arXiv subjects

Eleni C. Akrida

Publications and source records attributed to Eleni C. Akrida.

7 recordsLinked to original sources

arXiv2020

Matching in Stochastically Evolving Graphs

This paper studies the maximum cardinality matching problem in stochastically evolving graphs. We formally define the arrival-departure model with stochastic departures. There, a graph is sampled from a specific probability distribution and it is revealed as a series of snapshots. Our goal is to study algorithms that create a large matching in the sampled graphs. We define the price of stochasticity for this problem which intuitively captures the loss of any algorithm in the worst case in the size of the matching due to the uncertainty of the model. Furthermore, we prove the existence of a deterministic optimal algorithm for the problem. In our second set of results we show that we can efficiently approximate the expected size of a maximum cardinality matching by deriving a fully randomized approximation scheme (FPRAS) for it. The FPRAS is the backbone of a probabilistic algorithm that is optimal when the model is defined over two timesteps. Our last result is an upper bound of $\frac{2}{3}$ on the price of stochasticity. This means that there is no algorithm that can match more than $\frac{2}{3}$ of the edges of an optimal matching in hindsight.

cs.DS↗

arXiv2019

Temporal Vertex Cover with a Sliding Time Window

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge of $G$, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs has focused on the notion of a temporal path and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal graph problems. In this paper, motivated by applications in sensor and in transportation networks, we introduce and study two natural temporal extensions of the classical problem Vertex Cover. In both cases we wish to minimize the total number of "vertex appearances" that are needed to "cover" the whole temporal graph. In our first problem, Temporal Vertex Cover, the aim is to cover every edge at least once during the lifetime of the temporal graph, where an edge can be covered by one of its endpoints, only at a time step when it is active. In our second, more pragmatic variation Sliding Window Temporal Vertex Cover, we are also given a natural number $Δ$, and our aim is to cover every edge at least once at every $Δ$ consecutive time steps. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. In particular, we provide strong hardness results, complemented by various approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions.

cs.CC↗

arXiv2019

Connected Subgraph Defense Games

We study a security game over a network played between a $defender$ and $k$ $attackers$. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $λ$ nodes to scan and clean.Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. "The price of defense" (MFCS'06). We are interested in Nash equilibria (NE) of this game, as well as in characterizing $defense$-$optimal$ networks which allow for the best $equilibrium$ $defense$ $ratio$, termed Price of Defense; this is the ratio of $k$ over the expected number of attackers that the defender catches in a NE. We provide characterizations of the NEs of this game and defense-optimal networks. This allows us to show that the NEs of the game coincide independently from the coordination or not of the attackers. In addition, we give an algorithm for computing NEs. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $λ$ constantly close to 1 or $n$. For the special case of tree-networks, we refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal NE. On the other hand, we prove that it is NP-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any $λ$.

cs.GT↗

arXiv2019

How fast can we reach a target vertex in stochastic temporal graphs?

Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature. Given a static underlying graph $G=(V,E)$, a temporal graph on $G$ is a sequence of snapshots $G_t$, one for each time step $t\geq 1$. In this paper we study stochastic temporal graphs, i.e. stochastic processes $\mathcal{G}$ whose random variables are the snapshots of a temporal graph on $G$. A natural feature observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; i.e. the probability an edge $e\in E$ appears at time step $t$ depends on its appearance (or absence) at the previous $k$ steps. In this paper we study the hierarchy of models memory-$k$, addressing this memory effect in an edge-centric network evolution: every edge of $G$ has its own independent probability distribution for its appearance over time. Clearly, for every $k\geq 1$, memory-$(k-1)$ is a special case of memory-$k$. We make a clear distinction between the values $k=0$ ("no memory") and $k\geq 1$ ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior, as our results indicate. For every $k\geq 0$ we investigate the complexity of two naturally related, but fundamentally different, temporal path (journey) problems: MINIMUM ARRIVAL and BEST POLICY. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices $s,y$. In the second one we are looking for the arrival time of the best policy for actually choosing a particular $s$-$y$ journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory $k$. Among other results we prove that, surprisingly, MINIMUM ARRIVAL is strictly harder than BEST POLICY; in fact, for $k=0$, MINIMUM ARRIVAL is #P-hard while BEST POLICY is solvable in $O(n^2)$ time.

cs.CC↗

arXiv2018

The temporal explorer who returns to the base

In this paper we study the problem of exploring a temporal graph (i.e. a graph that changes over time), in the fundamental case where the underlying static graph is a star. The aim of the exploration problem in a temporal star is to find a temporal walk which starts at the center of the star, visits all leafs, and eventually returns back to the center. We initiate a systematic study of the computational complexity of this problem, depending on the number k of time-labels that every edge is allowed to have; that is, on the number k of time points where every edge can be present in the graph. To do so, we distinguish between the decision version StarExp(k) asking whether a complete exploration of the instance exists, and the maximization version MaxStarExp(k) of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We present here a collection of results establishing the computational complexity of these two problems. On the one hand, we show that both MaxStarExp(2) and StarExp(3) can be efficiently solved in O(n log n) time on a temporal star with n vertices. On the other hand, we show that, for every k >= 6, StarExp(k) is NP-complete and MaxStarExp(k) is APX-hard, and thus it does not admit a PTAS, unless P = NP. The latter result is complemented by a polynomial-time 2-approximation algorithm for MaxStarExp(k), for every k, thus proving that MaxStarExp(k) is APX-complete. Finally, we give a partial characterization of the classes of temporal stars with random labels which are, asymptotically almost surely, yes-instances and no-instances for StarExp(k) respectively.

cs.CC↗

arXiv2017

Temporal flows in Temporal networks

We introduce temporal flows on temporal networks, i.e., networks the links of which exist only at certain moments of time. Such networks are ephemeral in the sense that no link exists after some time. Our flow model is new and differs from the "flows over time" model, also called "dynamic flows" in the literature. We show that the problem of finding the maximum amount of flow that can pass from a source vertex s to a sink vertex t up to a given time is solvable in Polynomial time, even when node buffers are bounded. We then examine mainly the case of unbounded node buffers. We provide a simplified static Time-Extended network (STEG), which is of polynomial size to the input and whose static flow rates are equivalent to the respective temporal flow of the temporal network, using STEG, we prove that the maximum temporal flow is equal to the minimum temporal s-t cut. We further show that temporal flows can always be decomposed into flows, each of which moves only through a journey, i.e., a directed path whose successive edges have strictly increasing moments of existence. We partially characterise networks with random edge availabilities that tend to eliminate the s-t temporal flow. We then consider mixed temporal networks, which have some edges with specified availabilities and some edges with random availabilities, we show that it is #P-hard to compute the tails and expectations of the maximum temporal flow (which is now a random variable) in a mixed temporal network.

cs.DS↗

arXiv2016

The complexity of optimal design of temporally connected graphs

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of $n$ vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex $u$ to vertex $v$ is a path from $u$ to $v$ where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a $(u,v)$-journey for any pair of vertices $u,v,~u\not= v$. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can \emph{freely choose} availability instances for all edges and aims for temporal connectivity with very small \emph{cost}; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in $n$. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead \emph{given} a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless $P=NP$. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be \emph{minimal}, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least $n \log{n}$ labels.

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