Quasi-polynomial Time Research Articles

Simulating boson sampling in lossy architectures

Photon losses are among the strongest imperfections affecting multi-photon interference. Despite their importance, little is known about their effect on boson sampling experiments. In this work we show that using classical computers, one can efficiently simulate multi-photon interference in all architectures that suffer from an exponential decay of the transmission with the depth of the circuit, such as integrated photonic circuits or optical fibers. We prove that either the depth of the circuit is large enough that it can be simulated by thermal noise with an algorithm running in polynomial time, or it is shallow enough that a tensor network simulation runs in quasi-polynomial time. This result suggests that in order to implement a quantum advantage experiment with single-photons and linear optics new experimental platforms may be needed.

An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria

We focus on the problem of computing approximate Nash equilibria in bimatrix games. In particular, we consider the notion of approximate well-supported equilibria, which is one of the standard approaches for approximating equilibria. It is already known that one can compute an ε-well-supported Nash equilibrium in time nO (log n/ε2), for any ε > 0, in games with n pure strategies per player. Such a running time is referred to as quasi-polynomial. Regarding faster algorithms, it has remained an open problem for many years if we can have better running times for small values of the approximation parameter, and it is only known that we can compute in polynomial-time a 0.6528-well-supported Nash equilibrium. In this paper, we investigate further this question and propose a much better quasi-polynomial time algorithm that computes a (1/2 + ε)-well-supported Nash equilibrium in time nO(log logn1/ε/ε2), for any ε > 0. Our algorithm is based on appropriately combining sampling arguments, support enumeration, and solutions to systems of linear inequalities.

Bootstrapping variables in algebraic circuits

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the first [Formula: see text] variables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior-a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a "near"-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits in [Formula: see text] time, for [Formula: see text], will lead to a near-complete derandomization of PIT (in contrast, [Formula: see text] time is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable, [Formula: see text]-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003) Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC '03].

Nonredundant Information Collection in Rescue Applications via an Energy-Constrained UAV

Unmanned aerial vehicles (UAVs) are emerging as promising devices to provide valuable information in rescue applications, which can be dispatched to take photographs for points of interests in disaster areas where humans are hard to approach. Most existing studies focused on the limited energy capacity issue of UAVs when they take photographs, which however ignored an important fact, that is, the photographs taken by the UAVs usually are highly redundant. In this paper we study a novel monitoring quality maximization problem to find a flying tour for an energy-constrained UAV, such that the amount of nonredundant information of the photographs taken by the UAV in its tour is maximized. Due to NP-hardness of the problem, we first propose an approximation algorithm with a quasi-polynomial time complexity. We then devise a fast yet scalable heuristic algorithm for the problem. We finally evaluate the performance of the proposed algorithms via both a real dataset and extensive simulations. Experimental results show that the proposed algorithms are very promising. Especially, the amounts of nonredundant information by the proposed approximation and heuristic algorithms are about 11% and 8% larger than that by the state-of-the-art, respectively. To the best of our knowledge, we are the first to consider the novel problem of collecting nonredundant information with an energy-constrained UAV.

An ordered approach to solving parity games in quasi-polynomial time and quasi-linear space

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have recently shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi-bi-linear complexity in the number of states and edges for a fixed number of colours, matching lower bounds for the algorithm of Calude et al., and a complexity index associated to our approach, which we compare to the recently proposed register index.

