arXiv subjects

Elisabet Burjons

Publications and source records attributed to Elisabet Burjons.

8 recordsLinked to original sources

arXiv2025

Uniform Universal Sets, Splitters, and Bisectors

Given a subset of size $k$ of a very large universe a randomized way to find this subset could consist of deleting half of the universe and then searching the remaining part. With a probability of $2^{-k}$ one will succeed. By probability amplification, a randomized algorithm needs about $2^k$ rounds until it succeeds. We construct bisectors that derandomize this process and have size~$2^{k+o(k)}$. One application is derandomization of reductions between average case complexity classes. We also construct uniform $(n,k)$-universal sets that generalize universal sets in such a way that they are bisectors at the same time. This construction needs only linear time and produces families of asymptotically optimal size without using advanced combinatorial constructions as subroutines, which previous families did, but are basedmainly on modulo functions and refined brute force search.

cs.DS↗

arXiv2025

Online General Knapsack with Reservation Costs

In the online general knapsack problem, an algorithm is presented with an item $x=(s,v)$ of size $s$ and value $v$ and must irrevocably choose to pack such an item into the knapsack or reject it before the next item appears. The goal is to maximize the total value of the packed items without overflowing the knapsack's capacity. As this classical setting is way too harsh for many real-life applications, we will analyze the online general knapsack problem under the reservation model. Here, instead of accepting or rejecting an item immediately, an algorithm can delay the decision of whether to pack the item by paying a fraction $0\le α$ of the size or the value of the item. This models many practical applications, where, for example, decisions can be delayed for some costs e.g. cancellation fees. We present results for both variants: First, for costs depending on the size of the items and then for costs depending on the value of the items. If the reservation costs depend on the size of the items, we find a matching upper and lower bound of $2$ for every $α$. On the other hand, if the reservation costs depend on the value of the items, we find that no algorithm is competitive for reservation costs larger than $1/2$ of the item value, and we find upper and lower bounds for the rest of the reservation range $0\leα< 1/2$.

cs.DS↗

arXiv2023

Delaying Decisions and Reservation Costs

We study the Feedback Vertex Set and the Vertex Cover problem in a natural variant of the classical online model that allows for delayed decisions and reservations. Both problems can be characterized by an obstruction set of subgraphs that the online graph needs to avoid. In the case of the Vertex Cover problem, the obstruction set consists of an edge (i.e., the graph of two adjacent vertices), while for the Feedback Vertex Set problem, the obstruction set contains all cycles. In the delayed-decision model, an algorithm needs to maintain a valid partial solution after every request, thus allowing it to postpone decisions until the current partial solution is no longer valid for the current request. The reservation model grants an online algorithm the new and additional option to pay a so-called reservation cost for any given element in order to delay the decision of adding or rejecting it until the end of the instance. For the Feedback Vertex Set problem, we first analyze the variant with only delayed decisions, proving a lower bound of $4$ and an upper bound of $5$ on the competitive ratio. Then we look at the variant with both delayed decisions and reservation. We show that given bounds on the competitive ratio of a problem with delayed decisions impliy lower and upper bounds for the same problem when adding the option of reservations. This observation allows us to give a lower bound of $\min{\{1+3α,4\}}$ and an upper bound of $\min{\{1+5α,5\}}$ for the Feedback Vertex Set problem. Finally, we show that the online Vertex Cover problem, when both delayed decisions and reservations are allowed, is $\min{\{1+2α, 2\}}$-competitive, where $α\in \mathbb{R}_{\geq 0}$ is the reservation cost per reserved vertex.

cs.DS↗

arXiv2022

Online Simple Knapsack with Reservation Costs

In the online simple knapsack problem items are presented in an iterative fashion and an algorithm has to decide for each item whether to reject or permanently include it into the knapsack without any knowledge about the rest of the instance. The goal is to pack the knapsack as full as possible. In this work, we introduce the option of reserving items for the cost of a fixed fraction $α$ of their size. An algorithm may pay this fraction in order to postpone its decision on whether to include or reject these items until after the last item of the instance was presented. While the classical online simple knapsack problem does not admit any constantly bounded competitive ratio in the deterministic setting, we find that adding the possibility of reservation makes the problem constantly competitive. We give tight bounds for the whole range of $α$ from $0$ to $1$.

cs.DS↗

arXiv2022

The Slotted Online One-Sided Crossing Minimization Problem on 2-Regular Graphs

In the area of graph drawing, the One-Sided Crossing Minimization Problem (OSCM) is defined on a bipartite graph with both vertex sets aligned parallel to each other and all edges being drawn as straight lines. The task is to find a permutation of one of the node sets such that the total number of all edge-edge intersections, called crossings, is minimized. Usually, the degree of the nodes of one set is limited by some constant k, with the problem then abbreviated to OSCM-k. In this work, we study an online variant of this problem, in which one of the node sets is already given. The other node set and the incident edges are revealed iteratively and each node has to be inserted into placeholders, which we call slots. The goal is again to minimize the number of crossings in the final graph. Minimizing crossings in an online way is related to the more empirical field of dynamic graph drawing. Note the slotted OSCM problem makes instances harder to solve for an online algorithm but in the offline case it is equivalent to the version without slots. We show that the online slotted OSCM-k is not competitive for any k greater or equal 2 and subsequently limit the graph class to that of 2-regular graphs, for which we show a lower bound of 4/3 and an upper bound of 5 on the competitive ratio.

cs.DS↗

arXiv2021

From Finite-Valued Nondeterministic Transducers to Deterministic Two-Tape Automata

The question whether P equals NP revolves around the discrepancy between active production and mere verification by Turing machines. In this paper, we examine the analogous problem for finite transducers and automata. Every nondeterministic finite transducer defines a binary relation associating each input word with all output words that the transducer can successfully produce on the given input. Finite-valued transducers are those for which there is a finite upper bound on the number of output words that the relation associates with every input word. We characterize finite-valued, functional, and unambiguous nondeterministic transducers whose relations can be verified by a deterministic two-tape automaton, show how to construct such an automaton if one exists, and prove the undecidability of the criterion.

cs.CC↗

arXiv2019

Reoptimization of Parameterized Problems

Parameterized complexity allows us to analyze the time complexity of problems with respect to a natural parameter depending on the problem. Reoptimization looks for solutions or approximations for problem instances when given solutions to neighboring instances. We try to combine both techniques, in order to better classify the complexity of problems in the parameterized setting. Specifically, we see that some problems in the class of compositional problems, which do not have polynomial kernels under standard complexity-theoretic assumptions, do have polynomial kernels under reoptimization for some local modifications. Moreover, we find that the reoptimization version of Vertex Cover has a polynomial kernel of size 2k using crown decomposition. Finally, in a negative result, we prove that the reoptimization version of Connected Vertex Cover does not have a Turing kernelization unless Set Cover has a polynomial kernel

cs.CC↗

arXiv2019

Finding Optimal Solutions With Neighborly Help

Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory's core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.

cs.CC↗