NP Sets Research Articles

Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

Cai and Hemachandra used iterative constant-setting to prove that Few ⊆ ⊕ P (and thus that Few P ⊆ ⊕P). In this article, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed “nongappiness”) of the easy-to-find “targets” used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant’s unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra–Pomerance–Wagstaff Conjecture implies that all \((\mathcal {O}(1) + \log \log n)\) -ambiguity NP sets are in the restricted counting class RC PRIMES .

On some FPT problems without polynomial Turing compressions

An impressive hardness theory which can prove compression lower bounds for a large number of FPT problems has been established under the assumption that NP ⊈ coNP/poly. However, there are no problems in FPT for which the existence of polynomial Turing compressions under any widely believed complexity assumptions has been excluded. In this paper, we provide a technique which can be used to prove that some FPT problems have no small Turing compressions under the assumption that there exists a problem in NP which does not have small-sized circuits. These FPT problems, which include edge clique cover parameterized by the number of cliques, integer linear programming and a-choosability parameterized by some structural parameters, etc. have the property that they remain NP-hard under Cook reductions even if their parameter values are small. Moreover, a trade-off between the size of the Turing compression lower bound and the robustness of the complexity assumption is obtained. In particular, we demonstrate that these FPT problems have no polynomial Turing compressions unless every set in NP has quasi-polynomial-sized circuits, and have no 2o(k) Turing compressions unless every set in NP has sub-exponential-sized circuits. Additionally, Turing kernelization lower bounds for these FPT problems are provided under some weaker complexity assumptions. Lastly, compression lower bounds for the above-mentioned FPT problems are proved under some complexity assumptions which are weaker than NP ⊈ coNP/poly, moreover, these results are proved under a method which is different from the previous hardness theory for compression lower bounds.

The canonical pairs of bounded depth Frege systems

The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number k, also called the depth. We show that the canonical pair of a depth d Frege system is polynomially equivalent to the pair (Ad+2,Bd+2) where Ad+2 (respectively, Bd+1) are depth d+1 games in which Player I (Player II) has a positional winning strategy. Although this characterization is stated in terms of games, we will show that these combinatorial structures can be viewed as generalizations of monotone Boolean circuits. In particular, depth 1 games are essentially monotone Boolean circuits. Thus we get a generalization of the monotone feasible interpolation for Resolution, which is a property that enables one to reduce the task of proving lower bounds on the size of refutations to lower bounds on the size of monotone Boolean circuits. However, we do not have a method yet for proving lower bounds on the size of depth d games for d>1.

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Ten years ago, Gla'er, Pavan, Selman, and Zhang [GPSZ08] proved that if P 6= NP, then all NP-complete sets can be simply split into two NP-complete sets. That advance might naturally make one wonder about a quite di erent potential consequence of NP-completeness: Can the union of easy NP sets ever be hard? In particular, can the union of two non-NP-complete NP sets ever be NP-complete? Amazingly, Ladner [Lad75] resolved this more than forty years ago: If P 6= NP, then all NP-complete sets can be simply split into two non-NP-complete NP sets. Indeed, this holds even when one requires the two non-NP-complete NP sets to be disjoint. We present this result as a mini-tutorial. We give a relatively detailed proof of this result, using the same technique and idea Ladner [Lad75] invented and used in proving a rich collection of results that include many that are more general than this result: delayed diagonalization. In particular, the proof presented is based on what one can call team diagonalization (or if one is being playful, perhaps even tag-team diagonalization): Multiple sets are formed separately by delayed diagonalization, yet those diagonalizations are mutually aware and delay some of their actions until their partner(s) have also succeeded in some coordinated action. We relatedly note that, as a consequence of Ladner's result, if P 6= NP, there exist OptP functions f and g whose composition is NP-hard yet neither f nor g is NP-hard.

Autoreducibility of NP-Complete Sets under Strong Hypotheses

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: Under the stronger assumption that there is a p-generic set in NP $${\cap}$$ coNP, we show: Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility.

Autoreducibility and mitoticity of logspace-complete sets for NP and other classes

We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain:•For NP and all other classes of the PH: each ≤mlog-complete set is ≤Tlog-autoreducible.•For P, Δkp, NEXP: each ≤mlog-complete set is a disjoint union of two ≤2-ttlog-complete sets.•For PSPACE: each ≤dttp-complete set is a disjoint union of two ≤dttp-complete sets.

Strong Reductions and Isomorphism of Complete Sets

We study the structure of the polynomial-time complete sets for NP and PSPACE under strong nondeterministic polynomial-time reductions (SNP-reductions). We show the following results. -- If NP contains a p-random language, then all polynomial-time complete sets for PSPACE are SNP-isomorphic. -- If NP ∩ co-NP contains a p-random language, then all polynomial-time complete sets for NP are SNP-isomorphic.

