arXiv subjects

Konstantinos Georgiou

Publications and source records attributed to Konstantinos Georgiou.

At least 37 records · Page 2Linked to original sources

arXiv2020

The Unit Acquisition Number of Binomial Random Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The unit acquisition number of $G$, denoted by $a_u(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdős-Rényi random graph process $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choose 2}$. We show that asymptotically almost surely $a_u(\mathcal{G}(n,m)) = 1$ right at the time step the random graph process creates a connected graph. Since trivially $a_u(\mathcal{G}(n,m)) \ge 2$ if the graphs is disconnected, the result holds in the strongest possible sense.

math.CO↗

arXiv2020

A Study of Knowledge Sharing related to Covid-19 Pandemic in Stack Overflow

The Covid-19 outbreak, beyond its tragic effects, has changed to an unprecedented extent almost every aspect of human activity throughout the world. At the same time, the pandemic has stimulated enormous amount of research by scientists across various disciplines, seeking to study the phenomenon itself, its epidemiological characteristics and ways to confront its consequences. Information Technology, and particularly Data Science, drive innovation in all related to Covid-19 biomedical fields. Acknowledging that software developers routinely resort to open question and answer communities like Stack Overflow to seek advice on solving technical issues, we have performed an empirical study to investigate the extent, evolution and characteristics of Covid-19 related posts. In particular, through the study of 464 Stack Overflow questions posted mainly in February and March 2020 and leveraging the power of text mining, we attempt to shed light into the interest of developers in Covid-19 related topics and the most popular technological problems for which the users seek information. The findings reveal that indeed this global crisis sparked off an intense and increasing activity in Stack Overflow with most post topics reflecting a strong interest on the analysis of Covid-19 data, primarily using Python technologies.

cs.SE↗

arXiv2020

Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope

We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).

cs.DS↗

arXiv2020

Probabilistically Faulty Searching on a Half-Line

We study $p$-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or $1$-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success $p$ is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal in 1980. Then in 2003, Alpern and Gal [The Theory of Search Games and Rendezvous] proposed a so-called monotone solution for searching the line ($2$-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the $2$-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line ($1$-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call $t$-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with $t$, and for all $p \in (0,1)$. The value of $t$ quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.

cs.DS↗

arXiv2020

Lower Bounds for Shoreline Searching with 2 or More Robots

Searching for a line on the plane with $n$ unit speed robots is a classic online problem that dates back to the 50's, and for which competitive ratio upper bounds are known for every $n\geq 1$. In this work we improve the best lower bound known for $n=2$ robots from 1.5993 to 3. Moreover we prove that the competitive ratio is at least $\sqrt{3}$ for $n=3$ robots, and at least $1/\cos(π/n)$ for $n\geq 4$ robots. Our lower bounds match the best upper bounds known for $n\geq 4$, hence resolving these cases. To the best of our knowledge, these are the first lower bounds proven for the cases $n\geq 3$ of this several decades old problem.

cs.DM↗

arXiv2019

Search-and-Fetch with 2 Robots on a Disk: Wireless and Face-to-Face Communication Models

We initiate the study of a new problem on searching and fetching in a distributed environment concerning treasure-evacuation from a unit disk. A treasure and an exit are located at unknown positions on the perimeter of a disk and at known arc distance. A team of two robots start from the center of the disk, and their goal is to fetch the treasure to the exit. At any time the robots can move anywhere they choose on the disk, independently of each other, with the same speed. A robot detects an interesting point (treasure or exit) only if it passes over the exact location of that point. We are interested in designing distributed algorithms that minimize the worst-case treasure-evacuation time, i.e. the time it takes for the treasure to be discovered and brought (fetched) to the exit by any of the robots. The communication protocol between the robots is either wireless, where information is shared at any time, or face-to-face (i.e. non-wireless), where information can be shared only if the robots meet. For both models we obtain upper bounds for fetching the treasure to the exit. Our main technical contribution pertains to the face-to-face model. More specifically, we demonstrate how robots can exchange information without meeting, effectively achieving a highly efficient treasure-evacuation protocol which is minimally affected by the lack of distant communication. Finally, we complement our positive results above by providing a lower bound in the face-to-face model.

