Quasi-polynomial Time Research Articles

On Short Expressions for Cosets of Permutation Subgroups

Following Babai’s algorithm (Graph isomorphism in quasipolynomial time, arXiv:1512.03547v2, 2016) for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups. This is feasible both in the original version of the algorithm and in its CFSG-free version, by Babai (2016, §13.1) and Pyber (A CFSG-free analysis of Babai’s quasipolynomial GI algorithm, arXiv:1605.08266, 2016). The existence of such descriptions gives a weak form of the Cameron–Maróti classification, even without assuming CFSG. This is applicable to proofs of diameter bounds for \(\mathrm {Alt}(n)\) as in Helfgott (Growth in linear algebraic groups and permutation groups: towards a unified perspective, arXiv:1804.03049, 2018): our main result is used in Dona (Towards a CFSG-free diameter bound for \({\mathrm{Alt}}(n)\), arXiv:1810.02710v3, 2018) to free Helfgott’s proof from the use of CFSG. We also thoroughly explicate Babai’s recursion process (as given in Helfgott et al. in Graph isomorphisms in quasi-polynomial time, arXiv:1710.04574, 2017) and obtain explicit constants for the runtime of the algorithm, both with and without the use of CFSG.

Translating between the representations of a ranked convex geometry

It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements. In Horn logic, these are respectively known as the Horn expressions and the characteristic models. In this paper, we consider the problem of translating between the two representations in acyclic convex geometries. Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem for which the existence of an output quasi-polynomial time algorithm is open. In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets. For this class, we provide output quasi-polynomial time algorithms based on hypergraph dualization for translating between the two representations.

Computational Complexity of Computing Symmetries in Finite-Domain Planning

Symmetry-based pruning is a powerful method for reducing the search effort in finitedomain planning. This method is based on exploiting an automorphism group connected to the ground description of the planning task { these automorphisms are known as structural symmetries. In particular, we are interested in the StructSym problem where the generators of this group are to be computed. It has been observed in practice that the StructSym problem is surprisingly easy to solve. We explain this phenomenon by showing that StructSym is GI-complete, i.e., the graph isomorphism problem is polynomial-time equivalent to it and, consequently, solvable in quasi-polynomial time. This implies that it is solvable substantially faster than most computationally hard problems encountered in AI. We accompany this result by identifying natural restrictions of the planning task and its causal graph that ensure that StructSym can be solved in polynomial time. Given that the StructSym problem is GI-complete and thus solvable quite efficiently, it is interesting to analyse if other symmetries (than those that are encompassed by the StructSym problem) can be computed and/or analysed efficiently, too. To this end, we present a highly negative result: checking whether there exists an automorphism of the state transition graph that maps one state s into another state t is a PSPACE-hard problem and, consequently, at least as hard as the planning problem itself.

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter roughly measures how many central nodes are visited by all shortest paths of a certain length. It has been shown that transportation networks, on which TSP and STP naturally occur for various applications in logistics, typically have a small highway dimension. While it was previously shown that these problems admit a quasi-polynomial time approximation scheme on graphs of constant highway dimension, we demonstrate that a significant improvement is possible in the special case when the highway dimension is 1. Specifically, we present a fully-polynomial time approximation scheme (FPTAS). We also prove that both TSP and STP are weakly \({\mathsf {NP}}\)-hard for these restricted graphs.

Linear Matroid Intersection is in Quasi-NC

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size $$n^{O(\log n)}$$ and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2 n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form $$A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}$$ , for an arbitrary matrix A0 and rank-1 matrices $$A_{1}, A_{2}, \dots, A_{m}$$ . This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.

Probabilistic automata of bounded ambiguity

Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whether a given probabilistic automaton accepts some word with probability greater than a given threshold. We consider finitely ambiguous probabilistic automata.Our main contributions are to construct efficient algorithms for analysing finitely ambiguous probabilistic automata through a reduction to a multi-objective optimisation problem called the stochastic path problem. We obtain a polynomial time algorithm for approximating the value of probabilistic automata of fixed ambiguity and a quasi-polynomial time algorithm for the emptiness problem for 2-ambiguous probabilistic automata.We complement these positive results by an inapproximability result stating that the value of finitely ambiguous probabilistic automata cannot be approximated unless P=NP.

