Arithmetical Congruence Preservation: from Finite to Infinite

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $\pi\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $\pi\circ F=f\circ\pi$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions

An EPTAS for Scheduling on Unrelated Machines of Few Different Types

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than $1.5$ unless P$=$NP. We consider the case that there are only a constant number $K$ of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $K=1$. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $\varepsilon > 0$ an assignment with makespan of length at most $(1+\varepsilon)$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $1/\varepsilon$. In particular we achieve a running time of $2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|)$, where $|I|$ denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques

Gradual sub-lattice reduction and a new complexity for factoring polynomials

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well

Analysis and optimization of the TWINKLE factoring device

We describe an enhanced version of the TWINKLE factoring device and analyse to what extent it can be expected to speed up the sieving step of the quadratic Sieve and number field Sieve factoring algorithms. The bottom line of our analysis is that the TWINKLE-assisted factorization of 768 bit numbers is difficult but doable in about 9 months (including the sieving and matrix parts) by a large organization which can use 80000 standard Pentium II PC's and 5000 TWINKLE device

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles with possibly different speeds, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot

On the String Consensus Problem and the Manhattan Sequence Consensus Problem

In the Manhattan Sequence Consensus problem (MSC problem) we are given $k$ integer sequences, each of length $l$, and we are to find an integer sequence $x$ of length $l$ (called a consensus sequence), such that the maximum Manhattan distance of $x$ from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequences. Practicality of our algorithms has been verified experimentally. It improves upon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensus problem for $k=5$ binary strings. Similarly as in Amir's algorithm we use a column-based framework. We replace the implied general integer linear programming by its easy special cases, due to combinatorial properties of the MSC for $k\le 5$. We also show that for a general parameter $k$ any instance can be reduced in linear time to a kernel of size $k!$, so the problem is fixed-parameter tractable. Nevertheless, for $k\ge 4$ this is still too large for any naive solution to be feasible in practice.Comment: accepted to SPIRE 201

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Let $F/E$ be a finite Galois extension of fields with abelian Galois group $\Gamma$. A self-dual normal basis for $F/E$ is a normal basis with the additional property that $Tr_{F/E}(g(x),h(x))=\delta_{g,h}$ for $g,h\in\Gamma$. Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, then $F$ admits a self-dual normal basis if and only if $[F:E]$ is odd. If $F/E$ is an extension of finite fields and $char(E)=2$, then $F$ admits a self-dual normal basis if and only if the exponent of $\Gamma$ is not divisible by $4$. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let $K$ be a finite extension of \Q_p, let $L/K$ be a finite abelian Galois extension of odd degree and let \bo_L be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional \bo_L-ideal with square equal to the inverse different of $L/K$. It is known that a self-dual integral normal basis exists for $A_{L/K}$ if and only if $L/K$ is weakly ramified. Assuming $p\neq 2$, we construct such bases whenever they exist