Complexity-theoretic Assumptions Research Articles

The moral hazard of complexity-theoretic assumptions

No abstract available.

Partially Polynomial Kernels for Set Cover and Test Cover

An instance of the $(n-k)$-Set Cover or the $(n-k)$-Test Cover problems is of the form $(\mathcal{U},\mathcal{S},k)$, where $\mathcal{U}$ is a set with $n$ elements, $\mathcal{S}\subseteq 2^\mathcal{U}$ with $|\mathcal{S}|=m$, and $k$ is the parameter. The instance is a Yes-instance of $(n-k)$-Set Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that every element of $\mathcal{U}$ is contained in some set in $\mathcal{S}'$. Similarly, it is a Yes-instance of $(n-k)$-Test Cover if and only if there exists $\mathcal{S}'\subseteq\mathcal{S}$ with $|\mathcal{S}'|\leq n-k$ such that for any pair of elements from $\mathcal{U}$, there exists a set in $\mathcal{S}'$ that contains one of them but not the other. It is known in the literature that both $(n-k)$-Set Cover and $(n-k)$-Test Cover do not admit polynomial kernels (under some well-known complexity theoretic assumptions). However, in this paper we show that they do admit “partially polynomial kernels”: we give polynomial time algorithms that take as input an instance $(\mathcal{U},\mathcal{S},k)$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) and return an equivalent instance $(\tilde{\mathcal{U}},\tilde{\mathcal{S}}, \tilde{k})$ of $(n-k)$-Set Cover (respectively, $(n-k)$-Test Cover) with $\tilde{k} \leq k$ and $| \tilde{\mathcal{U}}|= \mathcal{O}(k^2)$ (respectively, $|\tilde{\mathcal{U}}|=\mathcal{O}(k^7)$). These results allow us to generalize, improve, and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain “sparsity properties.” Using a part of our partial kernelization algorithm for $(n-k)$-Set Cover, we also get an improved fixed-parameter tractable algorithm for this problem which runs in time $\mathcal{O}(4^kk^{\mathcal{O}(1)}(m+n)+mn)$ improving over the previous best of $\mathcal{O}(8^{k+o(k)}(m+n)^{\mathcal{O}(1)})$. On the other hand, the partially polynomial kernel for $(n-k)$-Test Cover gives an algorithm with running time $\mathcal{O}(2^{\mathcal{O}(k^2)}(m+n)^{\mathcal{O}(1)})$. We believe such an approach could also be useful for other covering problems.

Backdoors to Normality for Disjunctive Logic Programs

The main reasoning problems for disjunctive logic programs are complete for the second level of the polynomial hierarchy and hence considered harder than the same problems for normal (i.e., disjunction-free) programs, which are on the first level. We propose a new exact method for solving the disjunctive problems which exploits the small distance of a disjunctive programs from being normal. The distance is measured in terms of the size of a smallest “backdoor to normality,” which is the smallest number of atoms whose deletion makes the program normal. Our method consists of three phases. In the first phase, a smallest backdoor is computed. We show that this can be done using an efficient algorithm for computing a smallest vertex cover of a graph. In the second phase, the backdoor is used to transform the logic program into a quantified Boolean formula (QBF) where the number of universally quantified variables equals the size of the backdoor and where the total size of the quantified Boolean formula is quasilinear in the size of the given logic program. The quasilinearity is achieved by means of a characterization of the least model of a Horn program in terms of level numberings. In a third phase, the universal variables are eliminated using universal expansion yielding a propositional formula. The blowup in the last phase is confined to a factor that is exponential in the size of the backdoor but linear in the size of the quantified Boolean formula. By checking the satisfiability of the resulting formula with a S at solver (or by checking the satisfiability of the quantified Boolean formula by a Q bf -S at solver), we can decide the A sp reasoning problems on the input program. In consequence, we have a transformation from A sp problems to propositional satisfiability where the combinatorial explosion, which is expected when transforming a problem from the second level of the polynomial hierarchy to the first level, is confined to a function of the distance to normality of the input program. In terms of parameterized complexity, the transformation is fixed-parameter tractable. We complement this result by showing that (under plausible complexity-theoretic assumptions) such a fixed-parameter tractable transformation is not possible if we consider the distance to tightness instead of distance to normality.

