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We revisit the classic online portfolio selection problem, where at each round a learner selects a distribution over a set of portfolios to allocate its wealth. It is known that for this problem a logarithmic regret with respect to Cover's loss is achievable using the Universal Portfolio Selection algorithm, for example. However, all existing algorithms that achieve a logarithmic regret for this problem have per-round time and space complexities that scale polynomially with the total number of rounds, making them impractical. In this paper, we build on the recent work by Haipeng et al. 2018 and present the first practical online portfolio selection algorithm with a logarithmic regret and whose per-round time and space complexities depend only logarithmically on the horizon. Behind our approach are two key technical novelties of independent interest. We first show that the Damped Online Newton steps can approximate mirror descent iterates well, even when dealing with time-varying regularizers. Second, we present a new meta-algorithm that achieves an adaptive logarithmic regret (i.e. a logarithmic regret on any sub-interval) for mixable losses.

In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the adversary reveals the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each new node the algorithm can also inspect its radius-$T$ neighborhood before choosing the output. Instead of looking ahead in time, we have the power of looking around in space. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-$T$ neighborhood, its sequential counterpart SLOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-$T$ neighborhood of the point of change. SLOCAL and dynamic-LOCAL models are sandwiched between LOCAL and online-LOCAL models, with LOCAL being the weakest and online-LOCAL the strongest model. In this work, we seek to answer the following question: is the online-LOCAL model strictly stronger than the LOCAL model when we look at graph algorithms for solving locally checkable labeling problems (LCLs)? First, we show that for LCL problems in paths, cycles, and rooted trees, all four models are roughly equivalent: the locality of any LCL problem falls in the same broad class - $O(\log^* n)$, $\Theta(\log n)$, or $n^{\Theta(1)}$ - in all four models. In particular, prior work on the LOCAL model directly generalizes to all four models. Second, we show that this equivalence does not hold in two-dimensional grids. We show that the locality of the $3$-coloring problem is $O(\log n)$ in the online-LOCAL model, while it is known to be $\Omega(\sqrt{n})$ in the LOCAL model.

In this work we address the question of efficiency of distributed computing in anonymous, congested and highly dynamic and not-always-connected networks/systems. More precisely, the system consists of an unknown number of anonymous nodes with congestion on links and local computation. Links can change arbitrarily from round to round, with only limitation that the union of any T consecutive networks must form a temporarily connected (multi-)graph on all nodes (knowledge of T is the only information the nodes require, otherwise the communication would not be feasible). Nodes do not have any IDs, only some number l of them have a bit distinguishing them from nodes without such a bit. In each round a node can send and receive messages from its current neighbors. Links and nodes are congested, in the sense that the length of messages and local cache memory for local computation is (asymptotically) logarithmic. All-to-all communication is a fundamental principle in distributed computing - it assumes that each node has an input message to be delivered to all other nodes. Without loss of generality, the size of each input message is logarithmic to fit in the link and node congestion assumption; otherwise, they could be split in logarithmic batches and considered one-by-one. Because of anonymity, each node needs to receive only a set of all input messages, each accompanied by a number of initiating nodes (message multiplicity). We prove that this task can be done in time polynomial in the (initially unknown) number of nodes n and in the lower bound on the isoperimetric numbers of dynamically evolving graphs. This allows to efficiently emulate a popular Congested Clique model on top of Anonymous Dynamic Congested Systems (ADCS) with Opportunistic Connectivity, even if the number of nodes may arbitrarily change in the beginning of emulation.

The $\mathsf{HYBRID}$ model was introduced as a means for theoretical study of distributed networks that use various communication modes. Conceptually, it is a synchronous message passing model with a local communication mode, where in each round each node can send large messages to all its neighbors in a local network (a graph), and a global communication mode, where each node is allotted limited (polylogarithmic) bandwidth per round which it can use to communicate with any node in the network. Prior work has often focused on shortest paths problems in the local network, as their global nature makes these an interesting case study how combining communication modes in the $\mathsf{HYBRID}$ model can overcome the individual lower bounds of either mode. In this work we consider a similar problem, namely computation of distance oracles and routing schemes. In the former, all nodes have to compute local tables, which allows them to look up the distance (estimates) to any target node in the local network when provided with the label of the target. In the latter, it suffices that nodes give the next node on an (approximately) shortest path to the target. Our goal is to compute these local tables as fast as possible with labels as small as possible. We show that this can be done exactly in $\widetilde O(n^{1/3})$ communication rounds and labels of size $\Theta(n^{2/3})$ bits. For constant stretch approximations we achieve labels of size $O(\log n)$ in the same time. Further, as our main technical contribution, we provide computational lower bounds for a variety of problem parameters. For instance, we show that computing solutions with stretch below a certain constant takes $\widetilde \Omega(n^{1/3})$ rounds even for labels of size $O(n^{2/3})$.

