Time 2O Research Articles

Imbalance is fixed parameter tractable

In the Imbalance Minimization problem we are given a graph G=(V,E) and an integer b and asked whether there is an ordering v1…vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2O(blogb)⋅nO(1). This resolves an open problem of Kára et al. [On the complexity of the balanced vertex ordering problem, in: COCOON, in: Lecture Notes in Comput. Sci., vol. 3595, 2005, pp. 849–858].

Tight complexity bounds for FPT subgraph problems parameterized by the clique-width

We give tight algorithmic lower and upper bounds for some double-parameterized subgraph problems when the clique-width of the input graph is one of the parameters. Let G be an arbitrary input graph on n vertices with clique-width at most w. We prove the following results. •The Dense (Sparse)k-Subgraph problem, which asks whether there exists an induced subgraph of G with k vertices and at least q edges (at most q edges, respectively), can be solved in time kO(w)⋅n, but it cannot be solved in time 2o(wlogk)⋅nO(1) unless the Exponential Time Hypothesis (ETH) fails.•The d-Regular Induced Subgraph problem, which asks whether there exists ad-regular induced subgraph of G, and the Minimum Subgraph of Minimum Degree at leastd problem, which asks whether there exists a subgraph of G with k vertices and minimum degree at least d, can be solved in time dO(w)⋅n, but they cannot be solved in time 2o(wlogd)⋅nO(1) unless ETH fails.

On exact algorithms for the permutation CSP

In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables V and a set of constraints C, in which the constraints are tuples of elements of V. The goal is to find a total ordering of the variables, π:V→[1,…,|V|], which satisfies as many constraints as possible. A constraint (v1,v2,…,vk) is satisfied by an ordering π when π(v1)<π(v2)<⋯<π(vk). An instance has arity k if all the constraints involve at most k elements.This problem expresses a variety of permutation problems including Feedback Arc Set and Betweenness problems. A naive algorithm, listing all the n! permutations, requires 2O(nlogn) time. Interestingly, Permutation CSP for arity 2 or 3 can be solved by Held–Karp-type algorithms in time O∗(2n), but no algorithm is known for arity at least 4 with running time significantly better than 2O(nlogn). In this paper we resolve the gap by showing that Arity 4 Permutation CSP cannot be solved in time 2o(nlogn) unless the exponential time hypothesis fails.

Clustering with local restrictions

We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let μ be a function on the subsets of vertices of a graph G. In the (μ,p,q)-Partition problem, the task is to find a partition of the vertices into clusters where each cluster C satisfies the requirements that (1) at most q edges leave C and (2) μ(C)⩽p. Our first result shows that if μ is an arbitrary polynomial-time computable monotone function, then (μ,p,q)-Partition can be solved in time nO(q), i.e., it is polynomial-time solvable for every fixed q. We study in detail three concrete functions μ (the number of vertices in the cluster, number of nonedges in the cluster, maximum number of non-neighbors a vertex has in the cluster), which correspond to natural clustering problems. For these functions, we show that (μ,p,q)-Partition can be solved in time 2O(p)⋅nO(1) and in time 2O(q)⋅nO(1) on n-vertex graphs, i.e., the problem is fixed-parameter tractable parameterized by p or by q.

Bin packing with fixed number of bins revisited

As Bin Packing is NP-hard already for k=2 bins, it is unlikely to be solvable in polynomial time even if the number of bins is a fixed constant. However, if the sizes of the items are polynomially bounded integers, then the problem can be solved in time nO(k) for an input of length n by dynamic programming. We show, by proving the W[1]-hardness of Unary Bin Packing (where the sizes are given in unary encoding), that this running time cannot be improved to f(k)⋅nO(1) for any function f(k) (under standard complexity assumptions). On the other hand, we provide an algorithm for Bin Packing that obtains in time 2O(klog2k)+O(n) a solution with additive error at most 1, i.e., either finds a packing into k+1 bins or decides that k bins do not suffice.

Catalan structures and dynamic programming in H-minor-free graphs

We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2O(k)⋅nO(1) steps. Our approach builds on a combination of Demaine–Hajiaghayiʼs bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs. This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions. The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour. With a slight modification, similar combinatorial and algorithmic results can be derived for many other problems. Our approach can be viewed as a general framework for obtaining time 2O(k)⋅nO(1) algorithms on H-minor-free graph classes.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

Electrochemical codeposition of Mg-Li-Gd alloys from LiCl-KCl-MgCl 2-Gd 2O 3 melts