On the Complexity Landscape of Connected f-Factor Problems

Let G be an undirected simple graph having n vertices and let $$f:V(G)\rightarrow \{0,\dots , n-1\}$$ be a function. An f-factor of G is a spanning subgraph H such that $$d_H(v)=f(v)$$ for every vertex $$v\in V(G)$$ . The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte (Can J Math 6(1954):347–352, 1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connectedf-factor is easily seen to generalize Hamiltonian Cycle and hence is $$\mathsf {NP}$$ -complete. In fact, the Connected f-Factor problem remains $$\mathsf {NP}$$ -complete even when we restrict f(v) to be at least $$n^{\epsilon }$$ for each vertex v and constant $$0\le \epsilon <1$$ ; on the other side of the spectrum of nontrivial lower bounds on f, the problem is known to be polynomial time solvable when f(v) is at least $$\frac{n}{3}$$ for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restrictions on the function f. In particular, we show that when f(v) is restricted to be at least $$\frac{n}{(\log n)^c}$$ , the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if $$c\le 1$$ . Furthermore, we show that when $$c>1$$ , the problem is $$\mathsf {NP}$$ -intermediate.

A new perspective on the powers of two descent for discrete logarithms in finite fields

A new proof is given for the correctness of the powers of two descent method for computing discrete logarithms. The result is slightly stronger than the original work, but more importantly we provide a unified geometric argument, eliminating the need to analyse all possible subgroups of $\mathrm{PGL}_2(\mathbb F_q)$. Our approach sheds new light on the role of $\mathrm{PGL}_2$, in the hope to eventually lead to a complete proof that discrete logarithms can be computed in quasi-polynomial time in finite fields of fixed characteristic.

On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\ldots,v_k$ and each edge $uv$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been investigated as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are 1. a uniform random lift by a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues bounded by $\lambda+\mathcal{O}(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. The probability bounds as well as the dependency on $\lambda$ are almost optimal. As a special case, we obtain that there is a constant $c_1$ such that for every $k\leq 2^{c_1n/d^2}$, there exists a lift $H$ of every Ramanujan graph by a cyclic group of order $k$ such that $H$ is almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This result leads to a quasi-polynomial time deterministic algorithm to construct almost Ramanujan expanders; 2. there is a constant $c_2$ such that for every $k\geq 2^{c_2nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph such that $H$ is almost Ramanujan. This can be viewed as an analogue of the well-known nonexpansion result for constant degree abelian Cayley graphs. Suppose $k_0$ is the order of the largest abelian group that produces expanding lifts. Our two results highlight lower and upper bounds on $k_0$ that are tight up to a factor of $d^3$ in the exponent, thus suggesting a threshold phenomenon.

On the Selection of Polynomials for the DLP Quasi-Polynomial Time Algorithm for Finite Fields of Small Characteristic

In this paper we characterize the polynomials $f$ over a finite field $F$ satisfying the following property: there exists an extension field $L$ of $F$ such that for any positive integer $\ell$ les...

Layered Graphs: Applications and Algorithms

The computation of distances between strings has applications in molecular biology, music theory and pattern recognition. One such measure, called short reversal distance, has applications in evolutionary distance computation. It has been shown that this problem can be reduced to the computation of a maximum independent set on the corresponding graph that is constructed from the given input strings. The constructed graphs primarily fall into a class that we call layered graphs. In a layered graph, each layer refers to a subgraph containing, at most, some k vertices. The inter-layer edges are restricted to the vertices in adjacent layers. We study the MIS, MVC, MDS, MCV and MCD problems on layered graphs where MIS computes the maximum independent set; MVC computes the minimum vertex cover; MDS computes the minimum dominating set; MCV computes the minimum connected vertex cover; and MCD computes the minimum connected dominating set. MIS, MVC and MDS run in polynomial time if k=Θ(log|V|). MCV and MCD run in polynomial time ifk=O((log|V|)α), where α&lt;1. If k=Θ((log|V|)1+ϵ), for ϵ&gt;0, then MIS, MVC and MDS run in quasi-polynomial time. If k=Θ(log|V|), then MCV and MCD run in quasi-polynomial time.

