Generalized theorems on relationships among reducibility notions to certain complexity classes

In this paper we give several generalized theorems concerning reducibility notions to certain complexity classes. We study classes that are either (I) closed under NP many-one reductions and polynomial-time conjunctive reductions or (II) closed under coNP many-one reductions and polynomial-time disjunctive reductions. We prove that, for such a classK, (1) reducibility notions of sets toK under polynomial-time constant-round truth-table reducibility, polynomial-time log-Turing reducibility, logspace constant-round truth-table reducibility, logspace log-Turing reducibility, and logspace Turing reducibility are all equivalent and (2) every set that is polynomial-time positive Turing reducible to a set inK is already inK.From these results, we derive some observations on the reducibility notions to C₌P and NP.     

Some structural properties of SAT

The following four conjectures about structural of SAT are studied in this paper.(1) SAT∈P^SPARSE∩NP;(2)SAT∈SRTDtt;(3)SAT∈Ptt^bAPP;(4)FPtt^SAT=FTlog^SAT.It is proved that some pairs of these conjectures imply P=NP ,for example,if SAT∈P^SPARSE∩NP and SAT∈Ptt^bAPP,or if SAT∈SRTDtt and SAT∩PttbAPP,then P=NP.This improves previous results in literature.     

Model Checking Abilities of Agents: A Closer Look

Alternating-time temporal logic (atl) is a logic for reasoning about open computational systems and multi-agent systems. It is well known that atl model checking is linear in the size of the model. We point out, however, that the size of an atl model is usually exponential in the number of agents. When the size of models is defined in terms of states and agents rather than transitions, it turns out that the problem is (1) Δ ₃ ^P -complete for concurrent game structures, and (2) Δ ₂ ^P -complete for alternating transition systems. Moreover, for “Positive atl” that allows for negation only on the level of propositions, model checking is (1) Σ ₂ ^P -complete for concurrent game structures, and (2) NP-complete for alternating transition systems. We show a nondeterministic polynomial reduction from checking arbitrary alternating transition systems to checking turn-based transition systems, We also discuss the determinism assumption in alternating transition systems, and show that it can be easily removed. In the second part of the paper, we study the model checking complexity for formulae of atl with imperfect information (atl _ir ). We show that the problem is Δ ₂ ^P -complete in the number of transitions and the length of the formula (thereby closing a gap in previous work of Schobbens in Electron. Notes Theor. Comput. Sci. 85(2), 2004). Then, we take a closer look and use the same fine structure complexity measure as we did for atl with perfect information. We get the surprising result that checking formulae of atl _ir is also Δ ₃ ^P -complete in the general case, and Σ ₂ ^P -complete for “Positive atl _ir ”. Thus, model checking agents’ abilities for both perfect and imperfect information systems belongs to the same complexity class when a finer-grained analysis is used.     

A relationship between difference hierarchies and relativized polynomial hierarchies

Chang and Kadin have shown that if the difference hierarchy over NP collapses to levelk, then the polynomial hierarchy (PH) is equal to thekth level of the difference hierarchy over ₂ ^p . We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to levelk, then PH collapses to (P _(k–1) ^NP )^NP, the class of sets recognized in polynomial time withk – 1 nonadaptive queries to a set in NP^NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any classC that has _m ^p -complete sets and is closed under _conj ^p -and _m ^NP -reductions (alternatively, closed under _disj ^p -and _m ^co-NP -reductions), if the difference hierarchy overC collapses to levelk, then PH ^C = (P _(k–1)–tt ^NP ) ^C . Then we show that the exact counting class C_P is closed under _disj ^p - and _m ^co-NP -reductions. Consequently, if the difference hierarchy over C_P collapses to levelk, then PH^PP(= PH^C_P) is equal to (P _(k–1)–tt ^NP )^PP. In contrast, the difference hierarchy over the closely related class PP is known to collapse.Finally we consider two ways of relativizing the bounded query class P _k–tt ^NP : the restricted relativization P _k–tt ^{NP C} and the full relativization (P _k–tt ^NP ) ^C . IfC is NP-hard, then we show that the two relativizations are different unless PH ^C collapses.Richard Beigel was supported in part by NSF Grants CCR-8808949 and CCR-8958528. Richard Chang was supported in part by NSF Research Grant CCR 88-23053. This work was done while Mitsunori Ogiwara was at the Department of Information Science, Tokyo Institute of Technology, Tokyo, Japan.     

