Scheduling with AND/OR Precedence Constraints

Möhring, Rolf H.; Skutella, Martin; Stork, Frederik

doi:10.1137/s009753970037727x

Cited by 84 publications

(91 citation statements)

References 20 publications

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“…It is an open problem whether the max atoms problem is in P, but it is known to be polynomial-time equivalent to determining the winner in mean payoff games (Möhring, Skutella, and Stork [28]), which is known to be in NP ∩ co-NP. Note that here the assumption that c is represented in binary is important: when c is represented in unary, or when we drop all but a finite number of relations in Γ , the resulting problem is known to be in P.…”

Section: Csp(γ )mentioning

confidence: 99%

“…Such relations have been called tropically convex in the literature [12]. 2 This class is a non trivial extension of the max-atoms problem (for instance it contains relations such as x (y + z)/2), and it is not covered by the known reduction to mean payoff games [28,1,3]. Indeed, it is open whether the CSP for tropically convex semilinear relations can be reduced to mean payoff games (in fact, Zwick and Paterson [32] believe that mean payoff games are "strictly easier" than simple stochastic games, which reduce to our problem via the results presented in Section 4).…”

Section: R E S U Lt Smentioning

confidence: 99%

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Tropically Convex Constraint Satisfaction

Bodirsky

Mamino

2017

Theory Comput Syst

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A bstract. A semilinear relation S ⊆ Q n is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in NP ∩ co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinear constraints in general into NP ∩ co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the maxatoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.

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Section: Csp(γ )mentioning

confidence: 99%

Section: R E S U Lt Smentioning

confidence: 99%

Tropically Convex Constraint Satisfaction

Bodirsky

Mamino

2017

Theory Comput Syst

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“…The following are equivalent: Proof. The equivalence between 1 and 2 follows (essentially) from Lemma 7.5 in [22] (see also [4] for a proof of the exact statement). The equivalence between 2 and 3 follows from Lemma 2 in [8].…”

Section: Chaining Refutationsmentioning

confidence: 99%

“…We thank Manuel Bodirsky for bringing [22] to our attention, and to the anonymous referees for their comments. This work was done while visiting the Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona.…”

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confidence: 99%

Mean-Payoff Games and Propositional Proofs

Atserias

Maneva

2010

Automata, Languages and Programming

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We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in Σ 2 -Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of Σ 2 -Frege, and to the interpolation problem for Σ 2,2 -Frege. Since the interpolation problem for Σ 1 -Frege (i.e. resolution) is solvable in polynomial time, our result is close to optimal up to the computational complexity of solving mean-payoff games. The proof of the main result requires building low-depth formulas that compute the bits of the sum of a constant number of integers in binary notation, and lowcomplexity proofs of the required arithmetic properties.

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“…Furthermore, since max(e 1 Another less trivial equivalence of MAP is with the problem of deciding the existence of super fixpoints of min-max functions [8]. A min-max function is a function f : Z n → Z n whose coordinates are min-max expressions, i.e., terms in the grammar A more significant relationship is with the problem of computing earliest job start times for the systems of AND/OR precedence constraints of [9]. To show PTIMEequivalence with MAP, simple syntactic transformations suffice, like interchanging min with max and with .…”

Section: Simple Equivalences With Mapmentioning

confidence: 99%