Mg-Li-Gd alloys were prepared by electrochemical codeposition from LiCl-KCl-MgCl 2-Gd 2O 3 melts on molybdenum electrode with constant current density at 823 and 973 K. The microstructure of the Mg-Li-Gd alloys was analyzed by X-ray diffraction (XRD), optical microscopy (OM) and scanning electron microscopy (SEM). The results show that magnesium and gadolinium deposit mainly in the first 30 min, and the alloy obtained contains 96.53% Mg, 0.27% Li and 3.20% Gd (mass fraction). Then, the reduction of lithium ions occurs quickly. The composition of alloy can be adjusted by controlling electrolysis time or Gd 2O 3 concentration in LiCl-KCl melts. With the addition of Gd into Mg-Li alloys, the corrosion resistance of the alloys is enhanced. XRD results suggest that Mg 3Gd and Mg 2Gd can be formed in Mg-Li-Gd alloys. The distribution of Gd element in Mg-Li-Gd alloys indicates that Gd element mainly distributes at the grain boundaries of Mg-Li-Gd alloys.

Mechanochemical synthesis and characterisation of BaTa 2O 6 ceramic powders

Mechanochemical synthesis was used to prepare BaTa 2O 6 powders from BaCO 3 and Ta 2O 5 precursors in a planetary ball mill. Effect of milling time and heat treatment temperature on the formation of BaTa 2O 6 and on the microstructure was investigated. Intensive milling of starting materials resulted in crystallization of BaTa 2O 6 even after 1 h of milling time and single phase BaTa 2O 6 was obtained after 10 h of milling under optimal conditions. The powder derived from 10 h of mechanical activation had crystallite size of 22 nm. But the increase in milling time did not decrease the crystallite size further. High energy milling activated the powders that although 1 h of milling led to formation of single phase BaTa 2O 6 at 1200 °C, this temperature decreased to 900 °C after 5 h of milling. No significant grain growth was observed when the milled powders were heat treated below 900 °C. However, annealing at 1100 and 1200 °C gave an average BaTa 2O 6 grain size of 180 and 650 nm, respectively. An unidentified phase started to form at 1100 °C increasing to high amounts at 1200 °C and they had different shapes and sizes than BaTa 2O 6 grains. These elongated large grains were thought to be due to liquid phase formation caused by iron contamination.

Balanced families of perfect hash functions and their applications

The construction of perfect hash functions is a well-studied topic. In this article, this concept is generalized with the following definition. We say that a family of functions from [ n ] to [ k ] is a δ-balanced ( n,k )-family of perfect hash functions if for every S ⊆ [ n ], | S |= k , the number of functions that are 1-1 on S is between T /δ and δ T for some constant T &gt;0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on S , for each S of size k . In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking δ to be close to 1) for every such S . Our main result is that for any constant δ &gt; 1, a δ-balanced ( n,k )-family of perfect hash functions of size 2 O ( k log log k ) log n can be constructed in time 2 O ( k log log k ) n log n . Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial-time algorithm for approximating both the number of simple paths of length k and the number of simple cycles of size k for any k ≤ O (log n /log log log n ) in a graph with n vertices. The approximation is up to any fixed desirable relative error.

Kinetic investigation of the effect of milling time on the hydrogen sorption reaction of magnesium catalyzed with different Nb 2O 5 contents

The effect of milling time and Nb 2O 5 content on the magnesium hydrogen sorption reaction is investigated in this paper. The absorption and desorption kinetics of magnesium at 300 °C, with 0.05, 0.1, 0.2, 0.5, 1 mol% Nb 2O 5 contents and after 2, 5, 10, 20, 50, 100 h milling were measured. Reaction kinetics can be enhanced up to a certain limit by increasing the milling time or increasing the Nb 2O 5 content. The kinetic rate constants for different milling times and Nb 2O 5 contents were calculated and compared. Furthermore the rate-limiting step of the absorption and the desorption reactions was determined by fitting the respective kinetic equations. While the absorption is diffusion controlled, the rate-limiting step for desorption changes, with increasing of milling time or Nb 2O 5 content, from chemisorption to interface velocity of the transformed phase, demonstrating the effectiveness of the Nb 2O 5 catalyst.

Effects of ball-milling time and Bi 2O 3 addition on electrochemical performance of ball-milled La 2Mg 17 + 200 wt.% Ni composites

Effects of Bi 2O 3 addition and ball-milling time on the electrochemical performance of the ball-milled La 2Mg 17 + 200 wt.% Ni composites in 6 M KOH aqueous solution were investigated. The electrode characteristics of mechanically ground composites of La 2Mg 17 + 200 wt.% Ni with and without 1 wt.% Bi 2O 3 were examined such as discharge capacity, high-rate dischargeability and cycle life. It is demonstrated that for the original ball-milled composites, the high-rate dischargeability and cycling stability improved with the increase of ball-milling time, but the discharge capacity was just on the contrary. However, when the composites ball-milled with 1 wt.% Bi 2O 3 for different times, the effect on the discharge capacity, high-rate dischargeability and cycling stability can be very different depending on the mechanical grinding time.

