Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

Hemaspaandra, Lane A.; Rothe, Jörg

doi:10.1137/s0097539794261970

Cited by 23 publications

(10 citation statements)

References 49 publications

(67 reference statements)

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“…For example, a large number of definitions are known to be equivalent ( [CGH + 1], [KSW], [HR1], see also [Ha]). It is known that if the boolean hierarchy collapses to some finite level, then so does the polynomial hierarchy [Ka1], [CK], [BCO].…”

Section: Outline and Context Of Our Resultsmentioning

confidence: 99%

“…Hemaspaandra et al studied the question of whether and to what extent the order matters in which various oracle sets from the boolean hierarchy are accessed [HHW]. Boolean hierarchies over classes other than NP were intensely investigated as well: Gundermann et al [GNW] and Beigel et al [BCO] studied boolean hierarchies over counting classes, Bertoni et al [BBJ + ] studied boolean hierarchies over the class RP ("random polynomial time," see [Ad]), and Hemaspaandra and Rothe [HR1] studied the boolean hierarchy over UP ("unambigous polynomial time," introduced by Valiant [Va]) and over any set class closed under intersection.…”

Section: Outline and Context Of Our Resultsmentioning

confidence: 99%

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Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

Riege

Rothe

2004

Theory Comput Syst

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Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH 2k (NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH 2 (NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner's conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP.

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Section: Outline and Context Of Our Resultsmentioning

confidence: 99%

Section: Outline and Context Of Our Resultsmentioning

confidence: 99%

Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

Riege

Rothe

2004

Theory Comput Syst

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“…Their papers initiated an intensive work and many papers on the Boolean hierarchy; e.g., [20,15,13,21,2,5,1,6,12,17] to name just a few.…”

Section: Exact Colorability and The Boolean Hierarchy Over Npmentioning

confidence: 99%

Exact complexity of Exact-Four-Colorability

Rothe¹

2003

Information Processing Letters

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Let M k be a given set of k integers. Define Exact-M k -Colorability to be the problem of determining whether or not χ(G), the chromatic number of a given graph G, equals one of the k elements of the set M k exactly. In 1987, Wagner [Theoret. Comput. Sci. 51 (1987 proved that Exact-M k -Colorability is BH 2k (NP)-complete, where M k = {6k + 1, 6k + 3, . . . , 8k − 1} and BH 2k (NP) is the 2kth level of the Boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not χ(G) = 7, where DP = BH 2 (NP). Wagner raised the question of how small the numbers in a k-element set M k can be chosen such that Exact-M k -Colorability still is BH 2k (NP)-complete. In particular, for k = 1, he asked if it is DP-complete to determine whether or not χ(G) = 4.In this note, we solve Wagner's question and prove the optimal result: For each k 1, Exact-M k -Colorability is BH 2k (NP)-complete for M k = {3k + 1, 3k + 3, . . . , 5k − 1}. In particular, for k = 1, we determine the precise threshold of the parameter t ∈ {4, 5, 6, 7} for which the problem Exact-{t}-Colorability jumps from NP to DP-completeness: It is DP-complete to determine whether or not χ(G) = 4, yet Exact-{3}-Colorability is in NP. troduced DP, the class of differences of two NP problems. They showed that DP contains various interesting types of problems, including uniqueness problems, critical graph problems, and exact optimization problems. For example, Cai and Meyer [7] proved the DPcompleteness of Minimal-3-Uncolorability, a critical graph problem that asks whether a given graph is not 3-colorable, but deleting any of its vertices makes it 3-colorable. A graph is said to be k-colorable if its vertices can be colored using no more than k colors such that no two adjacent vertices receive the same color. The chromatic number of a graph G, denoted χ(G), is defined to be the smallest k such that G is k-colorable.

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“…The boolean hierarchy has been intensely investigated, and quite a bit has been learned about its structure (see, e.g., [10,11,9,25,24,30,14,3,5]). Recently, various results have also been developed regarding boolean hierarchies over classes other than NP [7,13,5,22]. For any language classes C 1 and C 2 , define P C 1 [1]:C 2 [1] to be the class of languages accepted by polynomial-time machines making one query to a C 1 oracle followed by one query to a C 2 oracle.…”

Section: Preliminariesmentioning

confidence: 99%

Query Order

Hemaspaandra¹,

Hempel²,

Wechsung³

1998

SIAM J. Comput.

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We study the effect of query order on computational power, and show that P BH j [1]:BH k [1] -the languages computable via a polynomial-time machine given one query to the jth level of the boolean hierarchy followed by one query to the kth level of the boolean hierarchy-equals R p j+2k−1-tt (NP) if j is even and k is odd, and equals R p j+2k-tt (NP) otherwise. Thus, unless the polynomial hierarchy collapses, it holds that for each 1 ≤ j ≤ k:We extend our analysis to apply to more general query classes. IntroductionThis paper studies the importance of query order. Everyone knows that it makes more sense to first look up in your on-line datebook the date of the yearly computer science conference and then phone your travel agent to get tickets, as opposed to first phoning your travel agent (without knowing the date) and then consulting your on-line datebook to find the date. In real life, order matters.This paper seeks to determine-for the first time to the best of our knowledge-whether one's everyday-life intuition that order matters carries over to complexity theory.In particular, for classes C 1 and C 2 from the boolean hierarchy [10,11], we ask whether one question to C 1 followed by one question to C 2 is more powerful than one question to C 2 followed by one question to C 1 . That is, we seek the relative powers of the classes P C 1 [1]:C 2 [1] and P C 2 [1]:C 1 [1] . As is standard [26], for any constant m we say A is m-truth-table reducible to B (A ≤ p m-tt B) if there is a polynomial-time computable function that, on each input x, computes both

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Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

Cited by 23 publications

References 49 publications

Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

Exact complexity of Exact-Four-Colorability

Query Order

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