On finding optimal polytrees

We study the NP-hard problem of finding a directed acyclic graph (DAG) on a given set of nodes so as to maximize a given scoring function. The problem models the task of inferring a probabilistic network from data, which has been studied extensively in the fields of artificial intelligence and machine learning. Several variants of the problem, where the output DAG is constrained in several ways, are NP-hard as well, for example when the DAG is required to have bounded in-degree, or when it is required to be a polytree. Polynomial-time algorithms are known only for rare special cases, perhaps most notably for branchings, that is, polytrees in which the in-degree of every node is at most one.In this paper, we generalize this polynomial-time result to polytrees that can be turned into a branching by deleting a constant number of arcs. Our algorithm stems from a matroid intersection formulation. As the order of the polynomial time bound depends on the number of deleted arcs, the algorithm does not establish fixed-parameter tractability when parameterized by that number. We show that certain additional constraints on the sought polytree render the problem fixed-parameter tractable. We contrast this positive result by showing that if we parameterize by the number of deleted nodes, a somewhat more powerful parameter, the problem is not fixed-parameter tractable, subject to a complexity-theoretic assumption.

Quantum Cryptography II: How to re-use a one-time pad safely even if P=NP.

When elementary quantum systems, such as polarized photons, are used to transmit digital information, the uncertainty principle gives rise to novel cryptographic phenomena unachievable with traditional transmission media, e.g. a communications channel on which it is impossible in principle to eavesdrop without a high probability of being detected. With such a channel, a one-time pad can safely be reused many times as long as no eavesdrop is detected, and, planning ahead, part of the capacity of these uncompromised transmissions can be used to send fresh random bits with which to replace the one-time pad when an eavesdrop finally is detected. Unlike other schemes for stretching a one-time pad, this scheme does not depend on complexity-theoretic assumptions such as the difficulty of factoring.

A Completeness Theory for Polynomial (Turing) Kernelization

The framework of Bodlaender et al. (J Comput Sys Sci 75(8):423---434, 2009) and Fortnow and Santhanam (J Comput Sys Sci 77(1):91---106, 2011) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, some issues are not addressed by this framework, including the existence of Turing kernels such as the "kernelization" of leaf out-branching $$(k)$$ ( k ) that outputs $$n$$ n instances each of size poly $$(k)$$ ( k ) . Observing that Turing kernels are preserved by polynomial parametric transformations (PPTs), we define two kernelization hardness hierarchies by the PPT-closure of problems that seem fundamentally unlikely to admit efficient Turing kernelizations. This gives rise to the MK- and WK-hierarchies which are akin to the M- and W-hierarchies of parameterized complexity. We find that several previously considered problems are complete for the fundamental hardness class WK[1], including Min Ones $$d$$ d -SAT $$(k)$$ ( k ) , Binary NDTM Halting $$(k)$$ ( k ) , Connected Vertex Cover $$(k)$$ ( k ) , and Clique parameterized by $$k \log n$$ k log n . We conjecture that no WK[1]-hard problem admits a polynomial Turing kernel. Our hierarchy subsumes an earlier hierarchy of Harnik and Naor that, from a parameterized perspective, is restricted to classical problems parameterized by witness size. Our results provide the first natural complete problems for, e.g., their class $$VC_1$$ V C 1 ; this had been left open.

Guarantees and limits of preprocessing in constraint satisfaction and reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.

On Detecting Nearly Structured Preference Profiles

Structured preference domains, such as, for example, the domains of single-peaked and single-crossing preferences, are known to admit efficient algorithms for many problems in computational social choice. Some of these algorithms extend to preferences that are close to having the respective structural property, i.e., can be made to enjoy this property by performing minor changes to voters' preferences, such as deleting a small number of voters or candidates. However, it has recently been shown that finding the optimal number of voters or candidates to delete in order to achieve the desired structural property is NP-hard for many such domains. In this paper, we show that these problems admit efficient approximation algorithms. Our results apply to all domains that can be characterized in terms of forbidden configurations; this includes, in particular, single-peaked and single-crossing elections. For a large range of scenarios, our approximation results are optimal under a plausible complexity-theoretic assumption. We also provide parameterized complexity results for this class of problems.