cs.DS↗

arXiv2019

Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model

Two robots stand at the origin of the infinite line and are tasked with searching collaboratively for an exit at an unknown location on the line. They can travel at maximum speed $b$ and can change speed or direction at any time. The two robots can communicate with each other at any distance and at any time. The task is completed when the last robot arrives at the exit and evacuates. We study time-energy tradeoffs for the above evacuation problem. The evacuation time is the time it takes the last robot to reach the exit. The energy it takes for a robot to travel a distance $x$ at speed $s$ is measured as $xs^2$. The total and makespan evacuation energies are respectively the sum and maximum of the energy consumption of the two robots while executing the evacuation algorithm. Assuming that the maximum speed is $b$, and the evacuation time is at most $cd$, where $d$ is the distance of the exit from the origin, we study the problem of minimizing the total energy consumption of the robots. We prove that the problem is solvable only for $bc \geq 3$. For the case $bc=3$, we give an optimal algorithm, and give upper bounds on the energy for the case $bc>3$. We also consider the problem of minimizing the evacuation time when the available energy is bounded by $Δ$. Surprisingly, when $Δ$ is a constant, independent of the distance $d$ of the exit from the origin, we prove that evacuation is possible in time $O(d^{3/2}\log d)$, and this is optimal up to a logarithmic factor. When $Δ$ is linear in $d$, we give upper bounds on the evacuation time.

cs.DS↗

arXiv2019

Energy Consumption of Group Search on a Line

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of $d$, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least $9d-o(d)$. It was shown recently that $k\geq2$ robots traveling at unit speed also require at least $9d$ group search time. We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance $x$ at constant speed $s$ is given by $s^2 x$. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple $c$ of the distance $d$ and the speed of the robots is bounded by $b$. Motivation for this study is that for the case when robots must complete the search in $9d$ time with maximum speed one, a single robot requires at least $9d$ energy, while for two robots, all previously proposed algorithms consume at least $28d/3$ energy. When the robots have bounded memory, we generalize existing algorithms to obtain a family of optimal (and in some cases nearly optimal) algorithms parametrized by pairs of $b,c$ values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. We also propose a novel search algorithm, with unbounded memory, that simultaneously achieves search time $9d$ and consumes energy $8.42588d$. Our result shows that two robots can search on the line in optimal time $9d$ while consuming less total energy than a single robot within the same search time.

cs.DM↗

arXiv2018

Average Case - Worst Case Tradeoffs for Evacuating 2 Robots from the Disk in the Face-to-Face Model

The problem of evacuating two robots from the disk in the face-to-face model was first introduced in [Czyzowicz et al., DISC'14], and extensively studied (along with many variations) ever since with respect to worst case analysis. We initiate the study of the same problem with respect to average case analysis, which is also equivalent to designing randomized algorithms for the problem. First we observe that algorithm $B_{2}$ of~[Czyzowicz et al., DISC'14] with worst case cost $WRS(B_{2}):=5.73906$ has average case cost $AVG(B_{2}):=5.1172$. Then we verify that none of the algorithms that induced worst case cost improvements in subsequent publications has better average case cost, hence concluding that our problem requires the invention of new algorithms. Then, we observe that a remarkable simple algorithm, $B_{1}$, has very small average case cost $AVG(B_{1}):=1+π$, but very high worst case cost $WRS(B_{1}):=1+2π$. Motivated by the above, we introduce constrained optimization problem $_2Evac_{F2F}^w$, in which one is trying to minimize the average case cost of the evacuation algorithm given that the worst case cost does not exceed $w$. The problem is of special interest with respect to practical applications, since a common objective in search-and-rescue operations is to minimize the average completion time, given that a certain worst case threshold is not exceeded, e.g. for safety or limited energy reasons. Our main contribution is the design and analysis of families of new evacuation parameterized algorithms $A(p)$ which can solve $_2Evac_{F2F}^w$, for every $w \in [WRS(B_{1}),WRS(B_{2})]$.

