Polynomial Inn Research Articles

Coexistence times in the Moran process with environmental heterogeneity

Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size N , each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate μ between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when μ is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in N . By contrast, when μ is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in N . Moreover, we precisely identify the threshold migration rate μ ⋆ that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results.

Detecting Feedback Vertex Sets of Size k in O ⋆ (2.7 k ) Time

In the Feedback Vertex Set (FVS) problem, one is given an undirected graph G and an integer k , and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f(k) in run times of the type O ⋆ (f(k)) , where the O ⋆ notation omits factors polynomial in n . This race seemed to have reached a conclusion in 2011, when a randomized O ⋆ (3 k ) time algorithm based on Cut&Count was introduced. In this work, we show the contrary and give a O ⋆ (2.7 k ) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1-Ω (1))k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.

Max Flows in Planar Graphs with Vertex Capacities

We consider the maximum flow problem in directed planar graphs with capacities on both vertices and arcs and with multiple sources and sinks. We present three algorithms when the capacities are integers. The first algorithm runs in O ( min { k 2 n , n log 3 n + kn }) time when all capacities are bounded by a constant, where n is the number of vertices in the graph, and k is the number of terminals. This algorithm is the first to solve the vertex-disjoint paths problem in linear time when k is fixed but larger than 2. The second algorithm runs in O ( k 5 Δ n polylog ( nU )) time, where each arc capacity and finite vertex capacity is bounded by U , and Δ is the maximum degree of the graph. Finally, when k = 3, we present an algorithm that runs in O ( n log n ) time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when k and Δ are fixed and U is bounded by a polynomial in n . Prior to this result, the fastest algorithms ran in O ( n 4/3+ o (1) ) time for unit capacities; in the smallest of O ( n 3/2 log n log U ), Õ( n 10/7 U 1/7 ), O ( n 11/8+o(1) U 1/4 ), and O ( n 4/3 + o(1) U 1/3 ) time for integer capacities; and in O ( n 2 /log n ) time for real capacities, even when k = 3.

Selection and Sorting in the “Restore” Model

We consider the classical selection and sorting problems in a model where the initial permutation of the input has to be restored after completing thecomputation. Such algorithms are useful for designing space-efficient algorithms, when one encounters subproblems that have to be solved by subroutines. It is important that these subroutines leave the array in its original state after they finish so that the computation can be properly resumed. Algorithms in this model can also be relevant for saving communication time, in case the data is distributed among several machines and would need to be copied to further machines for execution of the subroutine. Although the requirement of the restoration is stringent compared to the classicalversions of the problems, this model is more relaxed than a read-only memory where the input elements are not allowed to be moved within the input array. We first show that for a sequence of n integers, selection (finding the median or more generally the k -th smallest element for a given k ) can be done in O ( n ) time using O (lg n ) words 1 of extra space in this model. In contrast, no linear-time selection algorithm is known that uses polylogarithmic space in the read-only memory model. For sorting n integers in this model, we first present an O ( n lg n )-time algorithm using O (lg n ) words of extra space that outputs (in a write only tape) the given sequence in sorted order while restoring the order of the original input in the input tape. When the universe size U is polynomial in n , we give a faster O ( n )-time algorithm (analogous to radix sort) that uses O ( n ε ) words of extra space for an arbitrarily small constant ε > 0. More generally, we show how to match the time bound of any word-RAM integer sorting algorithms using O ( n ε ) words of extra space. In sharp contrast, there is an Ω ( n 2 / S )-time lower bound for integer sorting using O ( S ) bits of space in the read-only memory model. Extension of our results to arbitrary input types beyond integers is not possible: for “indivisible” input elements, we can prove the same Ω ( n 2 / S ) lower bound for sorting in our model. We also describe space-efficient algorithms to count the number of inversions in a given sequence in this model. En route, we develop linear-time in-place algorithms to extract leading bits of the input array and to compress and decompress strings with low entropy; these techniques may be of independent interest.