Pricing lotteries

We examine optimal mechanisms for a multi-product monopolist selling n substitutable goods to a buyer of unknown type randomly sampled from a known distribution. The optimal selling strategy assigns posted prices to lotteries (probability distributions over items), and it is known that pricing lotteries can be strictly superior to pricing individual items. Our results quantify the magnitude of this phenomenon and shed light on the computational hardness of determining (or approximating) the optimal selling strategy. In particular, we show that when the number of items is at least three, there is no finite upper bound on the ratio between the maximum revenue attainable by pricing lotteries and that which can be attained by pricing items. Furthermore, the time required to compute the optimal selling strategy is polynomial in the support size of the buyer's type distribution. We also show that these two results are reversed in a model in which the buyer is allowed to buy any desired number of lotteries and receive an independent sample from each: the ratio between the revenue of the optimal selling strategy and that which can be attained by pricing items is only logarithmic in the number of goods n, and under standard complexity-theoretic assumptions there is no polynomial-time algorithm to compute or even approximate the optimal selling strategy.

Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most $$d$$ , throughout the uniqueness region of the Gibbs measure on the infinite $$d$$ -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree $$d$$ for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.

Lower bounds on the complexity of [formula omitted] model-checking

Kreutzer and Tazari proved in 2010 that MSO2 model-checking is not polynomial (XP) w.r.t. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. We prove that MSO1 model-checking with a fixed set of vertex labels — i.e., without edge-set quantification — is not solvable even in quasi-polynomial time for fixed MSO1-formulas in such graph classes. Both the lower bounds hold modulo a certain complexity-theoretic assumption, namely, the Exponential-Time Hypothesis (ETH) in the former case and the non-uniform ETH in the latter case. In comparison to Kreutzer and Tazari, we show a different set of problems to be intractable, and our stronger complexity assumption of non-uniform ETH slightly weakens assumptions on the graph class and greatly simplifies important lengthy parts of the former proof. Our result also has an interesting consequence in the realm of digraph width measures.

Parameterized Complexity Results for Exact Bayesian Network Structure Learning

Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) super-structure. The super-structure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed super-structure as a natural generalization of its undirected counterpart. Our results apply to several variants of score-based Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth, and in linear time if in addition the super-structure has bounded maximum degree. Furthermore, we show that if the directed super-structure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Similarly, exact Bayesian network structure learning remains NP-hard for "almost acyclic" directed super-structures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).

Towards communication-efficient quantum oblivious key distribution

Oblivious Transfer, a fundamental problem in the field of secure multi-party computation is defined as follows: A database DB of N bits held by Bob is queried by a user Alice who is interested in the bit DB_b in such a way that (1) Alice learns DB_b and only DB_b and (2) Bob does not learn anything about Alice's choice b. While solutions to this problem in the classical domain rely largely on unproven computational complexity theoretic assumptions, it is also known that perfect solutions that guarantee both database and user privacy are impossible in the quantum domain. Jakobi et al. [Phys. Rev. A, 83(2), 022301, Feb 2011] proposed a protocol for Oblivious Transfer using well known QKD techniques to establish an Oblivious Key to solve this problem. Their solution provided a good degree of database and user privacy (using physical principles like impossibility of perfectly distinguishing non-orthogonal quantum states and the impossibility of superluminal communication) while being loss-resistant and implementable with commercial QKD devices (due to the use of SARG04). However, their Quantum Oblivious Key Distribution (QOKD) protocol requires a communication complexity of O(N log N). Since modern databases can be extremely large, it is important to reduce this communication as much as possible. In this paper, we first suggest a modification of their protocol wherein the number of qubits that need to be exchanged is reduced to O(N). A subsequent generalization reduces the quantum communication complexity even further in such a way that only a few hundred qubits are needed to be transferred even for very large databases.

A Polynomial-Time Algorithm for Estimating the Partition Function of the Ferromagnetic Ising Model on a Regular Matroid

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids, and is properly included in the class of binary matroids. Using Seymour's decomposition theorem, we give an FPRAS for the class of regular matroids.

