Subexponential Time Research Articles

Hardness results for approximate pure Horn CNF formulae minimization

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in n Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of 2 log 1 ? o ( 1 ) n $2^{\log^{1-o(1)} n}$ . This is the case even when the inputs are restricted to pure Horn 3-CNFs with O(n 1+? ) clauses, for some small positive constant ?. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.

MPI-based implementation of an enhanced algorithm to solve the LPN problem in a memory-constrained environment

Abstract In recent years, several lightweight cryptographic protocols whose security lies in the assumed intractability of the learning parity with noise (LPN) problem have been proposed. The LPN problem has been shown to be solvable in subexponential time by algorithms that have very large (subexponential) memory requirements, which limits their practical applicability. When the memory resources are constrained, a brute-force search is the only known way of solving the LPN problem. In this paper, we propose a new parallel implementation, called Parallel-LPN, of an enhanced algorithm to solve the LPN problem. We implemented the Parallel-LPN in C and MPI (Message Passing Interface), and it was tested on a cluster system, where we obtained a quasi-linear speedup of approximately 90%. We also proposed a new algorithm by using combinatorial objects that enhances the Parallel-LPN performance and its serial version.

On a connection between small set expansions and modularity clustering

In this paper we explore a connection between two seemingly different problems from two different domains: the small-set expansion problem studied in unique games conjecture, and a popular community finding approach for social networks known as the modularity clustering approach. We show that a sub-exponential time algorithm for the small-set expansion problem leads to a sub-exponential time constant factor approximation for some hard input instances of the modularity clustering problem.

Parameterized Algorithms for the 2-Clustering Problem with Minimum Sum and Minimum Sum of Squares Objective Functions

In the {\sc Min-Sum 2-Clustering} problem, we are given a graph and a parameter $k$, and the goal is to determine if there exists a 2-partition of the vertex set such that the total conflict number is at most $k$, where the conflict number of a vertex is the number of its non-neighbors in the same cluster and neighbors in the different cluster. The problem is equivalent to {\sc 2-Cluster Editing} and {\sc 2-Correlation Clustering} with an additional multiplicative factor two in the cost function. In this paper we show an algorithm for {\sc Min-Sum 2-Clustering} with time complexity $O(n\cdot 2.619^{r/(1-4r/n)}+n^3)$, where $n$ is the number of vertices and $r=k/n$. Particularly, the time complexity is $O^*(2.619^{k/n})$ for $k\in o(n^2)$ and polynomial for $k\in O(n\log n)$, which implies that the problem can be solved in subexponential time for $k\in o(n^2)$. We also design a parameterized algorithm for a variant in which the cost is the sum of the squared conflict-numbers. For $k\in o(n^3)$, the algorithm runs in subexponential $O(n^3\cdot 5.171^{\theta})$ time, where $\theta=\sqrt{k/n}$.

Dynamics and termination cost of spatially coupled mean-field models

This work is motivated by recent progress in information theory and signal processing where the so-called spatially coupled design of systems leads to considerably better performance. We address relevant open questions about spatially coupled systems through the study of a simple Ising model. In particular, we consider a chain of Curie-Weiss models that are coupled by interactions up to a certain range. Indeed, it is well known that the pure (uncoupled) Curie-Weiss model undergoes a first-order phase transition driven by the magnetic field, and furthermore in the spinodal region such systems are unable to reach equilibrium in subexponential time if initialized in the metastable state. In contrast, the spatially coupled system is instead able to reach the equilibrium even when initialized to the metastable state. The equilibrium phase propagates along the chain in the form of a traveling wave. Here we study the speed of the wave front and the so-called termination cost-i.e., the conditions necessary for the propagation to occur. We reach several interesting conclusions about optimization of the speed and the cost.

