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Many nations are promoting the green transition in the energy sector to attain neutral carbon emissions by 2050. Smart Grid 2.0 (SG2) is expected to explore data-driven analytics and enhance communication technologies to improve the efficiency and sustainability of distributed renewable energy systems. These features are beyond smart metering and electric surplus distribution in conventional smart grids. Given the high dependence on communication networks to connect distributed microgrids in SG2, potential cascading failures of connectivity can cause disruption to data synchronization to the remote control systems. This paper reviews security threats and defense tactics for three stakeholders: power grid operators, communication network providers, and consumers. Through the survey, we found that SG2's stakeholders are particularly vulnerable to substation attacks/vandalism, malware/ransomware threats, blockchain vulnerabilities and supply chain breakdowns. Furthermore, incorporating artificial intelligence (AI) into autonomous energy management in distributed energy resources of SG2 creates new challenges. Accordingly, adversarial samples and false data injection on electricity reading and measurement sensors at power plants can fool AI-powered control functions and cause messy error-checking operations in energy storage, wrong energy estimation in electric vehicle charging, and even fraudulent transactions in peer-to-peer energy trading models. Scalable blockchain-based models, physical unclonable function, interoperable security protocols, and trustworthy AI models designed for managing distributed microgrids in SG2 are typical promising protection models for future research., Comment: 30 pages, 21 figures, 5 tables, accepted to appear in IEEE COMST

Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of elements, of size $O(|S^{opt}| \cdot 1/\varepsilon)$, where $S^{opt}$ denotes the optimal solution, that is guaranteed to contain a $3/2 + \varepsilon$ approximation, while guaranteeing certain robustness properties. We call such a small subset a Density Constrained Subset (DCS), which is inspired by the Edge-Degree Constrained Subgraph (EDCS) [Bernstein and Stein, 2015], originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the density-based decomposition. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models.

In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied max $k$-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint. First, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if $k$ is the rank of the matroid, we obtain $(1 - \varepsilon)$ approximation using $(1/\varepsilon)^{O(k)}n^{O(1)}$ time for partition and laminar matroids and using $(1/\varepsilon+k)^{O(k)}n^{O(1)}$ time for transversal matroids. This extends a result of Manurangsi for uniform matroids [Manurangsi, 2018]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics. In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees $2/3 \approx 0.66$ approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a $2/3 \cdot (1 - 1/(p+1))$ approximation in $n^{O(p)}$ time, for any integer $p$. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of $2/3$ is likely the best possible, using the currently known techniques of local search.

We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory of $O(\max(|M_G|, n) \cdot poly(\log(m),W,1/\varepsilon))$, where $|M_G|$ denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Bernstein, 2015], which achieves a $3/2 + \varepsilon$ approximation for the maximum cardinality simple matching. When $W$ is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a $2 - \delta$ approximation (for some small constant $\delta \sim 10^{-17}$) for the maximum weight simple matching. In particular, for the weighted $b$-matching problem, ours is the first result beating the approximation ratio of $2$. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Bernstein and Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a $2 - \frac{1}{2W} + \varepsilon$ approximation of the maximum weight matching is proved using the classical K\H{o}nig-Egerv\'ary's duality theorem.

In the $k$-edge-connected spanning subgraph ($k$ECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to $k$ link failures: Given an $n$-node $m$-edge graph with a cost function on the edges, our goal is to compute a minimum-cost $k$-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of $2$ECSS. The fastest $2$-approximation algorithm is however rather slow, taking $O(mn k)$ time [Khuller, Vishkin, STOC'92]. A faster time complexity of $O(n^2)$ can be obtained, but with a higher approximation guarantee of $(2k-1)$ [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that $(1+\epsilon)$-approximates the optimal fractional solution in $\tilde O(m/\epsilon^2)$ time (independent of $k$), which can be turned into a $(2+\epsilon)$ approximation algorithm that runs in time $\tilde O\left(\frac{m}{\epsilon^2} + \frac{k^2n^{1.5}}{\epsilon^2}\right)$ for (integral) $k$ECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{\Delta}{d} \rceil, d)$-edge colourable if $d$ is even and $(\lceil \frac{3\Delta - 1}{3d - 1} \rceil, d)$-edge colourable if $d$ is odd and these bounds are tight. We also prove that for every simple graph $G$, $\chi'_{d}(G) \in \{ \lceil \frac{\Delta}{d} \rceil, \lceil \frac{\Delta+1}{d} \rceil \}$ and characterize the values of $d$ and $\Delta$ for which it is NP-complete to compute $\chi'_d(G)$. These results generalize several classic results on the chromatic index of a graph by Shannon, Vizing, Holyer, Leven and Galil.