cs.DM↗

arXiv2018

Symmetric Rendezvous With Advice: How to Rendezvous in a Disk

In the classic Symmetric Rendezvous problem on a Line (SRL), two robots at known distance 2 but unknown direction execute the same randomized algorithm trying to minimize the expected rendezvous time. A long standing conjecture is that the best possible rendezvous time is 4.25 with known upper and lower bounds being very close to that value. We introduce and study a geometric variation of SRL that we call Symmetric Rendezvous in a Disk (SRD) where two robots at distance 2 have a common reference point at distance $ρ$. We show that even when $ρ$ is not too small, the two robots can meet in expected time that is less than $4.25$. Part of our contribution is that we demonstrate how to adjust known, even simple and provably non-optimal, algorithms for SRL, effectively improving their performance in the presence of a reference point. Special to our algorithms for SRD is that, unlike in SRL, for every fixed $ρ$ the worst case distance traveled, i.e. energy that is used, in our algorithms is finite. In particular, we show that the energy of our algorithms is $O\left(ρ^2\right)$, while we also explore time-energy tradeoffs, concluding that one may be efficient both with respect to time and energy, with only a minor compromise on the optimal termination time.

cs.DM↗

arXiv2018

God Save the Queen

Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in $1 + \frac{2π}{3} + \sqrt{3} \approx 4.8264$ time units and this is optimal~[Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most $4.81854$. Furthermore, we show that any strategy for saving the queen requires time at least $3 + π/6 + \sqrt{3}/2 \approx 4.3896$ in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to $3.8327$ for two servants, and $3.3738$ for three servants. Finally we show lower bounds for these cases of $3.6307$ (two servants) and $3.2017$ (three servants). The case of $n\geq 4$ is the subject of an independent study by Queen Daniela's Royal Scientific Team.

cs.MA↗

arXiv2017

Patrolling a Path Connecting a Set of Points with Unbalanced Frequencies of Visits

Patrolling consists of scheduling perpetual movements of a collection of mobile robots, so that each point of the environment is regularly revisited by any robot in the collection. In previous research, it was assumed that all points of the environment needed to be revisited with the same minimal frequency. In this paper we study efficient patrolling protocols for points located on a path, where each point may have a different constraint on frequency of visits. The problem of visiting such divergent points was recently posed by Gasieniec et al. in [13], where the authors study protocols using a single robot patrolling a set of $n$ points located in nodes of a complete graph and in Euclidean spaces. The focus in this paper is on patrolling with two robots. We adopt a scenario in which all points to be patrolled are located on a line. We provide several approximation algorithms concluding with the best currently known $\sqrt 3$-approximation.

cs.DC↗

arXiv2016

Search on a Line by Byzantine Robots

We consider the problem of fault-tolerant parallel search on an infinite line by $n$ robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed $1$ and can communicate in wireless mode among themselves. However, among the $n$ robots, there are $f$ robots that exhibit {\em byzantine faults}. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient verification that the target has been found. We aim to design algorithms that minimize the value of $S_d(n,f)$, the time to find a target at a distance $d$ from the origin by $n$ robots among which $f$ are faulty. We give several different algorithms whose running time depends on the ratio $f/n$, the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

cs.DS↗

arXiv2016

Searching with Advice: Robot Fence-Jumping

We study a new search problem on the plane involving a robot and an immobile treasure, initially placed at distance $1$ from each other. The length $β$ of an arc (a fence) within the perimeter of the corresponding circle, as well as the promise that the treasure is outside the fence, is given as part of the input. The goal is to device movement trajectories so that the robot locates the treasure in minimum time. Notably, although the presence of the fence limits searching uncertainty, the location of the fence is unknown, and in the worst case analysis is determined adversarially. Nevertheless, the robot has the ability to move in the interior of the circle. In particular the robot can attempt a number of chord-jump moves if it happens to be within the fence or if an endpoint of the fence is discovered. The optimal solution to our question can be obtained as a solution to a complicated optimization problem, which involves trigonometric functions, and trigonometric equations that do not admit closed form solutions. For the 1-Jump Algorithm, we fully describe the optimal trajectory, and provide an analysis of the associated cost as a function of $β$. Our analysis indicates that the optimal k-Jump Algorithm requires that the robot has enough memory and computation power to compute the optimal chord-jumps. Motivated by this, we give an abstract performance analysis for every k-Jump Algorithm. Subsequently, we present a highly efficient Halving Heuristic k-Jump Algorithm that can effectively approximate the optimal k-Jump Algorithm, with very limited memory and computation requirements.