Subexponential class group and unit group computation in large degree number fields

AbstractWe describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs

In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, k-Leaf Out-Branching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely k-Internal Out-Branching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that on these classes of graphs, the k-Directed Path problem is solvable in time O((1+ε)knf(ε)), for any ε>0, where f is some function of ε.Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem-specific combinatorial structures, and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

Polynomial-Time Algorithms for Energy Games with Special Weight Structures

Energy games belong to a class of turn-based two-player infinite-duration games played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in NP∩co-NP, but are not known to be in P. The existence of polynomial-time algorithms has been a major open problem for decades and apart from pseudopolynomial algorithms there is no algorithm that solves any non-trivial subclass in polynomial time. In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several worst-case instances on which previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting the graph structure does not necessarily help.

Constructing elliptic curve isogenies in quantum subexponential time

Abstract. Given two ordinary elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit a nonzero isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest known classical algorithm takes exponential time, and prior to our work no faster quantum algorithm was known. Recently, public-key cryptosystems based on the presumed hardness of this problem have been proposed as candidates for post-quantum cryptography. In this paper, we give a new subexponential-time quantum algorithm for constructing nonzero isogenies between two such elliptic curves, assuming the Generalized Riemann Hypothesis (but with no other assumptions). Our algorithm is based on a reduction to a hidden shift problem, and represents the first nontrivial application of Kuperberg's quantum algorithm for finding hidden shifts. This result suggests that isogeny-based cryptosystems may be uncompetitive with more mainstream quantum-resistant cryptosystems such as lattice-based cryptosystems. As part of this work, we also present the first classical algorithm for evaluating isogenies having provably subexponential running time in the cardinality of the base field under GRH.

Langevin Dynamics with a Tilted Periodic Potential

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity γ and subject to a further external field α. For a suitable choice of the parameters α and γ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale.

On Canonical Forms for Zero-Sum Stochastic Mean Payoff Games

We consider two-person zero-sum mean payoff undiscounted stochastic games and obtain sufficient conditions for the existence of a saddle point in uniformly optimal stationary strategies. Namely, these conditions enable us to bring the game, by applying potential transformations, to a canonical form in which locally optimal strategies are globally optimal, and hence the value for every initial position and the optimal strategies of both players can be obtained by playing the local game at each state. We show that these conditions hold for the class of additive transition (AT) games, that is, the special case when the transitions at each state can be decomposed into two parts, each controlled completely by one of the two players. An important special case of AT-games form the so-called BWR-games which are played by two players on a directed graph with positions of three types: Black, White and Random. We give an independent proof for the existence of a canonical form in such games, and use this result to derive the existence of a canonical form (and hence, of a saddle point in uniformly optimal stationary strategies) in a wide class of games, which includes stochastic games with perfect information (PI), switching controller (SC) games and additive rewards, additive transition (ARAT) games. Unlike the proof for AT-games, our proof for the BWR-case does not rely on the existence of a saddle point in stationary strategies. We also derive some algorithmic consequences from these our reductions to BWR-games, in terms of solving PI-, and ARAT-games in sub-exponential time.

Local List-Decoding and Testing of Random Linear Codes from High Error

In this paper, we give efficient algorithms for list-decoding and testing random linear codes. Our main result is that random sparse linear codes are locally list-decodable and locally testable in the high-error regime with only a constant number of queries. More precisely, we show that for all constants $c> 0$ and $\gamma > 0$, and for every linear code $\mathcal C \subseteq \{0,1\}^N$ which is (1) sparse: $|\mathcal C| \leq N^c$, and (2) unbiased: each nonzero codeword in $\mathcal C$ has weight $\in (\frac{1}{2} - N^{-\gamma}, \frac{1}{2} + N^{-\gamma})$, then $\mathcal C$ is locally testable and locally list-decodable from $(\frac{1}{2} - \epsilon)$-fraction worst-case errors using only ${ {poly}}(\frac{1}{\epsilon})$ queries to a received word. We also give subexponential time algorithms for list-decoding arbitrary unbiased (but not necessarily sparse) linear codes in the high-error regime. In particular, this yields the first subexponential time algorithm even for the problem of (unique) decoding ra...