Physical Portrayal of Computational Complexity

Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because information requires physical representations and because many natural processes complete in nondeterministic polynomial time (NP). The irreversible process with three or more degrees of freedom is found intractable when, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving a problem in the class NP, decisions among alternatives will affect subsequently available sets of decisions. Thus the state space of a nondeterministic finite automaton is evolving due to the computation itself, hence it cannot be efficiently contracted using a deterministic finite automaton. Conversely when solving problems in the class P, the set of states does not depend on computational history, hence it can be efficiently contracted to the accepting state by a deterministic sequence of dissipative transformations. Thus it is concluded that the state set of class P is inherently smaller than the state set of class NP. Since the computational time needed to contract a given set is proportional to dissipation, the computational complexity class P is a proper (strict) subset of NP.

ON THE PROOF COMPLEXITY OF THE NISAN–WIGDERSON GENERATOR BASED ON A HARD NP ∩ coNP FUNCTION

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τ(g)b expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement (S) that the tautologies have no polynomial size proofs in any propositional proof system. This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement (S) is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory (properly included in Buss's theory [Formula: see text]) then Razborov's conjecture would follow, and if TPV could be added too then Statement (S) would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems.

Avoiding Simplicity is Complex

It is a trivial observation that every decidable set has strings of length n with Kolmogorov complexity log n+O(1) if it has any strings of length n at all. Things become much more interesting when one asks whether a similar property holds when one considers resource-bounded Kolmogorov complexity. This is the question considered here: Can a feasible set A avoid accepting strings of low resource-bounded Kolmogorov complexity, while still accepting some (or many) strings of length n? More specifically, this paper deals with two notions of resource-bounded Kolmogorov complexity: Kt and KNt. The measure Kt was defined by Levin more than three decades ago and has been studied extensively since then. The measure KNt is a nondeterministic analog of Kt. For all strings x, Kt(x)≥KNt(x); the two measures are polynomially related if and only if NEXP⊆EXP/poly (Allender et al. in J. Comput. Syst. Sci. 77:14–40, 2011). Many longstanding open questions in complexity theory boil down to the question of whether there are sets in P that avoid all strings of low Kt complexity. For example, the EXP vs ZPP question is equivalent to (one version of) the question of whether avoiding simple strings is difficult: (EXP=ZPP if and only if there exist ϵ>0 and a “dense” set in P having no strings x with Kt(x)≤|x| ϵ (Allender et al. in SIAM J. Comput. 35:1467–1493, 2006)). Surprisingly, we are able to show unconditionally that avoiding simple strings (in the sense of KNt complexity) is difficult. Every dense set in NP∩coNP contains infinitely many strings x such that KNt(x)≤|x| ϵ for every ϵ>0. The proof does not relativize. As an application, we are able to show that if E=NE, then accepting paths for nondeterministic exponential time machines can be found somewhat more quickly than the brute-force upper bound, if there are many accepting paths.

THE THOMPSON–HIGMAN MONOIDS Mk,i: THE ${\mathcal J}$-ORDER, THE ${\mathcal D}$-RELATION, AND THEIR COMPLEXITY

The Thompson–Higman groups Gk,ihave a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,iand Invk,iand investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity.The maximal subgroups of Mk,1and Invk,1are isomorphic to the groups Gk,j(1 ≤ j ≤ k - 1); so we rediscover all the Thompson–Higman groups within Mk,1.Deciding the Green relations [Formula: see text] and [Formula: see text] of Mk,1, when the inputs are words over a finite generating set of Mk,1, is in P.When a circuit-like generating set is used for Mk,1then deciding [Formula: see text] is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for [Formula: see text] is xNPsearch-complete, whereas the multiplier search problems of [Formula: see text] and [Formula: see text] are not in xNPsearch unless NP = coNP. The class of search problems xNPsearch is introduced as a slight generalization of NPsearch.Deciding [Formula: see text] for Mk,1when the inputs are words over a circuit-like generating set, is ⊕k-1• NP -complete; for any h ≥ 2, ⊕h•NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # • NP-complete), and the image size modulo h problem (which is ⊕h•NP-complete). For Invk,1over a circuit-like generating set, deciding [Formula: see text] is ⊕k-1P-complete. It is interesting that the little known complexity classes coDP and ⊕k-1•NP play a central role in Mk,1.

On the Extensions of Krasnoselskii‐Type Theorems to p‐Cyclic Self‐Mappings in Banach Spaces

A set of np(≥2)‐cyclic and either continuous or contractive self‐mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self‐mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii‐type theorems.