All Natural NP-Complete Problems Have Average-Case Complete Versions

The theory of average case complexity studies the expected complexity of computational tasks under various specific distributions on the instances, rather than their worst case complexity. Thus, this theory deals with distributional problems, defined as pairs each consisting of a decision problem and a probability distribution over the instances. While for applications utilizing hardness, such as cryptography, one seeks an efficient algorithm that outputs random instances of some problem that are hard for any algorithm with high probability, the resulting hard distributions in these cases are typically highly artificial, and do not establish the hardness of the problem under “interesting” or “natural” distributions. This paper studies the possibility of proving generic hardness results (i.e., for a wide class of NP{\mathcal{NP}} -complete problems), under “natural” distributions. Since it is not clear how to define a class of “natural” distributions for general NP{\mathcal{NP}} -complete problems, one possibility is to impose some strong computational constraint on the distributions, with the intention of this constraint being to force the distributions to “look natural”. Levin, in his seminal paper on average case complexity from 1984, defined such a class of distributions, which he called P-computable distributions. He then showed that the NP{\mathcal{NP}} -complete Tiling problem, under some P-computable distribution, is hard for the complexity class of distributional NP{\mathcal{NP}} problems (i.e. NP{\mathcal{NP}} with P-computable distributions). However, since then very few NP{\mathcal{NP}}- complete problems (coupled with P-computable distributions), and in particular “natural” problems, were shown to be hard in this sense. In this paper we show that all natural NP{\mathcal{NP}} -complete problems can be coupled with P-computable distributions such that the resulting distributional problem is hard for distributional NP{\mathcal{NP}}.     

P-selective sets,tally languages,and the behavior of polynomial time reducibilities onNP

The notion ofp-selective sets, and tally languages, are used to study polynomial time reducibilities onNP. P-selectivity has the property that a setA belongs to the classP if and only if both _m ^p A andA isp-selective. We prove that for every tally language set inNP there exists a polynomial time equivalent set inNP that isp-selective. From this result it follows that if NEXT DEXT, then polynomial time Turing and many-one reducibilities differ onNP. This research was supported in part by the National Science Foundation under grant MCS 77-23493     

On the power of generalized Mod-classes

We investigate the computational power of the new counting class ModP which generalizes the classes Mod _p P,p prime. We show that ModP is polynomialtime truth-table equivalent in power to #P and that ModP is contained in the class AmpMP. As a consequence, the classes PP, ModP, and AmpMP are all Turing equivalent, and thus AmpMP and ModP are not low for MP unless the counting hierarchy collapses to MP. Furthermore, we show that every set in C=P is reducible to some set in ModP via a random many-one reduction that uses only logarithmically many random bits. Hence, ModP and AmpMP are not closed under polynomial-time conjunctive reductions unless the counting hierarchy collapses.     

