The classes PPA-k: Existence from arguments modulo k

Hollender, Alexandros

doi:10.1016/j.tcs.2021.06.016

Cited by 9 publications

(16 citation statements)

References 30 publications

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“…In fact, Filos-Ratsikas and Goldberg [2019] conjectured that the complexities of the problems for different values of k are incomparable, and are characterized by different complexity classes. The complexity classes that are believed to be the most related are called PPA-k, defined also by Papadimitriou [1994] in his original paper; we refer the reader to the recent papers of [G ö ös et al, 2019;Hollender, 2019] for a more detailed discussion of these classes.…”

Section: Discussion and Related Workmentioning

confidence: 99%

Consensus-Halving: Does It Ever Get Easier?

Filos-Ratsikas¹,

Hollender²,

Sotiraki³

et al. 2020

Preprint

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In the ε-Consensus-Halving problem, a fundamental problem in fair division, there are n agents with valuations over the interval [0, 1], and the goal is to divide the interval into pieces and assign a label "+" or "−" to each piece, such that every agent values the total amount of "+" and the total amount of "−" almost equally. The problem was recently proven by Goldberg [2018, 2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question.In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving ε-Consensus-Halving for any ε, as well as a polynomial-time algorithm for ε = 1/2; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k-Division problem [Simmons and Su, 2003]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.

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Section: Discussion and Related Workmentioning

confidence: 99%

Consensus-Halving: Does It Ever Get Easier?

Filos-Ratsikas¹,

Hollender²,

Sotiraki³

et al. 2020

Preprint

Self Cite

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“…In Section 1.6, we include a list of open problems that illustrate the broader relevance of PPA q . Finally, a concurrent and independent work by Hollender [Hol19] also establishes the structural properties of PPA q corresponding to §1.1 and §1.5.…”

mentioning

confidence: 84%

On the Complexity of Modulo-q Arguments and the Chevalley-Warning Theorem

Göös,

Kamath,

Sotiraki

et al. 2019

Preprint

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We study the search problem class PPA q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA p for prime p.Our main result is to establish that an explicit version of a search problem associated to the Chevalley-Warning theorem is complete for PPA p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA p when p ≥ 3.Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA q .

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“…The complexity classes PPA-k are defined as follows [Pap94,Hol21]. For any integer k ≥ 2, PPA-k is the set of problems reducible in polynomial time to the problem Bipartite-mod-k: Definition 2.…”

Section: Preliminariesmentioning

confidence: 99%

“…• n cuts lies in the Turing closure of PPA-k. In particular, when k = p r for a prime p, PPA-k is equal to PPA-p and PPA-p is closed under Turing reduction [Hol21,GKSZ20].…”

Section: Upper Bound and Implications On Complexitymentioning

confidence: 99%

Consensus Division in an Arbitrary Ratio

Goldberg¹,

Li²

2022

Preprint

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We consider the problem of partitioning a line segment into two subsets, so that n finite measures all has the same ratio of values for the subsets. Letting α ∈ [0, 1] denote the desired ratio, this generalises the PPA-complete consensushalving problem, in which α = 1 2 . It is known that for any α, there exists a solution using 2n cuts of the segment. Here we show that if α is irrational, that upper bound is almost optimal. We also obtain bounds that are nearly exact for a large subset of rational values α. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1. when using the minimal number of cuts for each instance is required, the problem is NP-hard for any α;2. for a large subset of rational α = k , when k−1 k • 2n cuts are available, the problem is in the Turing closure of PPA-k; 3. when 2n cuts are allowed, the problem belongs to PPA for any α; furthermore, the problem belong to PPA-p for any prime p if 2(p − 1) • p/2 p/2 • n cuts are available.

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The classes PPA-k: Existence from arguments modulo k

Cited by 9 publications

References 30 publications

Consensus-Halving: Does It Ever Get Easier?

Consensus-Halving: Does It Ever Get Easier?

On the Complexity of Modulo-q Arguments and the Chevalley-Warning Theorem

Consensus Division in an Arbitrary Ratio

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