On Finding Bipartite Graphs With a Small Number of Short Cycles and Large Girth

The problem of finding bipartite (Tanner) graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low-density parity-check (LDPC) codes. In this paper, we prove that for given integers $\alpha $ , $\beta $ , and $\gamma $ , and degree sequences $\pi $ and $\pi '$ , the problem of determining whether there exists a simple bipartite graph with degree sequences $(\pi, \pi ')$ that has at most $\alpha $ ( $\beta $ and $\gamma $ ) cycles of length four (six and eight, respectively) is NP-complete. This is proved by a two-step polynomial-time reduction from the 3-Partition Problem. On the other hand, using connections to linear hypergraphs, we prove that given the degree sequence $\pi $ , a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is $\pi $ and has a girth of at least six. In addition to these complexity results, we devise a quasi-polynomial time algorithm that can construct a bipartite graph with given (irregular) degree sequences and a girth that increases logarithmically with the size of the graph. Compared to the well-known progressive-edge-growth (PEG) algorithm, the proposed method, in some cases, results in Tanner graphs with larger girth, particularly for scenarios where the degree sequences of the graph are strictly enforced.

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatable unless P = NP. Indeed, we show that it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively.

Algorithms for the quantitative Lock/Key model of cytoplasmic incompatibility

Cytoplasmic incompatibility (CI) relates to the manipulation by the parasite Wolbachia of its host reproduction. Despite its widespread occurrence, the molecular basis of CI remains unclear and theoretical models have been proposed to understand the phenomenon. We consider in this paper the quantitative Lock-Key model which currently represents a good hypothesis that is consistent with the data available. CI is in this case modelled as the problem of covering the edges of a bipartite graph with the minimum number of chain subgraphs. This problem is already known to be NP-hard, and we provide an exponential algorithm with a non trivial complexity. It is frequent that depending on the dataset, there may be many optimal solutions which can be biologically quite different among them. To rely on a single optimal solution may therefore be problematic. To this purpose, we address the problem of enumerating (listing) all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time. Interestingly, in order to solve the above problems, we considered also the problem of enumerating all the maximal chain subgraphs of a bipartite graph and improved on the current results in the literature for the latter. Finally, to demonstrate the usefulness of our methods we show an application on a real dataset.

Efficient Approximation of the Capacitated Vehicle Routing Problem in a Metric Space of an Arbitrary Fixed Doubling Dimension

In this paper, for the first time, we provide a quasi-polynomial time approximation scheme for the well-known capacitated vehicle routing problem formulated in metric spaces of an arbitrary fixed doubling dimension.

A separation theorem for joint sensor and actuator scheduling with guaranteed performance bounds

We study the problem of jointly designing a sparse sensor and actuator schedule for linear dynamical systems while guaranteeing a control/estimation performance that approximates the fully sensed/actuated setting. We further prove a separation principle, showing that the problem can be decomposed into finding sensor and actuator schedules separately. However, it is shown that this problem cannot be efficiently solved or approximated in polynomial, or even quasi-polynomial time for time-invariant sensor/actuator schedules; instead, we develop deterministic polynomial-time algorithms for a time-varying sensor/actuator schedule with guaranteed approximation bounds. Our main result is to provide a polynomial-time joint actuator and sensor schedule that on average selects only a constant number of sensors and actuators at each time step, irrespective of the dimension of the system. The key idea is to sparsify the controllability and observability Gramians while providing approximation guarantees for Hankel singular values. This idea is inspired by recent results in theoretical computer science literature on sparsification.

Lipschitz Continuity and Approximate Equilibria

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in L_1 and L_2^2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

Normalisers of primitive permutation groups in quasipolynomial time

We show that given generators for subgroups $G$ and $H$ of $\mathrm{S}_n$, if $G$ is primitive then generators for $\mathrm{N}_H(G)$ may be computed in quasipolynomial time, namely $2^{O(\log^3 n)}$. The previous best known bound was simply exponential.

Broadcasting on Random Directed Acyclic Graphs

We study the following generalization of the wellknown model of broadcasting on trees. Consider an infinite directed acyclic graph (DAG) with a unique source vertex X. Let the collection of vertices at distance k from X be called the kth layer, and suppose every non-source vertex has indegree d ≥ 2. At layer 0, the source vertex is given a random bit. At layer k ≥ 1, each vertex receives d bits from its parents in the (k-1)th layer, which are transmitted along edges that are independent binary symmetric channels (BSCs) with crossover probability δ ∈ (0, 1/2). Each vertex combines its d noisy inputs using a 2 deterministic d-ary Boolean processing function that generates the value at the vertex. The goal is to be able to reconstruct the original bit X with probability of error bounded away from 1/2 using the values of all vertices at an arbitrarily deep layer k. This question is closely related to models of reliable computation and storage, and information flow in biological networks. In this paper, we treat the case of randomly constructed DAGs, for which we show that broadcasting is only possible if the BSC noise level δ is below a certain (degree and function dependent) critical threshold. For d ≥ 3, and random DAGs with layers of size Ω(log(k)) and majority processing functions, we identify the critical threshold. For d = 2, we establish a similar result for the NAND processing function. We also prove a partial converse result for odd d ≥ 3 illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex's value. Finally, for any BSC noise level δ, we construct explicit DAGs (using regular bipartite lossless expander graphs) with bounded degree and layers of size Θ(log(k)) admitting reconstruction. In particular, we show that the first r layers of such DAGs can be generated in either deterministic quasipolynomial time or randomized polylogarithmic time in r. These results portray a doubly-exponential advantage for storing a bit in bounded degree DAGs compared to trees, where d = 1 but layer sizes need to grow exponentially with depth in order for broadcasting to be possible.