On (1, $$\epsilon $$ ϵ )-Restricted Max–Min Fair Allocation Problem

We study the max–min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight $$w_j$$ . For this setting, Asadpour et al. (ACM Trans Algorithms 8(3):24, 2012) showed that a certain configuration-LP can be used to estimate the optimal value to within a factor of $$4+\delta $$ , for any $$\delta >0$$ , which was recently extended by Annamalai et al. (in: Indyk (ed) Proceedings of the twenty-sixth annual ACMSIAM symposium on discrete algorithms, SODA 2015, San Diego, CA, USA, January 4–6, 2015) to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezakova and Dani (SIGecom Exch 5(3):11–18, 2005) showed that it is $$\mathsf {NP}$$ -hard to approximate the problem within any ratio smaller than 2. In this paper we consider the $$(1,\epsilon )$$ -restricted max–min fair allocation problem in which each item j is either heavy $$(w_j = 1)$$ or light $$(w_j = \epsilon )$$ , for some parameter $$\epsilon \in (0,1)$$ . We show that the $$(1,\epsilon )$$ -restricted case is also $$\mathsf {NP}$$ -hard to approximate within any ratio smaller than 2. Using the configuration-LP, we are able to estimate the optimal value of the problem to within a factor of $$3+\delta $$ , for any $$\delta >0$$ . Extending this idea, we also obtain a quasi-polynomial time $$(3+4\epsilon )$$ -approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as $$\epsilon $$ tends to 0, the approximation ratio of our polynomial-time algorithm approaches $$3+2\sqrt{2}\approx 5.83$$ .

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Coloring (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected).We show that Grundy Coloring can be solved in time O∗(2.443n) and Weak Grundy Coloring in time O∗(2.716n) on graphs of order n. While Grundy Coloring and Weak Grundy Coloring are known to be solvable in time O∗(2O(wk)) for graphs of treewidth w (where k is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O∗(2o(wlogw)). We also describe an O∗(22O(k)) algorithm for Weak Grundy Coloring, which is therefore FPT for the parameter k. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for Grundy Coloring). Although we do not know whether Grundy Coloring is in FPT, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for Grundy Coloring and Weak Grundy Coloring on apex-minor graphs. In stark contrast with the two other problems, we show that Connected Grundy Coloring is NP-complete already for k=7 colors.

Approximation Schemes for Capacitated Geometric Network Design

We study a capacitated network design problem in a geometric setting. The input consists of an integral edge capacity k and two sets of points on the Euclidean plane, sources, and sinks, with an integral demand for each point. The demand of each source specifies the amount of flow that has to be shipped from the source, and the demand of each sink specifies the amount of flow that has to be shipped to the sink. The goal is to construct a minimum-length network that allows one to route the requested flow from the sources to the sinks and where each edge in the network has capacity k. The vertices of the network are not constrained to the sets of sinks and sources-any point on the Euclidean plane can be used as a vertex. The flow is splittable and parallel edges are allowed. The capacitated geometric network design problem generalizes, among others, the geometric Steiner tree problem, and as such it is NP-hard. We show that if the demands are polynomially bounded and the edge capacity k is not too large, the single-sink capacitated geometric network design problem admits a polynomial time approximation scheme. If the capacity is arbitrarily large, then we design a quasi-polynomial time approximation scheme for the capacitated geometric network design problem allowing for an arbitrary number of sinks. Our results rely on a derivation of an upper bound on the number of vertices different from sources and sinks (the so-called Steiner vertices) in an optimal network. The bound is polynomial in the total demand of the sources. (Less)

A $(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.

Independence and Efficient Domination on P 6 -free Graphs

In the M aximum W eight I ndependent S et problem, the input is a graph G , every vertex has a non-negative integer weight, and the task is to find a set S of pairwise nonadjacent vertices, maximizing the total weight of the vertices in S . We give an n O (log 2 n ) time algorithm for this problem on graphs excluding the path P 6 on 6 vertices as an induced subgraph. Currently, there is no constant k known for which M aximum W eight I ndependent S et on P k -free graphs becomes NP-hard, and our result implies that if such a k exists, then k &gt; 6 unless all problems in NP can be decided in quasi-polynomial time. Using the combinatorial tools that we develop for this algorithm, we also give a polynomial-time algorithm for M aximum W eight E fficient D ominating S et on P 6 -free graphs. In this problem, the input is a graph G , every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood or to determine that no such set exists. Prior to our work, the class of P 6 -free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of M aximum W eight E fficient D ominating S et was unknown.