We consider the problem of maximizing a non-negative submodular function under the $b$-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of $2+\varepsilon$, $3 + 2 \sqrt{2} \approx 5.828$, and $4 + 2 \sqrt{3} \approx 7.464$, respectively. We also consider a generalized problem, where a $k$-uniform hypergraph is given, along with an extra matroid or a $k'$-matchoid constraint imposed on the edges, with the same goal of finding a $b$-matching that maximizes a submodular function. When the extra constraint is a matroid, we obtain the approximation ratios of $k + 1 + \varepsilon$, $k + 2\sqrt{k+1} + 2$, and $k + 2\sqrt{k + 2} + 3$ for linear, monotone and non-monotone submodular functions, respectively. When the extra constraint is a $k'$-matchoid, we attain the approximation ratio $\frac{8}{3}k+ \frac{64}{9}k' + O(1)$ for general submodular functions.

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum size of feasible solution. This immediately gives an $O(1/\varepsilon)$-pass $(2+\varepsilon)$-approximation algorithms for monotone submodular maximization in a matroid and $(3+\varepsilon)$-approximation for monotone submodular matching. Our algorithm is oblivious to the choice $\varepsilon$ and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithm for general, non-negative submodular functions subject to a $p$-matchoid constraint with a number of passes independent of the size of the ground set and $k$. We show that a randomized $O(p/\varepsilon)$-pass algorithm storing $O(p^3k\log(k)/\varepsilon^3)$ elements gives a $(p+1+\bar{\gamma}+O(\varepsilon))$-approximation, where $\bar{gamma}$ is the guarantee of the best-known offline algorithm for the same problem., Comment: Accepted at APPROX 2020, 25 pages

We consider the problem of optimizing a coverage function under a $\ell$-matchoid of rank $k$. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by $\ell$ and $k$. This result, combined with the lower bounds of Lovasz, and Jensen and Korte demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-parity problems in the setting of fixed-parameter tractability. For a general coverage function, we give both deterministic and randomized fixed-parameter algorithms, parameterized by $\ell$ and $z$, where $z$ is the number of points covered in an optimal solution. The resulting algorithms can be directly translated into streaming algorithms. For unweighted coverage functions, we show that we can find an exact solution even when the function is given in the form of a value oracle (and so we do not have access to an explicit representation of the set system). Our result can be implemented in the streaming setting and stores a number of elements depending only on $\ell$ and $z$, but completely indpendent of the total size $n$ of the ground set. This shows that it is possible to circumvent the recent space lower bound of Feldman et al, by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model.

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.

Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is much gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio $\frac{2}{2+\sqrt{2}}+\varepsilon$ requires $\Omega\left(\frac{n}{K^2}\right)$ space for any $\varepsilon>0$, where $K$ is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio $\frac{K}{2K-1}+\varepsilon$ requires $\Omega\left(\frac{n}{K}\right)$ space for any $\varepsilon>0$, where $K$ is the rank of the given matroid. In addition, we give streaming algorithms when we only have a weak oracle with which we can only evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios $\frac{K}{2K-1}$ and $\frac{1}{2}$, respectively, whose space complexity is exponential in $K$ but is independent of $n$. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of $\frac{K}{2K-1}$ for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of $n$.

