Deterministic Polynomial Time Algorithm Research Articles

On the modular inversion hidden number problem

We give a rigorous deterministic polynomial time algorithm for the modular inversion hidden number problem introduced by D. Boneh, S. Halevi and N.A. Howgrave-Graham in 2001. For our algorithm, we need to be given about 2/3 of the bits of the output, which matches one of the heuristic algorithms of D. Boneh, S. Halevi and N.A. Howgrave-Graham and answers one of their open questions. However their more efficient algorithm that requires only 1/3 of the bits of the output still remains heuristic.

Toward a deterministic polynomial time algorithm with optimal additive query complexity

In this paper, we study two combinatorial search problems: the coin weighing problem with a spring scale (also known as the vector reconstructing problem using additive queries) and the problem of reconstructing weighted graphs using additive queries. Suppose we are given n identical looking coins. Suppose that m out of the n coins are counterfeit and the rest are authentic. Assume that we are allowed to weigh subsets of coins with a spring scale. It is known that the optimal number of weighings for identifying the counterfeit coins and their weights is at least Ω ( m log n log m ) . We give a deterministic polynomial time adaptive algorithm for identifying the counterfeit coins and their weights using O ( m log n log m + m log log m ) weighings, assuming that the weight of the counterfeit coins are greater than the weight of the authentic coins. This algorithm is optimal when m ≤ n c / log log n , where c is any constant. Also our weighing complexity is within log log m times the optimal complexity for all m . To obtain this result, our algorithm makes use of search matrices, the divide and conquer approach and the guess and check approach. When combining these methods with the technique introduced in H. Mazzawi (2008) [20], we get a similar positive result for the problem of reconstructing a hidden weighted graph using additive queries.

Trading GRH for algebra: Algorithms for factoring polynomials and related structures

In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n log n , log|k|) time either a nontrivial factor of f(x) or a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRH-free results, most striking of which are: 1. Given a noncommutative algebra A of dimension n over a finite field k. There is a deterministic poly(n logn , log|k|) time algorithm to find a zero divisor in A. This is the best known deterministic GRH-free result since R´ (1990) first studied the problem of finding zero divisors in finite algebras and showed that this problem has the same complexity as factoring polynomials over finite fields. 2. Given a positive integer r > 4 such that either 4|r or r has two distinct prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of ther-th cyclotomic polynomial over a finite field. This is the best known deterministic GRH-free result since Evdokimov (1989) showed that cyclotomic polynomials can be factored over finite fields in deterministic polynomial time assuming GRH. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs

Almost Exact Matchings

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K 3,3-minor free graphs (these include all planar graphs as well as many others) in O(n 3.19) worst case time. Our algorithm can also count the number of perfect matchings in K 3,3-minor free graphs in O(n 2.19) time.

Polynomial-time algorithms for a new model of representation of algebraic varieties (in characteristic zero)

We suggest a model of representation of algebraic varieties based on representative systems of points of their irreducible components. Deterministic polynomial-time algorithms that substantiate this model are described in characteristic zero. The main result here is a construction of the intersection of algebraic varieties. As applications, we obtain efficient algorithms for constructing a smooth stratification and a smooth cover of an algebraic variety, which were introduced by the author earlier. Bibliography: 16 titles.

Hardness amplification within NP against deterministic algorithms

We study the average-case hardness of the class NP against algorithms in P. We prove that there exists some constant μ > 0 such that if there is some language in NP for which no deterministic polynomial time algorithm can decide L correctly on a 1 − ( log n ) − μ fraction of inputs of length n, then there is a language L ′ in NP for which no deterministic polynomial time algorithm can decide L ′ correctly on a 3 / 4 + ( log n ) − μ fraction of inputs of length n. In coding theoretic terms, we give a construction of a monotone code that can be uniquely decoded up to error rate 1 4 by a deterministic local decoder.

Typically-correct derandomization

A fundamental question in complexity theory is whether every randomized polynomial time algorithm can be simulated by a deterministic polynomial time algorithm (that is, whether BPP=P). A beautiful theory of derandomization was developed in recent years in attempt to solve this problem. In this article we survey some recent work on relaxed notions of derandomization that allow the deterministic simulation to err on some inputs. We use this opportunity to also provide a brief overview to some results and research directions in "classical derandomization".