cs.CG↗

arXiv2016

Know When to Persist: Deriving Value from a Stream Buffer

We consider \textsc{Persistence}, a new online problem concerning optimizing weighted observations in a stream of data when the observer has limited buffer capacity. A stream of weighted items arrive one at a time at the entrance of a buffer with two holding locations. A processor (or observer) can process (observe) an item at the buffer location it chooses, deriving this way the weight of the observed item as profit. The main constraint is that the processor can only move {\em synchronously} with the item stream; as a result, moving from the end of the buffer to the entrance, it crosses paths with the item already there, and will never have the chance to process or even identify it. \textsc{Persistence}\ is the online problem of scheduling the processor movements through the buffer so that its total derived value is maximized under this constraint. We study the performance of the straight-forward heuristic {\em Threshold}, i.e., forcing the processor to "follow" an item through the whole buffer only if its value is above a threshold. We analyze both the optimal offline and Threshold algorithms in case the input stream is a random permutation, or its items are iid valued. We show that in both cases the competitive ratio achieved by the Threshold algorithm is at least $2/3$ when the only statistical knowledge of the items is the median of all possible values. We generalize our results by showing that Threshold, equipped with some minimal statistical advice about the input, achieves competitive ratios in the whole spectrum between $2/3$ and $1$, following the variation of a newly defined density-like measure of the input. This result is a significant improvement over the case of arbitrary input streams, since in this case we show that no online algorithm can achieve a competitive ratio better than $1/2$.

cs.DS↗

arXiv2014

Stable marriage with general preferences

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.

cs.GT↗

arXiv2014

On Integrality Ratios for Asymmetric TSP in the Sherali-Adams Hierarchy

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer $t\geq 0$ and small $ε$, $0<ε\ll{1}$, there exists a digraph $G$ on $ν=ν(t,ε)=O(t/ε)$ vertices such that the integrality ratio for level~$t$ of the SA system starting with the standard LP on $G$ is $\ge 1+\frac{1-ε}{2t+3} \approx \frac43, \frac65, \frac87, \dots$. Thus, in terms of the input size, the result holds for any $t = 0,1,\dots,Θ(ν)$ levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level~$t$ of the SA system starting from the standard~LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced~LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of $1 +\frac{1-ε}{t+1}$ for any level~$t$ of the SA system, where $0<ε\ll{1}$ and the number of vertices is $ν(t,ε)=O((t/ε)^{(t/ε)})$. Also, our results for the standard~LP extend to the Path-ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

cs.DS↗

arXiv2014

Excuse Me! or The Courteous Theatregoers' Problem

Consider a theatre consisting of $m$ rows each containing $n$ seats. Theatregoers enter the theatre along aisles and pick a row which they enter along one of its two entrances so as to occupy a seat. Assume they select their seats uniformly and independently at random among the empty ones. A row of seats is narrow and an occupant who is already occupying a seat is blocking passage to new incoming theatregoers. As a consequence, occupying a specific seat depends on the courtesy of theatregoers and their willingness to get up so as to create free space that will allow passage to others. Thus, courtesy facilitates and may well increase the overall seat occupancy of the theatre. We say a theatregoer is courteous if (s)he will get up to let others pass. Otherwise, the theatregoer is selfish. A set of theatregoers is $p$-courteous if each theatregoer in the set is courteous with probability $p$, randomly and independently. It is assumed that the behaviour of a theatregoer does not change during the occupancy of the row. In this paper, we are interested in the following question: what is the expected number of occupied seats as a function of the total number of seats in a theatre, $n$, and the probability that a theatregoer is courteous, $p$? We study and analyze interesting variants of this problem reflecting behaviour of the theatregoers as entirely selfish, and $p$-courteous for a row of seats with one or two entrances and as a consequence for a theatre with $m$ rows of seats with multiple aisles. We also consider the case where seats in a row are chosen according to the geometric distribution and the Zipf distibrution (as opposed to the uniform distribution) and provide bounds on the occupancy of a row (and thus the theatre) in each case.

cs.DM↗