Improved Approximation Algorithm for Steiner k -Forest with Nearly Uniform Weights

In the Steiner k -Forest problem, we are given an edge weighted graph, a collection D of node pairs, and an integer k ⩽ | D |. The goal is to find a min-weight subgraph that connects at least k pairs. The best known ratio for this problem is min { O (√ n ), O (√ k )} [Gupta et al. 2010]. In Gupta et al. [2010], it is also shown that ratio ρ for Steiner k -Forest implies ratio O (ρ · log 2 n ) for the related Dial-a-Ride problem. The only other algorithm known for Dial-a-Ride, besides the one resulting from Gupta et al. [2010], has ratio O (√ n ) [Charikar and Raghavachari 1998]. We obtain approximation ratio n 0.448 for Steiner k -Forest and Dial-a-Ride with unit weights, breaking the O (√ n ) approximation barrier for this natural case. We also show that if the maximum edge-weight is O ( n ϵ ), then one can achieve ratio O ( n (1 + ϵ) · 0.448 ), which is less than √ n if ϵ is small enough. The improvement for Dial-a-Ride is the first progress for this problem in 15 years. To prove our main result, we consider the following generalization of the Minimum k -Edge Subgraph (M k -ES) problem, which we call Min-Cost ℓ-Edge-Profit Subgraph (MCℓ-EPS): Given a graph G = ( V , E ) with edge-profits p = { p e : e ∈ E } and node-costs c = { c v : v ∈ V }, and a lower profit bound ℓ, find a minimum node-cost subgraph of G of edge-profit at least ℓ. The M k -ES problem is a special case of MCℓ-EPS with unit node costs and unit edge profits. The currently best known ratio for M k -ES is n 3-2√2 + ϵ [Chlamtac et al. 2012]. We extend this ratio to MCℓ-EPS for general node costs and profits bounded by a polynomial in n , which may be of independent interest.

Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension

We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ϵ ∈ (0, 1) and an n -node edge-weighted network of doubling dimension α ∈ O (loglog n ), we present —a (1 + ϵ)-stretch labeled compact routing scheme with ⌈log n ⌉-bit routing labels, O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node; —a (9 + ϵ)-stretch name-independent compact routing scheme with O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o ( n (ϵ/60) 2 ) bits of storage at each node has stretch no less than 9 − ϵ for any ϵ ∈ (0, 8). Therefore, our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter Δ of the network. We also present a simpler nonscale-free (9 + ϵ)-stretch name-independent compact routing scheme with improved space requirements if Δ is polynomial in n .

Distributed Selfish Load Balancing on Networks

We study distributed load balancing in networks with selfish agents. In the simplest model considered here, there are n identical machines represented by vertices in a network and m > n selfish agents that unilaterally decide to move from one vertex to another if this improves their experienced load. We present several protocols for concurrent migration that satisfy desirable properties such as being based only on local information and computation and the absence of global coordination or cooperation of agents. Our main contribution is to show rapid convergence of the resulting migration process to states that satisfy different stability or balance criteria. In particular, the convergence time to a Nash equilibrium is only logarithmic in m and polynomial in n , where the polynomial depends on the graph structure. In addition, we show reduced convergence times to approximate Nash equilibria. Finally, we extend our results to networks of machines with different speeds or to agents that have different weights and show similar results for convergence to approximate and exact Nash equilibria.

Less haste, less waste: on recycling and its limits in strand displacement systems

We study the potential for molecule recycling in chemical reaction systems and their DNA strand displacement realizations. Recycling happens when a product of one reaction is a reactant in a later reaction. Recycling has the benefits of reducing consumption, or waste, of molecules and of avoiding fuel depletion. We present a binary counter that recycles molecules efficiently while incurring just a moderate slowdown compared with alternative counters that do not recycle strands. This counter is an n-bit binary reflecting Gray code counter that advances through 2n states. In the strand displacement realization of this counter, the waste—total number of nucleotides of the DNA strands consumed—is polynomial in n, the number of bits of the counter, while the waste of alternative counters grows exponentially in n. We also show that our n-bit counter fails to work correctly when many (Θ(n)) copies of the species that represent the bits of the counter are present initially. The proof applies more generally to show that in chemical reaction systems where all but one reactant of each reaction are catalysts, computations longer than a polynomial function of the size of the system are not possible when there are polynomially many copies of the system present.

Algorithms for pure Nash equilibria in weighted congestion games

In large-scale or evolving networks, such as the Internet, there is no authority possible to enforce a centralized traffic management. In such situations, game theory, and especially the concepts of Nash equilibria and congestion games [Rosenthal 1973] are a suitable framework for analyzing the equilibrium effects of selfish routes selection to network delays. We focus here on single-commodity networks where selfish users select paths to route their loads (represented by arbitrary integer weights ). We assume that individual link delays are equal to the total load of the link. We then focus on the algorithm suggested in Fotakis et al. [2005], i.e., a potential-based method for finding pure Nash equilibria in such networks. A superficial analysis of this algorithm gives an upper bound on its time, which is polynomial in n (the number of users) and the sum of their weights W . This bound can be exponential in n when some weights are exponential. We provide strong experimental evidence that this algorithm actually converges to a pure Nash equilibrium in polynomial time . More specifically, our experimental findings suggest that the running time is a polynomial function of n and log W . In addition, we propose an initial allocation of users to paths that dramatically accelerates this algorithm, compared to an arbitrary initial allocation. A by-product of our research is the discovery of a weighted potential function when link delays are exponential to their loads. This asserts the existence of pure Nash equilibria for these delay functions and extends the result of Fotakis et al. [2005].