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 >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 > 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.

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e., the problem of assigning values to the variables in a given symbolic matrix to maximize the resulting matrix rank. Matrix completion is one of the fundamental problems in computational complexity. It has numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [N. J. A. Harvey, D. R. Karger, and K. Murota, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 489–498; N. J. A. Harvey, D. R. Karger, and S. Yekhanin, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 1103–1111]. We design efficient deterministic algorithms for common generalizations of the results of Lovász and Geelen on this problem by allowing linear polynomials in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. Our methods are algebraic and quite different from those of Lovász and Geelen. We look at the problem of matrix completion in the more general setting of linear spaces of linear transformations and find a maximum rank element there using a greedy method. Matrix algebras and modules play a crucial role in the algorithm. We show (hardness) results for special instances of matrix completion naturally related to matrix algebras; i.e., in contrast to computing isomorphisms of modules (for which there is a known deterministic polynomial time algorithm), finding a surjective or an injective homomorphism between two given modules is as hard as the general matrix completion problem. The same hardness holds for finding a maximum dimension cyclic submodule (i.e., generated by a single element). For the “dual” task, i.e., finding the minimal number of generators of a given module, we present a deterministic polynomial time algorithm. The proof methods developed in this paper apply to fairly general modules and could also be of independent interest.

Algorithms for Generating Random Channel Instances for Channel Routing Problem

In this paper we have developed algorithms for generating random channel instances for their use in computing channel routing solutions in VLSI physical design. Channel instances are usually of two types: simple and general, and there are usually two kinds of inherent constraints involving channel routing problem: horizontal constraint and vertical constraint. Simple channel instances do not contain any vertical constraint, whereas, general channel instances contain both horizontal as well as vertical constraints. Most of the optimization problems in two-, three-, and multi-layer channel routing are NP-hard and, in fact, very few are polynomial time computable. Hence for each of the NP-hard problems in channel routing, it is unlikely to design a polynomial time deterministic algorithm. Developing heuristic algorithm may be a probable way out that hopefully provides good solutions for most of the instances available in literature. Novelty of a heuristic algorithm is judged better if it works for a variety of large number of randomly generated instances of the problem.

A Tight Upper Bound on the Probabilistic Embedding of Series-Parallel Graphs

We prove that every unweighted series-parallel graph can be probabilistically embedded into its spanning trees with logarithmic distortion. This is tight due to an $\Omega(\log n)$ lower bound established by Gupta, Newman, Rabinovich, and Sinclair on the distortion required to probabilistically embed the n-vertex diamond graph into a collection of dominating trees. Our upper bound is gained by presenting a polynomial time probabilistic algorithm that constructs spanning trees with low expected stretch. This probabilistic algorithm can be derandomized to yield a deterministic polynomial time algorithm for constructing a spanning tree of a given (unweighted) series-parallel graph G, whose communication cost is at most $O(\log n)$ times larger than that of G.

Knowledge Based Image Annotation Refinement

Recently, images on the Web and personal computers are prevalent around the human's life. To retrieve effectively these images, there are many (Automatic Image Annotation) AIA algorithms. However, it still suffers from low-level accuracy since it couldn't overcome the semantic-gap between low-level features (`color', `texture' and `shape') and high-level semantic meanings (e.g., `sky', `beach'). Namely, AIA techniques annotates images with many noisy keywords. In this paper, we propose a novel approach that augments the classical model with generic knowledge-based, WordNet. Our novel approach strives to prune irrelevant keywords by the usage of WordNet. To identify irrelevant keywords, we investigate various semantic similarity measures between keywords and finally fuse outcomes of all these measures together to make a final decision using Dempster-Shafer evidence combination. Furthermore, We can re-formulate the removal of erroneous keywords from image annotation problem into graph-partitioning problem, which is weighted MAX-CUT problem. It is possible that we have too many candidate keywords for web-images. Hence, we need to have deterministic polynomial time algorithm for MAX-CUT problem. We show that finding optimal solution for removing noisy keywords in the graph is NP-Complete problem and propose a new methodology for Knowledge Based Image Annotation Refinement (KBIAR) using a deterministic polynomial time algorithm, namely, randomized approximation graph algorithm. Finally, we demonstrate the superiority of this algorithm over traditional one including the most recent work for a benchmark dataset.