Dualization in lattices given by implicational bases

It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP. In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.

Deciding Parity Games in Quasi-polynomial Time

It is shown that the parity game can be solved in quasi-polynomial time. The parameterized parity game---with $n$ nodes and $m$ distinct values (a.k.a. colors or priorities)---is proven to be in th...

Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph G and a family $${\mathcal {D}}$$ of objects, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of $${\mathcal {D}}$$ consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given a graph G in which the edges might have different lengths, two sets $${\mathcal {D}},{\mathcal {C}}$$ of vertices of G, where vertices of $${\mathcal {D}}$$ have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of $${\mathcal {D}}$$ such that every vertex of $${\mathcal {C}}$$ is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably Maximum Weight Independent Set for polygons in the plane and Weighted Geometric Set Cover for unit disks and unit squares. We present quasi-polynomial time approximation schemes (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter $$\epsilon >0$$ we can compute a solution whose weight is within multiplicative factor of $$(1+\epsilon )$$ from the optimum in time $$2^{{\mathrm {poly}}(1/\epsilon ,\log |{\mathcal {D}}|)}\cdot n^{{\mathcal {O}}(1)}$$, where n is the number of vertices of the input graph. We note that a QPTAS for Maximum Weight Independent Set of Objects would follow from existing work. However, our main contribution is to provide a unified framework that works for both problems in both a planar and geometric setting and to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek and Wiese (in Proceedings of the FOCS 2013, IEEE, 2013; in Proceedings of the SODA 2014, SIAM, 2014) and Har-Peled and Sariel (in Proceedings of the SOCG 2014, SIAM, 2014) to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods as a phenomenon in planar graphs.

A Quasi-Polynomial Approximation for the Restricted Assignment Problem

The Restricted Assignment problem is a prominent special case of Scheduling on Unrelated Parallel Machines. For the strongest known linear programming relaxation, the configuration LP, we improve the nonconstructive bound on its integrality gap from 1.9412 to 1.8334 and significantly simplify the proof. Then we give a constructive variant, yielding a 1.8334-approximation in quasi-polynomial time. This is the first quasi-polynomial algorithm for this problem improving on the long-standing approximation rate of 2.

Synthesizing optimally resilient controllers

Recently, Dallal, Neider, and Tabuada studied a generalization of the classical game-theoretic model used in program synthesis, which additionally accounts for unmodeled intermittent disturbances. In this extended framework, one is interested in computing optimally resilient strategies, i.e., strategies that are resilient against as many disturbances as possible. Dallal, Neider, and Tabuada showed how to compute such strategies for safety specifications. In this work, we compute optimally resilient strategies for a much wider range of winning conditions and show that they do not require more memory than winning strategies in the classical model. Our algorithms only have a polynomial overhead in comparison to the ones computing winning strategies. In particular, for parity conditions, optimally resilient strategies are positional and can be computed in quasipolynomial time.

Feasibly constructive proofs of succinct weak circuit lower bounds

We ask for feasibly constructive proofs of known circuit lower bounds for explicit functions on bit strings of length n. In 1995 Razborov showed that many can be proved in PV1, a bounded arithmetic formalizing polynomial time reasoning. He formalized circuit lower bound statements for small n of doubly logarithmic order. It is open whether PV1 proves known lower bounds in succinct formalizations for n of logarithmic order. We give such proofs in APC1, an extension of PV1 formalizing probabilistic polynomial time reasoning: for parity and AC0, for mod q and AC0[p] (only for n slightly smaller than logarithmic), and for k-clique and monotone circuits. We also formalize Razborov and Rudich's natural proof barrier.We ask for short propositional proofs of circuit lower bounds expressed succinctly by propositional formulas of size nO(1) or at least much smaller than the 2O(n) size of the common “truth table” formula. We discuss two such expressions: one via feasible functions witnessing errors of circuits, and one via the anticheckers of Lipton and Young 1994. Our APC1 formalizations yield conditional upper bounds for the succinct formulas obtained by witnessing: we get short Extended Frege proofs from general circuit lower bounds expressed by the common “truth-table” formulas. We also show how to construct in quasipolynomial time propositional proofs of quasipolynomial size tautologies expressing AC0[p] quasipolynomial size lower bounds; these proofs are in Jeřábek's system WF.