Special Section on the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010)

This issue of SICOMP contains eight selected papers from the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010), held June 6--8, 2010, in Cambridge, Massachusetts. The STOC proceedings contained 78 papers, which the program committee selected from 279 submissions. The program committee consisted of Timothy Chan, Ken Clarkson, Constantinos Daskalakis, Irit Dinur, Faith Ellen, Alan Frieze, Parikshit Gopalan, Piotr Indyk, Valentine Kabanets, Yael Tauman Kalai, Howard Karloff, Robert Kleinberg, Assaf Naor, Noam Nisan, Chris Peikert, Jaikumar Radhakrishnan, Oded Regev, Alexander Russell, Leonard Schulman (chair), Aravind Srinivasan, Santosh Vempala, and Andrew Yao. Eight of the STOC papers appear in this special section, each expanded and subjected to the standard thorough reviewing process of the journal. They cover a diverse collection of topics: In “Improving Exhaustive Search Implies Superpolynomial Lower Bounds," R. Ryan Williams shows that there are natural problems in NP and BPP for which algorithms that improve over the naïve deterministic simulation even quite slightly, imply lower bounds such as NEXP $\not\in$ P/poly and LOGSPACE $\neq$ NP. Williams also proves certain unconditional time-space lower bounds for improving on exhaustive search; the length of the witness-string in some standard verification protocol is a key parameter here. In “An Effective Dichotomy for the Counting Constraint Satisfaction Problem," Martin Dyer and David Richerby consider the counting constraint satisfaction problem (\#CSP). This problem asks how many ways there are to satisfy a system of constraints on a set of variables, where a constraint is a relation chosen from a fixed finite set. This class is shown to have a decidable dichotomy, depending on the form of the relations. The dichotomy is that each problem in the class either is in FP or is \#P-complete, with no intermediate cases. In “Pseudorandom Generators for Polynomial Threshold Functions," Raghu Meka and David Zuckerman develop improved (and in many cases the first nontrivial) pseudorandom generators for low-degree polynomial threshold functions; related explicit constructions are also developed. A key ingredient is the use of invariance principles to construct pseudorandom generators. In “Local List-Decoding and Testing of Random Linear Codes from High Error," Swastik Kopparty and Shubhangi Saraf give efficient local list-decoding and testing algorithms for “sparse" random linear codes, and subexponential time algorithms for list-decoding random linear codes, which tolerate error rates approaching $1/2$. In “How to Compress Interactive Communication," Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao attack the important direct sum problem in communication complexity: is the complexity of evaluating $n$ copies of a function ever significantly less than $n$ times the complexity of evaluating it once? By defining a new notion of information cost for protocols --- the so-called internal information cost --- and providing new protocol compression schemes, they prove that computing $n$ copies of any function requires communicating at least $\sqrt{n}$ times as many bits as computing one copy of the function. In “A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations," Daniele Micciancio and Panagiotis Voulgaris provide the first $\exp(O(n))$-time algorithms for the closest vector problem (CVP) and shortest independent vectors problem (SIVP); their algorithm is, moreover, deterministic. Likewise they provide a deterministic algorithm for the shortest vector problem (SVP), whose $\exp(O(n))$ runtime is an improvement over the best known bounds for randomized algorithms. In “Perfect Matchings in $O(n \log n)$ Time in Regular Bipartite Graphs," Ashish Goel, Michael Kapralov, and Sanjeev Khanna provide a randomized algorithm that finds a perfect matching in a $d$-regular $n$-node bipartite graph in time $O(n \log n)$, notably, within time that may be sublinear in the input size and is independent of the degree. In “Efficiency Improvements in Constructing Pseudorandom Generators from One-Way Functions," Iftach Haitner, Omer Reingold, and Salil Vadhan give a new construction of pseudorandom generators from one-way functions that both simplifies and tightens the acclaimed original construction of Hastad, Impagliazzo, Levin, and Luby. We thank the authors, the STOC program committee, the STOC external reviewers, and the journal referees for all their work to make this special issue possible.