Cluster editing with locally bounded modifications

Given an undirected graph G=(V,E) and a nonnegative integer k, the NP-hard Cluster Editing problem asks whether G can be transformed into a disjoint union of cliques by modifying at most k edges. In this work, we study how “local degree bounds” influence the complexity of Cluster Editing and of the related Cluster Deletion problem which allows only edge deletions. We show that even for graphs with constant maximum degree Cluster Editing and Cluster Deletion are NP-hard and that this implies NP-hardness even if every vertex is incident with only a constant number of edge modifications. We further show that under some complexity-theoretic assumptions both Cluster Editing and Cluster Deletion cannot be solved within a running time that is subexponential in k, |V|, or |E|. Finally, we present a problem kernelization for the combined parameter “number d of clusters and maximum number t of modifications incident with a vertex” thus showing that Cluster Editing and Cluster Deletion become easier in case the number of clusters is upper-bounded.

Limits of Preprocessing

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a complexity theoretic assumption, none of the considered problems can be reduced by polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, such as induced width or backdoor size. Our results provide a firm theoretical boundary for the performance of polynomial-time preprocessing algorithms for the considered problems.

The shrinking property for NP and coNP

We study the shrinking and separation properties (two notions well-known in descriptive set theory) for NP and coNP and show that under reasonable complexity-theoretic assumptions, both properties do not hold for NP and the shrinking property does not hold for coNP . In particular we obtain the following results. 1. NP and coNP do not have the shrinking property unless PH is finite. In general, Σ n P and Π n P do not have the shrinking property unless PH is finite. This solves an open question posed by Selivanov (1994) [33]. 2. The separation property does not hold for NP unless UP ⊆ coNP . 3. The shrinking property does not hold for NP unless there exist NP -hard disjoint NP -pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]). 4. The shrinking property does not hold for NP unless there exist complete disjoint NP -pairs. Moreover, we prove that the assumption NP ≠ coNP is too weak to refute the shrinking property for NP in a relativizable way. For this we construct an oracle relative to which P = NP ∩ coNP , NP ≠ coNP , and NP has the shrinking property. This solves an open question posed by Blass and Gurevich (1984) [3] who explicitly ask for such an oracle.

On the expressive power of CNF formulas of bounded tree- and clique-width

We study representations of polynomials over a field K from the point of view of their expressive power. Three important examples for the paper are polynomials arising as permanents of bounded tree-width matrices, polynomials given via arithmetic formulas, and families of so called CNF polynomials. The latter arise in a canonical way from families of Boolean formulas in conjunctive normal form. To each such CNF formula there is a canonically attached incidence graph. Of particular interest to us are CNF polynomials arising from formulas with an incidence graph of bounded tree- or clique-width. We show that the class of polynomials arising from families of polynomial size CNF formulas of bounded tree-width is the same as those represented by polynomial size arithmetic formulas, or permanents of bounded tree-width matrices of polynomial size. Then, applying arguments from communication complexity we show that general permanent polynomials cannot be expressed by CNF polynomials of bounded tree-width. We give a similar result in the case where the clique-width of the incidence graph is bounded, but for this we need to rely on the widely believed complexity theoretic assumption # P ⊈ F P / p o l y .

Causal graphs and structurally restricted planning

The causal graph is a directed graph that describes the variable dependencies present in a planning instance. A number of papers have studied the causal graph in both practical and theoretical settings. In this work, we systematically study the complexity of planning restricted by the causal graph. In particular, any set of causal graphs gives rise to a subcase of the planning problem. We give a complete classification theorem on causal graphs, showing that a set of graphs is either polynomial-time tractable, or is not polynomial-time tractable unless an established complexity-theoretic assumption fails; our theorem describes which graph sets correspond to each of the two cases. We also give a classification theorem for the case of reversible planning, and discuss the general direction of structurally restricted planning.

Quantum Algorithm for Linear Systems of Equations

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition number kappa, the fastest known classical algorithms can find x(-->) and estimate x(-->)(dagger) Mx(-->) in time scaling roughly as N square root(kappa). Here, we exhibit a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa. Indeed, for small values of kappa [i.e., poly log(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.