest in para-AC 0 [13,5,11,6]. Just like whether p-HALT ∈ FPT, the question of whether p-HALT ∈ para-AC 0 can be related to open problems in proof complexity and descriptive complexity as well.…”

mentioning

confidence: 99%

A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

Chen

Müller

Yokoyama

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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The complexity of the parameterized halting problem for nondeterministic Turing machines p-HALT is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-HALT is in para-AC 0 , the parameterized version of the circuit complexity class AC 0 , then AC 0 , or equivalently, (+, ×)-invariant FO, has a logic. Although it is widely believed that p-HALT / ∈ para-AC 0 , we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊆ LINH. Here, LINH denotes the linear time hierarchy.On the other hand, we suggest an approach toward proving NE ⊆ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-DavisPutnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊆ LINH. Interestingly, central to this result is a para-AC 0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.

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On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

Cited by 45 publications

References 24 publications

Parameterized circuit complexity of model-checking on sparse structures

Parameterized circuit complexity of model-checking on sparse structures

Some lower bounds in parameterized AC0

A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

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