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Joerg Rothe

Publications and source records attributed to Joerg Rothe.

34 records · Page 2Linked to original sources
arXiv2007

On Approximating Optimal Weighted Lobbying, and Frequency of Correctness versus Average-Case Polynomial Time

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant--optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a ``frequently self-knowingly correct algorithm.'' We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein's junta distributions [PR07].

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arXiv2006

A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs

We show that computing the lexicographically first four-coloring for planar graphs is P^{NP}-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P \neq NP. We discuss this application to non-self-reducibility and provide a general related result.

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arXiv2001

Computing Complete Graph Isomorphisms and Hamiltonian Cycles from Partial Ones

We prove that computing a single pair of vertices that are mapped onto each other by an isomorphism $ϕ$ between two isomorphic graphs is as hard as computing $ϕ$ itself. This result optimally improves upon a result of Gál et al. We establish a similar, albeit slightly weaker, result about computing complete Hamiltonian cycles of a graph from partial Hamiltonian cycles.

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arXiv1999

Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

It is known that for any class C closed under union and intersection, the Boolean closure of C, the Boolean hierarchy over C, and the symmetric difference hierarchy over C all are equal. We prove that these equalities hold for any complexity class closed under intersection; in particular, they thus hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove that the Hausdorff hierarchy and the nested difference hierarchy over UP both fail to capture the Boolean closure of UP in some relativized worlds. Karp and Lipton proved that if nondeterministic polynomial time has sparse Turing-complete sets, then the polynomial hierarchy collapses. We establish the first consequences from the assumption that unambiguous polynomial time has sparse Turing-complete sets: (a) UP is in Low_2, where Low_2 is the second level of the low hierarchy, and (b) each level of the unambiguous polynomial hierarchy is contained one level lower in the promise unambiguous polynomial hierarchy than is otherwise known to be the case.

cs.CC↗
arXiv1999

Characterizations of the Existence of Partial and Total One-Way Permutations

In this note, we study the easy certificate classes introduced by Hemaspaandra, Rothe, and Wechsung, with regard to the question of whether or not surjective one-way functions exist. This is an important open question in cryptology. We show that the existence of partial one-way permutations can be characterized by separating P from the class of UP sets that, for all unambiguous polynomial-time Turing machines accepting them, always have easy (i.e., polynomial-time computable) certificates. This extends results of Grollmann and Selman. By Grädel's recent results about one-way functions, this also links statements about easy certificates of NP sets with statements in finite model theory. Similarly, there exist surjective poly-one one-way functions if and only if there is a set L in P such that not all FewP machines accepting L always have easy certificates. We also establish a condition necessary and sufficient for the existence of (total) one-way permutations.

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arXiv1999

Restrictive Acceptance Suffices for Equivalence Problems

One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach---weaker in strength of evidence but more broadly applicable---to suggesting that concrete~NP problems are not NP-complete. In particular we introduce the class EP, the subclass of NP consisting of those languages accepted by NP machines that when they accept always have a number of accepting paths that is a power of two. Since if any NP-complete set is in EP then all NP sets are in EP, it follows---with whatever degree of strength one believes that EP differs from NP---that membership in EP can be viewed as evidence that a problem is not NP-complete. We show that the negation equivalence problem for OBDDs (ordered binary decision diagrams) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation the equivalence problem is in EP^{NP}, thus tightening the existing NP^{NP} upper bound. We show that FewP, bounded ambiguity polynomial time, is contained in EP, a result that is not known to follow from the previous SPP upper bound. For the three problems and classes just mentioned with regard to EP, no proof of membership/containment in UP is known, and for the problem just mentioned with regard to EP^{NP}, no proof of membership in UP^{NP} is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.

cs.CC↗
arXiv1999

Polynomial-Time Multi-Selectivity

We introduce a generalization of Selman's P-selectivity that yields a more flexible notion of selectivity, called (polynomial-time) multi-selectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used to prove the first known (and optimal) lower bounds for generalized selectivity-like classes in terms of EL_2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara's P-mc (polynomial-time membership-comparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the P-selective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low_2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that P-Sel, the class of P-selective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the P-selective sets, and yet, in contrast with the P-mc classes, is refined enough to distinguish them.

cs.CC↗
arXiv1999

Easy Sets and Hard Certificate Schemes

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions---regarding immunity and class collapses---that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical ``wide spacing'' oracle construction technique yields instant non-bi-immunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay's classical result that NP \neq P = NP \cap coNP holds in some relativized world.

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arXiv1999

Exact Analysis of Dodgson Elections: Lewis Carroll's 1876 Voting System is Complete for Parallel Access to NP

In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner---a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound---NP-hardness---on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, i.e., it is complete for $\thetatwo$, for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses.

cs.CC↗
arXiv1999

Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.

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arXiv1999

A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard. The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism.

cs.CC↗
arXiv1999

Raising NP Lower Bounds to Parallel NP Lower Bounds

A decade ago, a beautiful paper by Wagner developed a ``toolkit'' that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During the past year, problems that previously were known only to be NP-hard or coNP-hard have been shown to be hard even for the class of sets solvable via parallel access to NP. Many of these problems are longstanding and extremely natural, such as the Minimum Equivalent Expression problem (which was the original motivation for creating the polynomial hierarchy), the problem of determining the winner in the election system introduced by Lewis Carroll in 1876, and the problem of determining on which inputs heuristic algorithms perform well. In the present article, we survey this recent progress in raising lower bounds.

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arXiv1998

Immunity and Simplicity for Exact Counting and Other Counting Classes

Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some relativized world, PSPACE (in fact, ParityP) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C_{=}P, and ParityP in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C_{=}P contains a set that is immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A} and ParityP^{A}. Strengthening results of Torán [J.ACM 38, 1991] and Green [IPL 37, 1991], we also show that, in suitable relativizations, NP contains a C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the existence of a C_{=}P^{B}-simple set for some oracle B, which extends results of Balcázar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for ``exact counting'' that is derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.

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arXiv1998

Tally NP Sets and Easy Census Functions

We study the question of whether every P set has an easy (i.e., polynomial-time computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as the containment of #P_1 in FP, where #P_1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P_{1}^{PH} function can be computed in FP^{#P_{1}^{#P_{1}}}. Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption of #P_1 being contained in FP implies P = BPP and that PH is contained in MOD_{k}P for each k \geq 2, which provides further evidence that not all sets in P have an easy census function. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n^α-enumerated in time n^β for fixed αand β) than that it can be precisely computed in polynomial time.

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arXiv1998

Creating Strong Total Commutative Associative Complexity-Theoretic One-Way Functions from Any Complexity-Theoretic One-Way Function

Rabi and Sherman [RS97] presented novel digital signature and unauthenticated secret-key agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use ``strong,'' total, commutative (in the case of multi-party secret-key agreement), associative one-way functions as their key building blocks. Though Rabi and Sherman did prove that associative one-way functions exist if $\p \neq \np$, they left as an open question whether any natural complexity-theoretic assumption is sufficient to ensure the existence of ``strong,'' total, commutative, associative one-way functions. In this paper, we prove that if $\p \neq \np$ then ``strong,'' total, commutative, associative one-way functions exist.

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