The complexity of deterministic PRAM simulation on distributed memory machines

In this paper we present lower and upper bounds for the deterministic simulation of a Parallel Random Access Machine (PRAM) withn processors andm variables on a Distributed Memory Machine (DMM) withp ≤n processors. The bounds are expressed as a function of the redundancyr of the scheme (i.e., the number of copies used to represent each PRAM variable in the DMM), and become tight for anym polynomial inn andr=Θ(1).

Some Algorithms for Nilpotent Permutation Groups

LetG,HandEbe subgroups of a finite nilpotent permutation group of degreen. We describe the theory and implementation of an algorithm to compute the normalizerNG(H) in time polynomial inn, and we give a modified algorithm to determine whetherHandEare conjugate underGand, if so, to find a conjugating element ofG. Other algorithms produce the intersectionG∩Hand the centralizerCG(H). The underlying method uses the imprimitivity structure of 〈G,H〉 and an associated canonical chief series to reduce computation to linear operations. Implementations in GAP and Magma are practical for degrees large enough to present difficulties for general-purpose methods.

Asymptotic Results for Primitive Permutation Groups

We prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostncμ(n), where μ(n) denotes the maximal exponent occurring in the prime factorization ofn. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizerGαof a primitive groupGof degreendoes not have the alternating group Alt(d) as a section, then the order ofGis bounded by a polynomial inn. This result extends a well-known theorem of Babai, Cameron and Pálfy. It is used to prove, for example, that ifHis a subgroup of indexnin a groupG, andHis a product ofbcyclic groups, then |G:HG|≤ncwherecdepends onb.

Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces

LetP be a complex polynomial inn variables of degree 2 andP(D) the corresponding partial differential operator with constant coefficients. It is shown thatP (D) :C ∞(ℝ n ) →C ∞(ℝ n ) admits a continuous linear right inverse if and only if after a separation of variables and up to a complex factor for some c ∈ ℂ the polynomialP has the form $$P(x_1 ,...,x_n ) = Q(x_1 ,...,x_r ) + L(x_{r + 1} ,...,x_n ) + c$$ where eitherr=1 andL≡0 orr>1,Q andL are real andQ is indefinite. The proof of this characterization is based on the general solution of the right inverse problem for such operators and the fact that for each operatorP(D) of the given form and each characteristic vectorN there exists a fundamental solution forP(D) supported by {x ∈ ℝ n : 〈x, N〉 ⪰ 0 #x007D;, which can be constructed explicitely using partial Fourier transform. The existence of sufficiently many fundamental solutions with support in closed half spaces implies that some right inverse can be given by a concrete formula. An example shows that the present characterization is restricted to operators of order 2.

Oracles and Queries That Are Sufficient for Exact Learning

We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of anNP-oracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a ∑P3-oracle. The hypothesis class for the above learning algorithms is the class of circuits of larger—but polynomially related—size. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth-3 ∧-∨-∧ formulas (by the work of Angluin this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e., depth-2 ∨-∧ formulas). We also investigate the power of anNP-oracle in the context of learning withmembershipqueries. We show that there are deterministic learning algorithms that use membership queries and anNP-oracle to learn: monotone boolean functions in time polynomial in the DNF size and CNF size of the target formula; and the class ofO(logn)-DNF∩O(logn)-CNF formulas in time polynomial inn. We also show that, with anNP-oracle and membership queries, there is a randomized expected polynomial time algorithm that learns any class that is learnable from membership queries with unlimited computational power. Using similar techniques, we show the following both for membership and for equivalence queries (when the hypotheses allowed are precisely the concepts in the class); any class learnable with unbounded ?computational-power is learnable in deterministic polynomial time with a ∑:p5-oracle. Furthermore, we identify the combinatorial properties that completely determine learnability in this information-theoretic sense. Finally we point out a consequence of our result in structural complexity theory showing that if everyNPset has polynomial-size circuits then the polynomial hierarchy collapses to ZPPNP.