Deterministic Identity Testing of Depth-4 Multilinear Circuits with Bounded Top Fan-in

We give the first subexponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 multilinear circuits with a plus gate at the top (also known as $\Sigma\Pi\Sigma\Pi$ circuits) and has a running time of $\exp(\mathrm{poly}(\log(n),\log(s),k))$ where $n$ is the number of variables, $s$ is the size of the circuit, and $k$ is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. Prior to this work, sub-exponential time deterministic algorithms were known for depth-$3$ circuits with small top fan-in and for very restricted versions of depth-$4$ circuits. The main ingredient in our proof is a new structural theorem for multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits. Roughly, this theorem shows that any nonzero multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuit contains an “embedded” nonzero multilinear $\Sigma\Pi\Sigma(k)$ circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits.

Is Submodularity Testable?

We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a submodular function is assumed. But how does one check if this oracle indeed represents a submodular function? Consider a function f:{0,1}^n \rightarrow R. The distance to submodularity is the minimum fraction of values of $f$ that need to be modified to make f submodular. If this distance is more than epsilon > 0, then we say that f is epsilon-far from being submodular. The aim is to have an efficient procedure that, given input f that is epsilon-far from being submodular, certifies that f is not submodular. We analyze a very natural tester for this problem, and prove that it runs in subexponential time. This gives the first non-trivial tester for submodularity. On the other hand, we prove an interesting lower bound (that is, unfortunately, quite far from the upper bound) suggesting that this tester cannot be very efficient in terms of epsilon. This involves non-trivial examples of functions which are far from submodular and yet do not exhibit too many local violations. We also provide some constructions indicating the difficulty in designing a tester for submodularity. We construct a partial function defined on exponentially many points that cannot be extended to a submodular function, but any strict subset of these values can be extended to a submodular function.

Towards factoring in $${SL(2,\,\mathbb{F}_{2^n})}$$

The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in finite groups is one of them. Given a finite group G, a set of generators S for this group and an element g ∈ G, the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions, a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group G := SL(2, F2n ), a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we first give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efficiently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of SL(2, F2n ). Our results do not yet completely remove the factorization problem in SL(2, F2n ) from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in SL(2, F2n ).

Uniform derandomization from pathetic lower bounds

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where 'pathetic' lower bounds of the form n(1+ε) would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: -If the word problem over S(5) requires constant-depth threshold circuits of size n(1+ε) for some ε > 0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). -If there are no constant-depth arithmetic circuits of size n(1+ε) for the problem of multiplying a sequence of n 3×3 matrices, then, for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC(0) circuits of subexponential size).

Virtual Parallel Computing and a Search Algorithm Using Matrix Product States

We propose a form of parallel computing on classical computers that is based on matrix product states. The virtual parallelization is accomplished by representing bits with matrices and by evolving these matrices from an initial product state that encodes multiple inputs. Matrix evolution follows from the sequential application of gates, as in a logical circuit. The action by classical probabilistic one-bit and deterministic two-bit gates such as NAND are implemented in terms of matrix operations and, as opposed to quantum computing, it is possible to copy bits. We present a way to explore this method of computation to solve search problems and count the number of solutions. We argue that if the classical computational cost of testing solutions (witnesses) requires less than O(n2) local two-bit gates acting on n bits, the search problem can be fully solved in subexponential time. Therefore, for this restricted type of search problem, the virtual parallelization scheme is faster than Grover's quantum algorithm.

On the Parameterized Intractability of Monadic Second-Order Logic

One of Courcelle's celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized algorithms, where the parameter is the tree-width plus the size of the formula. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem cannot be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions and is such that the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.

On reducing factorization to the discrete logarithm problem modulo a composite

The discrete logarithm problem modulo a composite—abbreviate it as DLPC—is the following: given a (possibly) composite integer n ≥ 1 and elements $${a, b \in \mathbb{Z}_n^*}$$, determine an $${x \in \mathbb{N}}$$ satisfying a x = b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem $${{\rm DLPC}_\varepsilon}$$ obtained by adding in the DLPC the constraint $${x\le (1-\varepsilon)n}$$, where $${\varepsilon}$$ is an arbitrary fixed number, $${0 < \varepsilon\le\frac{1}{2}}$$. We prove that factoring n reduces in deterministic subexponential time to the $${{\rm DLPC}_\varepsilon}$$ with $${O_\varepsilon((\ln n)^2)}$$ queries for moduli less or equal to n.