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.

Finding central decompositions of p-groups

Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The methods introduce new group isomorphism invariants including new characteristic subgroups.

Can a Graph Have Distinct Regular Partitions?

The regularity lemma of Szemerédi gives a concise approximate description of a graph via a so-called regular partition of its vertex set. In this paper we address the following problem: Can a graph have two “distinct” regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand, we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very “similar.” En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rödl and Schacht and by Lovász and Szegedy from a previously known variant of the regularity lemma due to Alon et al. in 2000. The proof also provides a deterministic polynomial time algorithm for finding such partitions.

Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing

We present two deterministic polynomial time algorithms for the following problem: check whether a sparse polynomial f(x) vanishes at a given primitive nth root of unity ζ n . A priori f(ζ n ) may be nonzero and doubly exponentially small in the input size. The existence of a polynomial time procedure in the case of factored n was conjectured by D. Plaisted in 1984, but all previously known algorithms are either randomized, or do not run in polynomial time. We apply polynomial zero testing algorithms to construct a nondeterministic polynomial time algorithm for the torsion point problem (TP). The problem TP is a particular case of the feasibility problem for a system of polynomial equations in complex numbers (coefficients of polynomials are integers). In the problem TP all coordinates of a solution must be roots of unity.

A randomized competitive algorithm for evaluating priced AND/OR trees

Recently, Charikar et al. investigated the problem of evaluating AND/OR trees, with non-uniform costs on its leaves, from the perspective of the competitive analysis. For an AND/OR tree T they presented a μ ( T ) -competitive deterministic polynomial time algorithm, where μ ( T ) is the number of leaves that must be read, in the worst case, in order to determine the value of T . Furthermore, they proved that μ ( T ) is a lower bound on the deterministic competitiveness, which assures the optimality of their algorithm. The power of randomization in this context has remained as an open question. Here, we take a step towards solving this problem by presenting a 5 6 μ ( T ) -competitive randomized polynomial time algorithm. This contrasts with the best known lower bound μ ( T ) / 2 .

Distributions of points in the unit square and large k-gons

We consider a generalization of Heilbronn’s triangle problem by asking, given any integers n ≥ k , for the supremum Δ k ( n ) of the minimum area determined by the convex hull of some k of n points in the unit square [ 0 , 1 ] 2 , where the supremum is taken over all distributions of n points in [ 0 , 1 ] 2 . Improving the lower bound Δ k ( n ) = Ω ( 1 / n ( k − 1 ) / ( k − 2 ) ) from [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390] and from [W.M. Schmidt, On a problem of Heilbronn, Journal of the London Mathematical Society (2) 4 (1972) 545–550] for k = 4 , we show that Δ k ( n ) = Ω ( ( log n ) 1 / ( k − 2 ) / n ( k − 1 ) / ( k − 2 ) ) for fixed integers k ≥ 3 as asked for in [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390]. Moreover, we provide a deterministic polynomial time algorithm which finds n points in [ 0 , 1 ] 2 , which achieve this lower bound on Δ k ( n ) .

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R 1×R 2×⋅⋅⋅×R k where R i (for 1≤i≤k) is a closed interval of the form [a i ,b i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}$approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(Δ+2)ln n⌉ dimensions, where Δ is the maximum degree of G. This algorithm implies that box (G)≤⌈(Δ+2)ln n⌉ for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, their boxicity is O(d av ln n) where d av is the average degree.

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of \(\min\{1+\gamma,\frac{2\gamma^{2}}{2\gamma^{2}-2\gamma +1}\}+\varepsilon\) and a randomized approximation algorithm that achieves a ratio of \(\frac{2\gamma^{3}+2\gamma^{2}}{3\gamma^{2}-2\gamma +1}+\varepsilon\) . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio \(\frac{1+\gamma}{1+3\gamma -4\gamma^{2}}+\varepsilon\) ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio \(\frac{1}{2}+ \frac{\gamma^{3}}{1-3\gamma^{2}}+\varepsilon\) ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2).