Two remarks on the power of counting

Papadimitriou, Christos H.; Zachos, Stathis

doi:10.1007/bfb0036487

Cited by 91 publications

(42 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, we show that the problem to test irreducibility of a semilinear set given by a constant-free decision circuit is complete for the class P NP [log] . The latter class was first studied by Papadimitriou & Zachos (1983) and consists of the decision problems that can be solved in polynomial time by O(log n) queries to some NP language. It is known (Buss & Hay 1991;Hemachandra 1989) that equivalently, P NP [log] can also be characterized as the set of languages in P NP whose queries are nonadaptive.…”

Section: Resultsmentioning

confidence: 99%

The complexity of semilinear problems in succinct representation

2006

View full text Add to dashboard Cite

We prove completeness results for twenty-three problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the Blum-Shub-Smale additive model of computation. If, in contrast, the circuit is constant-free, then the completeness results are for the Turing model of computation. One such result, the P NP[log] -completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.

show abstract

Section: Resultsmentioning

confidence: 99%

The complexity of semilinear problems in succinct representation

2006

View full text Add to dashboard Cite

show abstract

“…For natural complete problems in the levels of the boolean hierarchy, and especially in DP, see the survey by Riege and Rothe [27] and, more recently, the work of Nguyen et al [22] on social welfare optimization in multiagent resource allocation and of Reisch et al [26] on the margin of victory in Schulze, cup, and Copeland elections. P NP [log] was introduced by Papadimitriou and Zachos [24] as the class of problems that can be solved in polynomial time by asking O(log n) sequential Turing queries to an NP oracle. This class is also known as capturing "parallel access to NP" (denoted by P NP ) where polynomially many oracle queries may be asked in parallel; the equality of P NP[log] and P NP has been shown independently by Hemachandra [13] and Köbler et al [20].…”

Section: Complexity Theorymentioning

confidence: 99%

Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

Rey

Rothe

Schadrack

et al. 2015

Ann Math Artif Intell

View full text Add to dashboard Cite

We study the computational complexity of the existence and the verification problem for wonderfully stable partitions (WSPE and WSPV) and of the existence problem for strictly core stable coalition structures (SCSCS) in enemy-oriented hedonic games. In this note, we show that WSPV is NP-complete and both WSPE and SCSCS are DP-hard, where DP is the second level of the boolean hierarchy, and we discuss an approach for classifying the latter two problems in terms of their complexity.

show abstract

“…The class MOD 2 P (also written as ⊕P) was introduced by Papadimitriou and Zachos [30], and the class MOD k P for arbitrary k was introduced by Cai and Hemachandra [8]. Beigel and Gill [6] showed several closure properties hold for these classes -for example, for all primes p, the class MOD p P is closed under union, intersection, and complement.…”

Section: Other Related Workmentioning

confidence: 99%

Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

Williams

2008

comput. complex.

View full text Add to dashboard Cite

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known time-space tradeoffs for Sat. Let m > 0 be an integer, and define MOD m -Sat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2 cos(π/7) ≈ 1.801, there is a d > 0 such that MOD p -Sat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD 6 -Sat, as well as MOD m -Sat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a "canonical" one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.

show abstract

Two remarks on the power of counting

Cited by 91 publications

References 6 publications

The complexity of semilinear problems in succinct representation

The complexity of semilinear problems in succinct representation

Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

Contact Info

Product

Resources

About