The fault tolerance of NP-hard problems

We study the effects of faulty data on NP-hard sets. We consider hard sets for several polynomial time reductions, add corrupt data and then analyze whether the resulting sets are still hard for NP. We explain that our results are related to a weakened deterministic variant of the notion of program self-correction by Blum, Luby, and Rubinfeld. Among other results, we use the Left-Set technique to prove that m-complete sets for NP are nonadaptively weakly deterministically self-correctable while btt-complete sets for NP are weakly deterministically self-correctable. Our results can also be applied to the study of Yesha’s p-closeness. In particular, we strengthen a result by Ogiwara and Fu.

Infeasibility of instance compression and succinct PCPs for NP

The OR-SAT problem asks, given Boolean formulae ϕ 1 , … , ϕ m each of size at most n, whether at least one of the ϕ i 's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP / poly , and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et al. (2008) [6] and Harnik and Naor (2006) [20] and has a number of implications. (i) A number of parametric NP problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP / poly . (ii) Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP / poly . (iii) An approach of Harnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (iv) (Buhrman–Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.

Non-mitotic sets

We study the question of the existence of non-mitotic sets in NP. We show under various hypotheses that • 1-tt-mitoticity and m-mitoticity differ on NP. • T-autoreducibility and T-mitoticity differ on NP (this contrasts the situation in the recursion theoretic setting, where Ladner showed that autoreducibility and mitoticity coincide). • 2-tt-autoreducibility does not imply weak 2-tt-mitoticity (from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2-tt and T, although the notions coincide for m- and 1-tt-reducibility).

A note on [formula omitted]-completeness of NP-witnessing relations

In this note, we study under which conditions various sets (even easy ones) can be associated with a witnessing relation that is # P complete. We show a sufficient condition for an NP set to have such a relation. This condition applies also to many NP -complete sets, as well as to many sets in P .

Properties of NP‐Complete Sets

We study several properties of sets that are complete for NP. We prove that if L is an NP‐complete set and S \not\supseteq L is a p‐selective sparse set, then $L - S$ is $\leq^{p}_{m}$‐hard for NP. We demonstrate the existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$ such that for every $L \in \mbox{NP} - \mbox{P}$, L - S is not $\leq^p_m$‐hard for NP. Moreover, we prove for every $L \in \mbox{NP} - \mbox{P}$ that there exists a sparse $S \in $ EXP such that L - S is not $\leq^p_m$‐hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one‐way permutations to derive consequences for longstanding open questions about whether NP‐complete sets are immune. For example, assuming that pseudorandom generators and secure one‐way permutations exist, it follows easily that NP‐complete sets are not p‐immune. Assuming only that secure one‐way permutations exist, we prove that no NP‐complete set is DTIME$(2^{n^{\epsilon}})$‐immune. Also, using these hypotheses we show that no NP‐complete set is quasi‐polynomial‐close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing‐complete sets for NP are not closed under union. Our hypothesis asserts the existence of a UP‐machine M that accepts $0^*$ such that for some $0 < \epsilon < 1$, no $2^{n^{\epsilon}}$ time‐bounded machine can correctly compute infinitely many accepting computations of M. We show that if $\UP \cap \co\UP$ contains DTIME$(2^{n^{\epsilon}})$‐bi‐immune sets, then this hypothesis is true.

Resource bounded symmetry of information revisited

The information contained in a string x about a string y is the difference between the Kolmogorov complexity of y and the conditional Kolmogorov complexity of y given x, i.e., I ( x : y ) = C ( y ) - C ( y | x ) . The Kolmogorov–Levin Theorem says that I ( x : y ) is symmetric up to a small additive term. We investigate if this property also holds for several versions of polynomial time-bounded Kolmogorov complexity. We study symmetry of information for some variants of distinguishing complexity CD where CD ( x ) is the length of a shortest program which accepts x and only x. We show relativized worlds where symmetry of information does not hold in a strong way for deterministic and nondeterministic polynomial time distinguishing complexities CD poly and CND poly . On the other hand, for nondeterministic polynomial time distinguishing complexity with randomness, CAMD poly , we show that symmetry of information holds for most pairs of strings in any set in NP. Our techniques extend work of Buhrman et al. (Language compression and pseudorandom generators, in: Proc. 19th IEEE Conf. on Computational Complexity, IEEE, New York, 2004, pp. 15–28) on language compression by AM algorithms, and have the following application to the compression of samplable sources, introduced in Trevisan et al. (Compression of sample sources, in: Proc. 19th IEEE Conf. on Computational Complexity, IEEE, New York, 2004, pp. 1–15): any element x in the support of a polynomial time samplable source X can be given a description of size - log Pr [ X = x ] + O ( log 3 n ) , from which x can be recovered by an AM algorithm.

Resource bounded immunity and simplicity

Revisiting the 30-years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k -immunity and feasible k -immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.

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.