Writing of crystal-dots and lines by YAG laser irradiation and their morphologies in samarium tellurite glasses

A continuous-wave Nd:YAG laser operating at a wavelength of 1064 nm is used to irradiate BaO–Sm 2O 3–TeO 2 glasses under various conditions (laser power: 0.5–0.7 W, irradiation time: 1–60 s), and the formation, behavior and morphology of crystal-dots and lines consisting of the Sm 2Te 6O 15 phase are examined. Crystal-dots with a corn-like shape are formed at the surface of the glasses. The diameters of crystal-dots are 3–45 μm and their lengths (depths) toward the interior of glasses are 50–110 μm. The size of crystal-dots depends on laser power, laser irradiation time and Sm 2O 3 content. The diameter and length of crystal-dots increase almost linearly with the cube root of laser irradiation time t 1/3, suggesting that the growth of crystal-dots in the YAG laser-induced crystallization is controlled by diffusion of constituent ions. Crystal-lines with a width of ∼10 μm are also written by scanning the YAG laser at a speed of 20–30 μm s −1. A model of YAG laser-induced crystallization is proposed.

A randomized subexponential algorithm for parity games

We describe a randomized algorithm for Parity Games (equivalent to the Mu-Calculus Model Checking), which runs in expected time 2O(k1/(1+2epsilon)) when k is Omega(n1/2+epsilon), n is the number of...

Pseudorandom Generators without the XOR Lemma

R. Impagliazzo and A. Wigderson (1997, in “Proceedings of the twenty-ninth Annual ACM Symposium on Theory of Computing,” pp. 220–229) have recently shown that if there exists a decision problem solvable in time 2O(n) and having circuit complexity 2Ω(n) (for all but finitely many n) then P=BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of “hardness amplification” (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with a pseudorandom generator construction of N. Nisan and A. Wigderson (1999, J. Comput. System Sci.49, 149–167). In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs. Our first result is that when (a modified version of) the Nisan–Wigderson generator construction is applied with a “mildly” hard predicate, the result is a generator that produces a distribution indistinguishable from having large min-entropy. An extractor can then be used to produce a distribution computationally indistinguishable from uniform. This is the first construction of a pseudorandom generator that works with a mildly hard predicate without doing hardness amplification. We then show that in the Impagliazzo–Wigderson construction only the first hardness-amplification phase (encoding with multivariate polynomial) is necessary, since it already gives the required average-case hardness. We prove this result by (i) establishing a connection between the hardness-amplification problem and a list-decoding problem for error-correcting codes; and (ii) presenting a list-decoding algorithm for error-correcting codes based on multivariate polynomials that improves and simplifies a previous one by S. Arora and M. Sudan (1997, in “Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing,” pp. 485–495).

Structural characterization of nickel tantalum oxide synthesized by sol–gel spin coating technique

For the first time, crack-free, dense and transparent tetragonal NiTa 2O 6 single-phase structure thin films have been prepared by a sol–gel method using tantalum isopropoxide and nickel acetate. The formation of the NiTa 2O 6 phase starts from 750°C onwards. The complete tetragonal structure of NiTa 2O 6 forms on silicon(100) substrates at an annealing temperature of 850°C for 5 h. As the annealing temperature increased from 750°C to 950°C, we have not observed any kind of silicate phases due to silicon diffusion at the interface of Si and NiTa 2O 6 phase. Structural, morphological and elemental evolution of these NiTa 2O 6 thin films produced by the sol–gel synthesis were characterized by grazing incidence X-ray diffraction (GIXRD), tapping mode atomic force microscopy (TMAFM) and X-ray photoelectron spectroscopy (XPS), respectively.

On Membership Comparable Sets

A set A is k(n) membership comparable if there is a polynomial-time computable function that, given k(n) instances of A of length at most n, excludes one of the 2k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(logn) membership comparable, then the NP-hard promise problem UniqueSAT can be solved in polynomial time. Our result settles an open question, suggested by Buhrman, Fortnow, and Torenvliet, and extends the work of Ogihara, Beigel, Kummer, Stephan, and Agrawal and Arvind. These authors showed that if SAT is clogn membership comparable for c<1, then NP=P, and that if SAT is O(logn) membership comparable, then UniqueSAT∈DTIME[2log2n]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2o(n). Our main technical tool is an algorithm of Madhu Sudan (building on the work of Ar, Lipton, Rubinfeld, and Sudan) to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.

Planning constrained motion

We consider the motion planning problem for a point constrained to move along a path with radius of curvature at least one. The point moves in a two-dimensional universe with polygonal obstacles. We show the decidability of the reachability question: “Given a source placement (position and direction pair) and a target placement, is there a curvature-constrained path from source to target avoiding obstacles?” The decision procedure has time and space complexity 2 o(poly(n, m)) wheren is the number of corners andm is the number of bits required to specify the position of corners.