On completeness for NP via projection translations

In this paper we investigate the logics obtained by augmenting first-order logic (with successor) with operators corresponding to some decision problems complete forNP via logspace reductions. We show that our encodings of the Satisfiability Problem and the 3-Colourability Problem, namely SAT and 3COL, respectively (as sets of finite structures over specific vocabularies), are complete forNP via projection translations (very weak reductions between problems): the fact that an encoding of the Satisfiability Problem (resp. the 3-Satisfiability Problem) is (resp. not) complete forNP via interpretative reductions (again, very weak reductions) had been shown previously by Dahlhaus. It is unknown whether our encoding of the 3-Satisfiability Problem, namely 3SAT, is complete forNP via projection translations, although we show that it is for iterated projection translations. However, if we consider an encoding of the version of the 3-Satisfiability Problem where each instance has at most three literals in each clause (as opposed to exactly three literals), namely 3SAT, then we show that 3SAT is complete forNP via projection translations. It appears to matter as to how we encode a decision problem as a set of finite structures over some vocabulary when considering weak reductions such as ours. Our proof that SAT is complete forNP via projection translations differs extremely from the proof of Dahlhaus that SAT is complete forNP via interpretative reductions, as he relies on Fagin's well-known characterization ofNP whereas our results do not: indeed, they give Fagin's result as a corollary. We use the problem 3COL to show the difference between interpretative reductions and projection translations as we show that 3COL is complete forNP via the latter but not via the former. The logics mentioned above are shown to be similar in expressibility but are shown to be much more expressible than that formed by extending first-order logic with an operator corresponding to the Hamiltonian path problem for digraphs (assumingNP co-NP).     

Relativized isomorphisms of NP-complete sets

In this paper, we present several results about collapsing and non-collapsing degrees ofNP-complete sets. The first, a relativized collapsing result, is interesting because it is the strongest known constructive approximation to a relativization of Berman and Hartmanis' 1977 conjecture that all _m ^P -complete sets forNP arep-isomorphic. In addition, the collapsing result explores new territory in oracle construction, particularly in combining overlapping and apparently incompatible coding and diagonalizing techniques. We also present non-collapsing results, which are notable for their technical simplicity, and which rely on no unproven assumptions such as one-way functions. The basic technique developed in these non-collapsing constructions is surprisingly robust, and can be combined with many different oracle constructions.     

On the Complexity of Matroid Isomorphism Problem

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in S₂^p\Sigma_{2}^{p}. In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be S₂^p\Sigma_{2}^{p}-complete and is co NP-hard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism, and matroid isomorphism are both polynomial time many-one equivalent to graph isomorphism.     

Polynomial Time Introreducibility

Abstract. We introduce the polynomial time version, in short ≤ ^P _T -introreducibility, of the notion of introreducibility studied in Recursion Theory. We give a simpler proof of the known fact that a set is ≤ ^P _T -introreducible if and only if it is in P. Our proof generalizes to virtually all the computation bounded reducibilities ≤ _r , showing that a set is ≤ _r -introreducible if and only if it belongs to the minimum ≤ _r -degree. It also holds for most unbounded reducibilities like ≤ _m , ≤ _c , ≤ _d , ≤ _p , ≤ _tt , etc., but it does not hold for ≤ _T . A very strong way for a set L to fail being ≤ _r -introreducible is that L is not ≤ _r -reducible to any coinfinite subset of L . We call such sets ≤ _r -introimmune. It is known that ≤ _T -introimmune sets exist but they are not arithmetical. In this paper we ask for which reducibilities ≤ _r there exist arithmetical ≤ _r -introimmune sets. We show that they exist for the reducibilities ≤ _c and ≤ ^N _m . Finally, we prove separation results between introimmunities.     

Crossing Numbers of Graphs with Rotation Systems

We show that computing the crossing number and the odd crossing number of a graph with a given rotation system is NP-complete. As a consequence we can show that many of the well-known crossing number notions are NP-complete even if restricted to cubic graphs (with or without rotation system). In particular, we can show that Tutte’s independent odd crossing number is NP-complete, and we obtain a new and simpler proof of Hliněny’s result that computing the crossing number of a cubic graph is NP-complete.     

An observation on probability versus randomness with applications to complexity classes

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Löf. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classesP, BPP, AM, andPH in terms of reducibility to such oracles. These characterizations lead to results like:P =NP if and only if an algorithmically random set exists that is _btt ^P -hard forNP.The work of the first author was supported in part by the Alexander-von-Humboldt-Stiftung and by the National Science Foundation under Grant CCR-8913584 while he visited the Lehrstuhl für Theoretische Informatik, Institut für Informatik, Universität Würzburg, Germany. The work of the second author was supported in part by the National Science Foundation under Grant CCR-8809238 and in part by DIMACS, where he was a visitor while a portion of his work was done.     

