65 research outputs found
The lattice of arithmetic progressions
In this paper we investigate properties of the lattice of subsets of
that are arithmetic progressions, under the inclusion
order. For , this poset is not graded and thus not semimodular. We
start by deriving properties of the number of arithmetic progressions
of length in . Next, we look at the set of chains in and study the order complex of
. Third, we determine the set of coatoms in to give a general
formula for the value of evaluated at an arbitrary interval of .
In each of these three sections, we give an independent proof of the fact that
for , , where is the M\"obius function
of and is the classical (number-theoretic) M\"obius function. We
conclude by computing the homology groups of , providing yet another
explanation for the formula of the M\"obius function of .Comment: 8 pages, 1 figure, 2 table
On point sets with many unit distances in few directions
We study the problem of the maximum number of unit distances among n points in the plane under the additional restriction that we count only those unit distances that occur in a xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and suciently large n n k the extremal sets are essentially sections of lattices bounded by edges parallel to the k directions and of equal lengt
On Near-Linear-Time Algorithms for Dense Subset Sum
In the Subset Sum problem we are given a set of positive integers and a target and are asked whether some subset of sums to . Natural parameters for this problem that have been studied in the literature are and as well as the maximum input number and the sum of all input numbers . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in . In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time . Our main question is: When can dense Subset Sum be solved in near-linear time ? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters for which dense Subset Sum is in time . For notational convenience we assume without loss of generality that (as larger numbers can be ignored) and (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time if . - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds
Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses
For a fixed prime , the maximum coefficient (in absolute value) of
the cyclotomic polynomial , where and are free primes
satisfying exists. Sister Beiter conjectured in 1968 that
. In 2009 Gallot and Moree showed that for every sufficiently large. In this article Kloosterman
sums (`cloister man sums') and other tools from the distribution of modular
inverses are applied to quantify the abundancy of counter-examples to Sister
Beiter's conjecture and sharpen the above lower bound for .Comment: 2 figures; 15 page
Top-k-Convolution and the Quest for Near-Linear Output-Sensitive Subset Sum
In the classical Subset Sum problem we are given a set and a target , and the task is to decide whether there exists a subset of which sums to . A recent line of research has resulted in -time algorithms, which are (near-)optimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time , where is the set of all subset sums of that are smaller than . Furthermore, all known pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set . As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time . In particular, we ask whether can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time . Central to our approach is the study of top--convolution, a natural problem of independent interest: given sparse polynomials with non-negative coefficients, compute the lowest non-zero monomials of their product. We design an algorithm running in time , by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets
A characterization of class groups via sets of lengths
Let be a Krull monoid with class group such that every class contains
a prime divisor. Then every nonunit can be written as a finite
product of irreducible elements. If , with
irreducibles , then is called the length of the
factorization and the set of all possible is called the set
of lengths of . It is well-known that the system depends only on the class group . In the present
paper we study the inverse question asking whether or not the system is characteristic for the class group. Consider a further Krull monoid
with class group such that every class contains a prime divisor and
suppose that . We show that, if one of the
groups and is finite and has rank at most two, then and are
isomorphic (apart from two well-known pairings).Comment: The current version is close to the one to appear in J. Korean Math.
Soc., yet it contains a detailed proof of Proposition 2.4. The content of
Chapter 4 of the first version had been split off and is presented in ' A
characterization of Krull monoids for which sets of lengths are (almost)
arithmetical progressions' by the same authors (see hal-01976941 and
arXiv:1901.03506
{SETH}-Based Lower Bounds for Subset Sum and Bicriteria Path
Subset-Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. One of the most intriguing open problems in this area is to base the hardness of one of these problems on the other. Our main result is a tight reduction from k-SAT to Subset-Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial -time algorithm for Subset-Sum on numbers and target cannot be improved to time for any , unless the Strong Exponential Time Hypothesis (SETH) fails. This is one of the strongest known connections between any two of the core problems of fine-grained complexity. As a corollary, we prove a "Direct-OR" theorem for Subset-Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of given instances of Subset-Sum is a YES instance requires time . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s,t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset-Sum: On graphs with edges and edge lengths bounded by , we show that the pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to , in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017)
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