A QPTAS for the base of the number of crossing-free structures on a planar point set

The number of triangulations of a planar n point set S is known to be cn, where the base c lies between 2.43 and 30. Similarly, the number of crossing-free spanning trees on S is known to be dn, where the base d lies between 6.75 and 141.07. The fastest known algorithm for counting triangulations of S runs in 2(1+o(1))nlog⁡n time while that for counting crossing-free spanning trees runs in O⁎(7.125n) time. The fastest known, non-trivial approximation algorithms for the number of triangulations of S and the number of crossing-free spanning trees of S, respectively, run in time subexponential in n. We present the first non-trivial approximation algorithms for these numbers running in quasi-polynomial time. They yield the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of crossing-free spanning trees on S, respectively.

Maximum Weight Independent Sets for ([formula omitted],triangle)-free graphs in polynomial time

The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. For a long time, its complexity was an open problem for Pk-free graphs, k≥5.Recently, Lokshtanov et al. (2014) proved that MWIS can be solved in polynomial time for P5-free graphs, Lokshtanov et al. (2015) proved that MWIS can be solved in quasi-polynomial time for P6-free graphs, and Bacsó et al. (2016), and independently, Brause (2017) extended this to Pk-free graphs for every fixed k. Then very recently, Grzesik et al. (2017) showed that MWIS can be solved in polynomial time for P6-free graphs. It still remains an open problem for Pk-free graphs, k≥7.In this paper, we show that MWIS can be solved in polynomial time for (P7,triangle)-free graphs. This extends the corresponding result for (P6,triangle)-free graphs and may provide some progress in the study of MWIS for P7-free graphs such as the recent result by Maffray and Pastor (2016) showing that MWIS can be solved in polynomial time for (P7,bull)-free graphs.

On the discrete logarithm problem in finite fields of fixed characteristic

For q q a prime power, the discrete logarithm problem (DLP) in F q \mathbb {F}_{q} consists of finding, for any g ∈ F q × g \in \mathbb {F}_{q}^{\times } and h ∈ ⟨ g ⟩ h \in \langle g \rangle , an integer x x such that g x = h g^x = h . We present an algorithm for computing discrete logarithms with which we prove that for each prime p p there exist infinitely many explicit extension fields F p n \mathbb {F}_{p^n} in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions F p n \mathbb {F}_{p^n} in expected quasi-polynomial time.

Hardness of Approximation for Strip Packing

Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, for example, in scheduling and stock-cutting, and has been studied extensively. When the dimensions of the objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than 3/2, unless P= NP. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a (1.4 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ)-approximation by two independent research groups [FSTTCS 2016, WALCOM 2017]. This raises a question whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack. In this article, we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP} ⊑ DTIME(2 polylog (n) ).

An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Nash equilibrium (NE) is a central concept in game theory. Here we prove formally a published theorem on existence of an NE in two proof assistants, Coq and Isabelle: starting from a game with finitely many outcomes, one may derive a game by rewriting each of these outcomes with either of two basic outcomes, namely that Player 1 wins or that Player 2 wins. If all ways of deriving such a win/lose game lead to a game where one player has a winning strategy, the original game also has a Nash equilibrium. This article makes three other contributions: first, while the original proof invoked linear extension of strict partial orders, here we avoid it by generalizing the relevant definition. Second, we notice that the theorem also implies the existence of a secure equilibrium, a stronger version of NE that was introduced for model checking. Third, we also notice that the constructive proof of the theorem computes secure equilibria for non-zero-sum priority games (generalizing parity games) in quasi-polynomial time.