Springer Science and Business Media LLC

We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.

In this paper we establish the complexity map for the Group Activity Selection Problem (GASP), along with two of its prominent variants called sGASP and gGASP, focusing on the case when the number of types of agents is the parameter. In all these problems, one is given a set of agents (each with their own preferences) and a set of activities, and the aim is to assign agents to activities in a way which satisfies certain global as well as preference-based conditions. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps that allow us to focus on solutions with a simpler, well-defined structure (in particular, they are “acyclic”). These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta et al. (in: Algorithmic game theory—10th international symposium, SAGT 2017, vol 10504 of lecture notes in computer science, Springer, 2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well.

Computing the LZ77 factorization is a fundamental task in text compression and indexing, being the size z of this compressed representation closely related to the self-repetitiveness of the text. A long-standing problem is to compute LZ77 using small working space. Considering that $$\mathcal {O}(z)$$ words of space can be significantly (up to exponentially) smaller than the size n of the input text, even succinct and entropy-compressed solutions are often unduly memory demanding. In this work we focus on an important measure of text repetitiveness: the number $$r$$ of equal-letter runs in the Burrows–Wheeler transform of the reversed input text. As z, the measure $$r$$ is closely related to the number of repetitions in the text and can be exponentially smaller than n. We describe two algorithms computing LZ77 in $$\mathcal {O}(r\log n)$$ bits of working space and $$\mathcal {O}(n\log r)$$ time. Roughly speaking, our algorithms store a constant number of memory words per BWT run to keep track of first-last run-positions and a suitable indexing mechanism to sample the runs of the BWT (instead of its positions). Important consequences of our results include (i) the possibility to convert from RLBWT- to LZ77-based compressed formats without first decompressing the text, and (ii) the existence of asymptotically-optimal construction algorithms for repetition-aware self-indexes based on these compression techniques. We finally describe an implementation of our solutions and present extensive experiments on highly repetitive datasets. Our algorithms use a working space as small as 1% of the dataset size and are two to three orders of magnitude more space-efficient (albeit slower) than existing solutions based, respectively, on entropy compression and suffix arrays.

The all scores matrix of a grid graph is a matrix containing the optimal scores of paths from every vertex on the first row of the graph to every vertex on its last row. This matrix is commonly used to solve diverse string comparison problems. All scores matrices have the Monge property, and this was exploited by previous works that used all scores matrices for solving various problems. In this paper, we study an extension of grid graphs that contain an additional set of edges, called bridges. Our main result is to show several properties of the all scores matrices of such graphs. We also apply these properties to obtain an $$O(r(nm+n^2))$$ time algorithm for constructing the all scores matrix of an $$m\times n$$ grid graph with r bridges and bounded integer weights.

We consider online coloring of intervals with bandwidth in a setting where colors have variable capacities. Whenever the algorithm opens a new color, it must choose the capacity for that color and cannot change it later. A set of intervals can be assigned the same color a of capacity C a if the sum of bandwidths of intervals at each point does not exceed C a . The goal is to minimize the total capacity of all the colors used. We consider the bounded model, where all capacities must be chosen in the range (0,1], and the unbounded model, where the algorithm may use colors of any positive capacity. For the absolute competitive ratio, we give an upper bound of 14 and a lower bound of 4.59 for the bounded model, and an upper bound of 4 and a matching lower bound of 4 for the unbounded model. We also consider the offline version of these problems and show that whereas the unbounded model is polynomially solvable, the bounded model is NP-hard in the strong sense and admits a 3.6-approximation algorithm.

An n-vertex graph is equitably k-colorable if there is a proper coloring of its vertices such that each color is used either $$\left\lfloor n/k\right\rfloor $$ or $$\left\lceil n/k\right\rceil $$ times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, equitable coloring is W[1]-hard. We present an extensive study of structural parameterizations of Equitable Coloring, tackling both tractability and kernelization questions. We begin by showing that the problem is fixed parameter tractable when parameterized by distance to cluster or by distance to co-cluster—improving on the FPT algorithm of Fiala et al. (Theor Comput Sci 412(23):2513–2523, 2011) parameterized by vertex cover—and also when parameterized by distance to disjoint paths of bounded length. To justify the latter result, we adapt a proof of Fellows et al. (Inf Comput 209(2):143–153, 2011) to show that Equitable Coloring is W[1]-hard when simultaneously parameterized by distance to disjoint paths and number of colors. In terms of kernelization, on the positive side we present a linear kernel for the distance to clique parameter and a cubic kernel when parameterized by the maximum leaf number; on the other hand, we show that, unlike Vertex Coloring, Equitable Coloring does not admit a polynomial kernel when jointly parameterized by vertex cover and number of colors, unless $$\textsf {NP}\subseteq \textsf {coNP}/\textsf {poly}$$ . We also revisit the literature and derive other results on the parameterized complexity of the problem through minor reductions or other observations.

The linear-time ham-sandwich cut algorithm of Lo, Matoušek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen vertical lines. We consider the setting where we are not given the coordinates of the point set, but only the orientation of each point triple (the order type) and give a deterministic linear-time algorithm for the mentioned sub-algorithm. This yields a linear-time ham-sandwich cut algorithm even in our restricted setting. We also show that our methods are applicable to abstract order types.

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $$x \oplus y$$ , is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.