Learning two-tape automata from queries and counterexamples

We investigate the learning problem of two-tape deterministic finite automata (2-tape DFAs) from queries and counterexamples. Instead of accepting a subset of ∑*, a 2-tape DFA over an alphabet ∑ accepts a subset of ∑* × ∑*, and therefore, it can specify a binary relation on ∑*. In [3] Angluin showed that the class of deterministic finite automata (DFAs) is learnable in polynomial time from membership queries and equivalence queries, namely, from a minimally adequate teacher (MAT). In this article we show that the class of 2-tape DFAs is learnable in polynomial time from MAT. More specifically, we show an algorithm that, given any languageL accepted by an unknown 2-tape DFAM, learns from MAT a two-tape nonde-terministic finite automaton (2-tape NFA)M′ acceptingL in time polynomial inn andl, wheren is the size ofM andl is the maximum length of any counterexample provided during the learning process.

Representing shared data on distributed-memory parallel computers

The problem of representing a setU≜{u 1,...,u m} of read-write variables on ann-node distributed-memory parallel computer is considered. It is shown thatU can be represented among then nodes of a variant of the mesh of trees usingO((m/n) polylog(m/n)) storage per node such that anyn-tuple of variables may be accessed inO(logn (log logn)2) time in the worst case form polynomial inn.

Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples

We consider the problem of learning a polynomial over an arbitrary fieldFdefined on a set of boolean variables. We present the first provably effective algorithm for exactly identifying such polynomials using membership and equivalence queries. Our algorithm runs in time polynomial inn, the number of variables, andt, the number of nonzero terms appearing in the polynomial. The algorithm makes at mostnt+2 equivalence queries, and at most (nt+1)(t2+3t)/2 membership queries. Our algorithm is equally effective for learning a generalized type of polynomial defined on certain kinds of semilattices. We also present an extension of our algorithm for learning multilinear polynomials when the domain of each variable is the entire fieldF.

Finite population corrections for ranked set sampling

Ranked set sampling (RSS) for estimating a population mean μ is studied when sampling is without replacement from a completely general finite populationx=(x 1,x 2,...,x N )′. Explicit expressions are obtained for the variance of the RSS estimator $$\hat \mu _{RSS} $$ and for its precision relative to that of simple random sampling without replacement. The critical term in these expressions involves a quantity γ=(x−γ)′Γ(x−μ) where Γ is anN × N matrix whose entries are functions of the population sizeN and the set-sizem, but where Γ does not depend on the population valuesx. A computer program is given to calculate Γ for arbitraryN andm. When the population follows a linear (resp., quadratic) trend, then γ is a polynomial inN of degree 2m+2 (resp., 2m+4). The coefficients of these polynomials are evaluated to yield explicit expressions for the variance and the relative precision of $$\hat \mu _{RSS} $$ for these populations. Unlike the case of sampling from an infinite population, here the relative precision depends upon the number of replications of the set sizem.

Deterministic on-line routing on area-universal networks

Two deterministic routing networks are presented: the pruned butterfly and the sorting fat-tree . Both networks are area-universal, that is, they can simulate any other routing network fitting in similar area with polylogarithmic slowdown. Previous area-universal networks were either for the off-line problem, where the message set to be routed is known in advance and substantial precomputation is permitted, or involved randomization, yielding results that hold only with high probability. The two networks introduced here are the first that are simultaneously deterministic and on-line, and they use two substantially different routing techniques. The performance of their routing algorithms depends on the difficulty of the problem instance, which is measured by a quantity λ known as the load factor. The pruned butterfly runs in time O (λlog 2 N ), is the number of possible sources and destinations for messages and λ is assumed to be polynomial in N . The sorting fat-tree algorithm runs in O (λ log N + log 2 N ) time for a restricted class of message sets including partial permutations. Other results of this work include a “flexible” circuit that is area-time optimal across a range of different input sizes and an area-time lower bound for routers based on wire-length arguments.

The complexity of cover graph recognition for some varieties of finite lattices

We address the following decision problem: Instance: an undirected graphG. Problem: IsG a cover graph of a lattice? We prove that this problem is NP-complete. This extends results of Brightwell [5] and Nesetřil and Rodl [12]. On the other hand, it follows from Alvarez theorem [2] that recognizing cover graphs of modular or distributive lattices is in P. An important tool in the proof of the first result is the following statement which may be of independent interest: Given an integerl, l⩾3, there exists an algorithm which for a graphG withn vertices yields, in time polynomial inn, a graphH with the number of vertices polynomial inn, and satisfying girth(H)⩾l and χ(H)=χ(G).