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking $O(\varepsilon^{-1})$ passes: ----a $(1-e^{-1}-\varepsilon)$-approximation algorithm for the cardinality-constrained problem ---- a $(0.5-\varepsilon)$-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run in $O^\ast(n)$ time, using $O^\ast(K)$ space, where $n$ is the size of the ground set and $K$ is the size of the knapsack. Here the term $O^\ast$ hides a polynomial of $\log K$ and $\varepsilon^{-1}$. Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes $O(n\varepsilon^{-1} \log (\varepsilon^{-1}\log K) )$ time, improving on the algorithm of Badanidiyuru and Vondr\'{a}k that takes $O(n \varepsilon^{-1} \log (\varepsilon^{-1} K) )$ time.

We study computing {\em all-pairs shortest paths} (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits $(1+o(1))$-approximation $\tilde O(n)$-time algorithms ~\cite{LenzenP-podc15,Nanongkai-STOC14}, which are matched with $\tilde \Omega(n)$-time lower bounds~\cite{Nanongkai-STOC14,LenzenP_stoc13,FrischknechtHW12}\footnote{$\tilde \Theta$, $\tilde O$ and $\tilde \Omega$ hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.}. No $\omega(n)$ lower bound or $o(m)$ upper bound were known for exact computation. In this paper, we present an $\tilde O(n^{5/4})$-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive $O(m)$-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are {\em asymmetric} (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an $\tilde O(n^{3/4}k^{1/2}+n)$-time algorithm for the {\em $k$-source shortest paths} problem where we want every node to know distances from $k$ sources; this improves Elkin's recent bound~\cite{Elkin-STOC17} when $k=\tilde \omega(n^{1/4})$., Comment: Minor corrections in Section 4

We are given a bipartite graph $G = (A \cup B, E)$ where each vertex has a preference list ranking its neighbors: in particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference lists of $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number of vertices that prefer $M$ to~$M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is NP-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b$'s list and $b$ desires to be matched to any of them. Our main result is an $O(n^2)$ algorithm (where $n = |A \cup B|$) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in $B$ have no preferences and do not care whether they are matched or not., Comment: A shortened version of this paper has appeared at ICALP 2015

Makespan minimization in restricted assignment $(R|p_{ij}\in \{p_j, \infty\}|C_{\max})$ is a classical problem in the field of machine scheduling. In a landmark paper in 1990 [8], Lenstra, Shmoys, and Tardos gave a 2-approximation algorithm and proved that the problem cannot be approximated within 1.5 unless P=NP. The upper and lower bounds of the problem have been essentially unimproved in the intervening 25 years, despite several remarkable successful attempts in some special cases of the problem [2,4,12] recently. In this paper, we consider a special case called graph-balancing with light hyper edges, where heavy jobs can be assigned to at most two machines while light jobs can be assigned to any number of machines. For this case, we present algorithms with approximation ratios strictly better than 2. Specifically, Two job sizes: Suppose that light jobs have weight $w$ and heavy jobs have weight $W$, and $w < W$. We give a $1.5$-approximation algorithm (note that the current 1.5 lower bound is established in an even more restrictive setting [1,3]). Indeed, depending on the specific values of $w$ and $W$, sometimes our algorithm guarantees sub-1.5 approximation ratios. Arbitrary job sizes: Suppose that $W$ is the largest given weight, heavy jobs have weights in the range of $(\beta W, W]$, where $4/7\leq \beta < 1$, and light jobs have weights in the range of $(0,\beta W]$. We present a $(5/3+\beta/3)$-approximation algorithm. Our algorithms are purely combinatorial, without the need of solving a linear program as required in most other known approaches.

We study classical deadline-based preemptive scheduling of tasks in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each task is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of tasks in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. [16], its exact computational complexity has been repeatedly posed as an open question (see e.g. [2,8,15]). The currently best known upper and lower bounds are a 4/3-approximation algorithm and NP-hardness due to [2] and [2,17], respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.