Abstract
In this paper, we study the relationship of parameterized enumeration complexity classes defined by Creignou et al. (MFCS 2013). Specifically, we introduce two hierarchies (IncFPTa and CapIncFPTa) of enumeration complexity classes for incremental fpt-time in terms of exponent slices and show how they interleave. Furthermore, we define several parameterized function classes and, in particular, introduce the parameterized counterpart of the class of nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist, TFNP, known from Megiddo and Papadimitriou (TCS 1991). We show that this class TF(para-NP), the restriction of the function variant of NP to total functions, collapsing to F(FPT), the function variant of FPT, is equivalent to the result that OutputFPT coincides with IncFPT. In addition, these collapses are shown to be equivalent to TFNP = FP, and also equivalent to P equals NP intersected with coNP. Finally, we show that these two collapses are equivalent to the collapse of IncP and OutputP in the classical setting. These results are the first direct connections of collapses in parameterized enumeration complexity to collapses in classical enumeration complexity, parameterized function complexity, classical function complexity, and computational complexity theory.
Highlights
In 1988, Johnson, Papadimitriou, and Yannakakis [1] laid the cornerstone of enumeration algorithms.In modern times of ubiquitous computing, such algorithms are of central importance in several areas of life and research such as combinatorics, computational geometry, and operations research [2]
In 2017, Capelli and Strozecki [12] investigated IncP and its relationship to other classical enumeration classes in depth, greatly improving the overall understanding of incremental polynomial delay. One of their central results for this paper shows that if the function variant of NP restricted to total functions coincides with FP, we have a collapse of incremental polynomial time with OutputP, the class of problems for which all solutions can be enumerated in time polynomial in the input length and in the solution set size
This is a significant question of research, and we improve the understanding of this question by pointing out connections to classical enumeration complexity where similar phenomena have been observed [12]
Summary
As already motivated in the beginning of this section, measuring the runtime of enumeration algorithms is usually abandoned beyond the study of OutputP. A CapIncPa -algorithm if and only if there exists a polynomial p such that for all x ∈ Q, algorithm A outputs i elements of Sol( x ) in time O( p(| x |) · i a ) (for every 0 ≤ i ≤ |Sol( x )|). We want to point out that Capelli and Strozecki use the definition of CapIncP for IncP (and use the name “UsualIncP” for IncP instead) They prove that the notions of CapIncP and IncP are equivalent up to an exponential space requirement when using a structured delay. After studying the definition of the class CapIncP, one could come to the opinion that the class is more general than IncP because it does not necessarily have a uniform delay. The previous proposition told us that this is not true
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