This paper presents algorithms for local inversion of maps and shows how several important computational problems such as cryptanalysis of symmetric encryption algorithms, RSA algorithm and solving the elliptic curve discrete log problem (ECDLP) can be addressed as local inversion problems. The methodology is termed as the \emph{Local Inversion Attack}. It utilizes the concept of \emph{Linear Complexity} (LC) of a recurrence sequence generated by the map defined by the cryptanalysis problem and the given data. It is shown that when the LC of the recurrence is bounded by a bound of polynomial order in the bit length of the input to the map, the local inversion can be accomplished in polynomial time. Hence an incomplete local inversion algorithm which searches a solution within a specified bound on computation can estimate the density of weak cases of cryptanalysis defined by such data causing low LC. Such cases can happen accidentally but cannot be avoided in practice and are fatal insecurity flaws of cryptographic primitives which are wrongly assumed to be secure on the basis of exponential average case complexity. An incomplete algorithm is proposed for solving problems such as key recovery of symmetric encryption algorithms, decryption of RSA ciphertext without factoring the modulus, decrypting any ciphertext of RSA given one plaintext ciphertext pair created with same private key in chosen ciphertext attack and solving the discrete logarithm on elliptic curves over finite fields (ECDLP) as local inversion problems. It is shown that when the LCs of the respective recurrences for given data are small, solutions of these problems are possible in practically feasible time and memory resources.

The fair $k$-median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fair violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. As far as we know, there is no available approximation algorithm for the problem without any fair violation. In this paper, we consider the fair $k$-median problem in bounded doubling metrics and general metrics. We provide the first QPTAS for fair $k$-median problem in doubling metrics. Based on the split-tree decomposition of doubling metrics, we present a dynamic programming process to find the candidate centers, and apply min-cost max-flow method to deal with the assignment of clients. Especially, for overcoming the difficulties caused by the fair constraints, we construct an auxiliary graph and use minimum weighted perfect matching to get part of the cost. For the fair $k$-median problem in general metrics, we present an approximation algorithm with ratio $O(\log k)$, which is based on the embedding of given space into tree metrics, and the dynamic programming method. Our two approximation algorithms for the fair $k$-median problem are the first results for the corresponding problems without any fair violation, respectively.

We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several homogeneous settings, but also leads to surprising results for some heterogeneous settings -- for example, while Hazan and Levy (2014) showed that $\widetilde{O}(d^{3/2}\sqrt{T})$ regret is achievable for a sequence of $T$ smooth and strongly convex $d$-dimensional functions, our algorithm reveals that the same is achievable even if $T^{3/4}$ of them are not strongly convex, and sometimes even if a constant fraction of them are not strongly convex. Our approach is inspired by the framework of Bartlett et al. (2007) who studied a similar heterogeneous setting but with stronger gradient feedback. Extending their framework to the bandit feedback setting requires novel ideas such as lifting the feasible domain and using a logarithmically homogeneous self-concordant barrier regularizer.

In the Strip Packing problem (SP), we are given a vertical half-strip $[0,W]\times[0,\infty)$ and a set of $n$ axis-aligned rectangles of width at most $W$. The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In this paper, we study approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e., the Strip Packing problem where we require additionally that the packing needs to be guillotine separable. This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly NP-hard. Moreover, due to a reduction from the Partition problem, it is NP-hard to obtain a polynomial-time $(3/2-\varepsilon)$-approximation algorithm for GSP for any $\varepsilon>0$ (exactly as Strip Packing). We provide a matching polynomial time $(3/2+\varepsilon)$-approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time $(1+\varepsilon)$-approximation algorithm for GSP. This is surprising as it is NP-hard to obtain a $(5/4-\varepsilon)$-approximation algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.