On a simple hard variant of Not-All-Equal 3-Sat

Darmann, Andreas; Döcker, Janosch

doi:10.1016/j.tcs.2020.02.010

Cited by 10 publications

(6 citation statements)

References 3 publications

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“…On the complexity side, the Sparsification lemma [19], the folklore linear reductions from bounded-occurrence 3-SAT to bounded-occurrence Monotone Not-All-Equal 3-SAT and to Monotone Not-All-Equal 3-SAT-E4 [6], and finally our quadratic reduction, imply that 2 Ω( √ n) time is required to solve PMC in n-vertex planar graphs. Our reduction (as we will see) indeed has a quadratic blow-up as it creates O(1) vertices per variable and clause, and O(1) vertices for each of the O(n 2 ) crossings in a (non-planar) drawing of the variable-clause incidence graph.…”

Section: Theorem 1 Perfect Matching Cut Is Np-hard In 3-connected Cub...mentioning

confidence: 89%

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Cutting Barnette graphs perfectly is hard

Bonnet¹,

Chakraborty²,

Duron³

2023

Preprint

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A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted.

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Section: Theorem 1 Perfect Matching Cut Is Np-hard In 3-connected Cub...mentioning

confidence: 89%

“…Outline of the proof. We reduce the NP-complete problem Monotone Not-All-Equal 3-SAT with exactly 4 occurrences of each variable [6] to PMC. Observe that flipping the value of every variable of a satisfying assignment results in another satisfying assignment.…”

Section: Theorem 1 Perfect Matching Cut Is Np-hard In 3-connected Cub...mentioning

confidence: 99%

Cutting Barnette graphs perfectly is hard

Bonnet¹,

Chakraborty²,

Duron³

2023

Preprint

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“…□ Lemma 2.1 leads to the following hardness result, where we lose a factor 2𝑞 in the degree bound due to the reduction. We note that for 𝑞 = 2, 𝐾 = 3, Δ = 4, and simple hypergraphs, NP-hardness is known [DD20]. However, the main point of the next theorem is that there is a degree bound that scales roughly as 𝑞 𝐾 and makes the problem NP-hard.…”

Section: It Follows Thatmentioning

confidence: 94%

“…For 𝑞 = 2 and 𝐾 > 2, using two copies of the hypergraph from Lemma 2.1, we build an "equality" gadget, i.e., a simple hypergraph 𝐻 of maximum degree Δ ≤ 2(𝐾𝑞 𝐾 −1 ln 𝑞 + 1) = 𝐾2 𝐾 ln 2 + 2 with distinct vertices 𝑢, 𝑣 which both have degree 1 such that for every 𝑞-colouring 𝜎 it holds that 𝜎 (𝑢) = 𝜎 (𝑣). It is well-known that finding 2-colourings of 𝐾-uniform simple hypergraphs is NP-hard (or we can use for example [DD20]), and using the equality gadget 𝐻 , for any 𝐾-uniform simple hypergraph 𝐹 , we can construct a 𝐾-uniform simple hypergraph 𝐹 ′ of maximum degree Δ such that 𝐹 is 2-colourable if and only if 𝐹 ′ is 2-colourable. One possible way to do so is replacing each degree-𝑑 vertex 𝑤 of 𝐹 with a cycle of length 𝑑 and then replacing each edge 𝑒 of the cycle with a distinct copy of the hypergraph 𝐻 using 𝑢, 𝑣 for the endpoints of the edge 𝑒; then, for each hyperedge of 𝐹 that uses 𝑤, in 𝐹 ′ we use instead one of the 𝑑 vertices of the cycle.…”

Section: It Follows Thatmentioning

confidence: 99%

Inapproximability of Counting Hypergraph Colourings

Galanis

Guo

Wang

2022

ACM Trans. Comput. Theory

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Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this paper is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ . An efficient algorithm exists if \(\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2} \) (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021). Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K /2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour on random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.

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“…In fact, for simplicity, we reduce from the NP-hard variant of MONOTONE NOT-ALL-EQUAL 3-SAT where each variable appears in exactly four clauses [Darmann and Döcker, 2020]. Given an instance ϕ = C 1 ∧ • • • ∧ C m over variables X of MONOTONE NOT-ALL-EQUAL 3-SAT, note that m needs to be even, as there are m = 4•|X| 3 and m needs to be an integer.…”

Section: Single-author-single-paper Settingmentioning

confidence: 99%

Combating Collusion Rings is Hard but Possible

Boehmer¹,

Bredereck²,

Nichterlein³

2021

Preprint

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A recent report of Littmann [Commun. ACM '21] outlines the existence and the fatal impact of collusion rings in academic peer reviewing. We introduce and analyze the problem CYCLE-FREE REVIEWING that aims at finding a review assignment without the following kind of collusion ring: A sequence of reviewers each reviewing a paper authored by the next reviewer in the sequence (with the last reviewer reviewing a paper of the first), thus creating a review cycle where each reviewer gives favorable reviews. As a result, all papers in that cycle have a high chance of acceptance independent of their respective scientific merit.We observe that review assignments computed using a standard Linear Programming approach typically admit many short review cycles. On the negative side, we show that CYCLE-FREE RE-VIEWING is NP-hard in various restricted cases (i.e., when every author is qualified to review all papers and one wants to prevent that authors review each other's or their own papers or when every author has only one paper and is only qualified to review few papers). On the positive side, among others, we show that, in some realistic settings, an assignment without any review cycles of small length always exists. This result also gives rise to an efficient heuristic for computing (weighted) cycle-free review assignments, which we show to be of excellent quality in practice.2 was a soft constraint [Leyton-Brown and Mausam, 2021]. Yet, there is a lack of systematic studies concerning the computation of such assignments. Motivated by this, we propose and analyze CYCLE-FREE REVIEWING, the problem of computing an assignment of papers to agents that is free of review cycles of length at most z, both from a theoretical and practical perspective.

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On a simple hard variant of Not-All-Equal 3-Sat

Cited by 10 publications

References 3 publications

Cutting Barnette graphs perfectly is hard

Cutting Barnette graphs perfectly is hard

Inapproximability of Counting Hypergraph Colourings

Combating Collusion Rings is Hard but Possible

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