Schemes for deterministic polynomial factoring

Deterministic polynomial factoring over finite fields: A uniform approach via [formula omitted]-schemes

This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.

Recently, images on the Web and personal computers are prevalent around the human’s life. To retrieve effectively those images, there are many AIA (Automatic Image Annotation) algorithms. However, it still suffers from low-level accuracy since it couldn’t overcome the semantic-gap be tween low-level features (‘color’,‘texture’ and ‘shape’) and high-level semantic meanings (e.g., ‘sky’,‘beach’). Namely, AIA techniques annotates images with many noisy key words. Refinement process has been appeared in these days and it tries to remove noisy keywords by using Knowledge-base and boosting candidate keywords. Because of limitless of candidate keywords and the incorrectness of web-image textual descriptions, this is the time we need to have deterministic polynomial time algorithm. We show that finding optimal solution for removing noisy keywords in the graph is NP-Complete problem and propose new methodology for KBIAR (Knowledge Based Image Annotation Refinement) using the randomized approximation graph algorithm as the general deterministic polynomial time algorithm.

Using shape grammars and extremal optimization in 3D IC layout design

We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the to one paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant fraction of his maximin share (MMS). The up to one item paradigm fails to give such a guarantee, and it is not difficult to present examples in which previous BoBW mechanisms give agents only a $\frac{1}{n}$ fraction of their MMS. Our main result is a deterministic polynomial time algorithm that computes a distribution over allocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least his proportional share up to one item, and more importantly, at least half of his MMS. Moreover, this last ex-post guarantee holds even with respect to a more demanding notion of a share, introduced in this paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly best possible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, and one cannot guarantee agents more than a $\frac{n}{2n-1}$ fraction of their TPS ex-post.

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

Given an n-dimensional lattice L and some target vector, this paper studies the algorithms for approximate closest vector problem (CVPγ) by using an approximate shortest independent vectors problem oracle (SIVPγ). More precisely, if the distance between the target vector and the lattice is no larger than c \( {\scriptscriptstyle \frac{c}{\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \) for arbitrary large but finite constant c > 0, we give randomized and deterministic polynomial time algorithms to find a closest vector, while previous reductions were only known for \( {\scriptscriptstyle \frac{c}{2\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \). Moreover, if the distance between the target vector and the lattice is larger than some quantity with respect to λn(L), using SIVPγ oracle and Babai’s nearest plane algorithm, we can solve \( \mathrm{CVP}\upgamma \sqrt{n} \) in deterministic polynomial time. Specially, if the approximate factor γ ϵ (1, 2) in the SIVPγ oracle, we obtain a better reduction factor for CVP.

Let f̃(X) ∈ ℤ[X] be a degree-n polynomial such that f(X): = f̃(X)od p factorizes into n distinct linear factors over 𝔽_p. We study the problem of deterministically factoring f(X) over 𝔽_p given f̃(X). Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of f(X) in the case that the Galois group of f̃(X) is (permutation isomorphic to) a linear group G ≤ GL(V) on the set S of roots of f̃(X), where V is a finite-dimensional vector space over a finite field 𝔽 and S is identified with a subset of V. In particular, when |S| = |V|^{Ω(1)}, the algorithm runs in time polynomial in n^{log n/(log log log log n)^{1/3}} and the size of the input, improving Evdokimov’s algorithm. Our result also applies to a general Galois group G when combined with a recent algorithm of the author. 
\nTo prove our main result, we introduce a family of objects called linear m-schemes and reduce the problem of factoring f(X) to a combinatorial problem about these objects. We then apply techniques from additive combinatorics to obtain an improved bound. Our techniques may be of independent interest.

In this paper we develop new algorithms for factoring polynomials over finite fields by exploring an interesting connection between the algebraic factoring problem and the combinatorial problem of stable coloring of tournaments. We present an algorithm which can be viewed as a recursive refinement scheme through which most cases of polynomials are completely factored in deterministic polynomial time within the first level of refinement, most of the remaining cases are factored completely before the end of the second level refinement, and so on. The algorithm has average polynomial time complexity and (n \(^{\rm log{\it n}}\) logp)O(1) worst case complexity. Under a purely combinatorial conjecture concerning tournaments, the algorithm has worst case complexity (n \(^{\rm loglog{\it n}}\) logp)O(1). Our approach is also useful in reducing the amount of randomness needed to factor a polynomial completely in expected polynomial time. We present a random polynomial time algorithm for factoring polynomials over finite fields which requires only log p random bits. All these results assume the Extended Riemann Hypothesis.KeywordsPolynomial TimeZero DivisorStable ColoringPrimitive IdempotentFactoring PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Bresar-Klep theorem [Bresar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q . Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Bresar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [ 7 ], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [ 25 ] and Kedlaya & Umans [ 16 ] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most \(d^{o(1)}\) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e., \(n \notin d^{o(1)}\) ) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field \(\mathbb {F}_{q}\) of characteristic p , which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time \(\begin{equation*} \left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, n, p)\right), \end{equation*}\) provided that p is at most d o (1) , and q is at most (exp (exp (exp (...(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: — We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [ 21 ] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. — We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [ 26 ]. More precisely, for every fixed prime p , we show that for every constant ɛ > 0, and large enough n , the rank of any \(n \times n\) Vandermonde matrix V over the field \(\mathbb {F}_{p^a}\) can be reduced to ( n /exp (Ω (poly(ɛ)log 0.53 n ))) by changing at most n Θ (ɛ) entries in every row of V , provided a ≤ poly(log n ). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [ 9 ].

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n).At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

Hybrid genetic algorithms are presented that use optimization heuristics and genetic techniques to outperform all existing programs for the timetabling problem. The timetabling problem is very hard (NP-complete) and a general polynomial time deterministic algorithm is not known. An artificial intelligence approach, in a logic programming environment, may be useful for such a problem. The decomposition and classification of constraints and the constraint ordering to obtain the minimization of the backtracking and the maximization of the parallelism are illustrated. The school timetabling problem is discussed in detail as a case study. The genetic algorithm approach is particularly well suited for this kind of problem, since there exists an easy way to assess a good timetable but not a well-structured automatic technique for constructing it. So, a population of timetables is created that evolves toward the best solutions. The evaluation function and the genetic operators are well separated from the domain-specific parts, such as the problem knowledge and the heuristics, i.e.. from the timetable builder. A fundamental issue and a general problem in the decision process and automated reasoning is how to efficiently obtain logic decisions under disjunctive constraints. Logic constraint satisfaction problems are in general NP-hard and a general deterministic polynomial time algorithm is not known. The present article illustrates an approach based on the hybridization of constrained heuristic search with novel genetic algorithm techniques. It compares favorably with the best-known programs to solve decisions problems under logic constraints. Complexity of the new algorithms and results of significant experiments are reported.

Polynomial factorization over finite fields by computing Euler–Poincaré characteristics of Drinfeld modules

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker's theorem. An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances $x_1,\dots,x_t$ to a single SAT-instance $y$ of size poly(max $|x_i|$) such that $y$ is satisfiable if and only if all $x_i$ are satisfiable. The "AND" in the name stems from the fact that the predicate "$y$ is satisfiable" can be written as the AND of all predicates "$x_i$ is satisfiable". Drucker's result complements the result by Bodlaender et al. (2009) and Fortnow and Santhanam (2010), who proved the analogous statement for OR-compressions, and Drucker's proof not only subsumes that result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure. Drucker (2012) presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko (1983) for P-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker's theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.