Some observations on NP real numbers and P-selective sets

We show that each standard left cut of a real number is a p-selective set. Classification results about NP real numbers and recursively enumerable real numbers follow from similar results about p-selective and semirecursive sets. In particular, it is proved that no left cut can be NP-hard unless the polynomial hierarchy collapses to ?₂^P. This result is surprising because it shows that the McLaughlin-Martin construction of a ?_tt-complete r.e. semirecursive set fails at the polynomial time complexity level.     

Reductions between disjoint NP-Pairs

Disjoint NP-pairs are pairs (A, B) of nonempty, disjoint sets in NP. We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A, B) and (C, D) is a Turing reduction with the additional property that if the input belongs to A B, then all queries belong to C D. We prove under the reasonable assumption that UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A, B) and (C, D) such that (A, B) is truth-table reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DistNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.     

Best Possible Approximation Algorithm for MAX SAT with Cardinality Constraint

We consider the MAX SAT problem with the additional constraint that at most P variables have a true value. We obtain a (1-e ^-1 ) -approximation algorithm for this problem. Feige [6] has proved that for MAX SAT with cardinality constraint with clauses without negations this is the best possible performance guarantee unless P=\NP . Received August 20, 1998; revised June 23, 1999, and April 17, 2000.     

Sharply Bounded Alternation and Quasilinear Time

We define the sharply bounded hierarchy, SBH(QL)}, a hierarchy of classes within P , using quasilinear-time computation and quantification over strings of length log n . It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH(QL) and its corresponding hierarchy of function classes, and present a variety of problems in these classes, including ≤ ^ql _m -complete problems for each class in SBH(QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P≠ PSPACE . We present characterizations of SBH(QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH(QL). Received January 1997, and in final form August 1997.     

A second step toward the strong polynomial-time hierarchy

In this paper we discuss the concepts ofimmunity andsimplicity in levels of the relativized Polynomial-time Hierarchy just aboveP. We consider various diagonalization techniques with which oracle sets can be constructed relative to which strong separations between language classes in the first two levels of this hierarchy are established. In particular, we build oracle sets for separation of relativized Σ ₂ ^P from relativizedNP with immunity, of relativized Σ ₂ ^P from relativizedNP with bi-immunity, of relativized Σ ₂ ^P from relativized Δ ₂ ^P with immunity, of relativized Π ₂ ^P from relativized Δ ₂ ^P with immunity, and finally of relativized Σ ₂ ^P from relativized Π ₂ ^P with simplicity.     

Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets

The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress σ in the optimal cantilevers is assumed to be bounded by −σ _C ≤σ≤σ _T , where σ _C and σ _T represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewiński et al. (Int J Mech Sci 36:375–398, 1994a) to the case of σ _T ≠σ _C . The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.     

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. Therefore, lower bounds for treelike resolution (known since the 1960s) apply to them. However, these lower bounds say nothing about the behavior of such algorithms on satisfiable formulas. Proving exponential lower bounds for them in the most general setting is impossible without proving P ≠ NP; therefore, to prove lower bounds, one has to restrict the power of branching heuristics. In this paper, we give exponential lower bounds for two families of DPLL algorithms: generalized myopic algorithms, which read up to n ^1−ε of clauses at each step and see the remaining part of the formula without negations, and drunk algorithms, which choose a variable using any complicated rule and then pick its value at random. ^★Extended abstract of this paper appeared in Proceedings of ICALP 2004, LNCS 3142, Springer, 2004, pp. 84–96. ^†Supported by CCR grant CCR-0324906. ^‡Supported in part by Russian Science Support Foundation, RAS program of fundamental research “Research in principal areas of contemporary mathematics,” and INTAS grant 04-77-7173. ^§Supported in part by INTAS grant 04-77-7173.