We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based on a simple graph decomposition followed by comparing with the smallest 2-edge cover.

We consider integer programming problems in standard form $\max \{c^Tx : Ax = b, \, xe0, \, x $^n\}$ where $A $^{m ×n}$, $b $^m$ and $c $^n$. We show that such an integer program can be solved in time $(m Å )^{O(m)} Å \|b\|_^2$, where $$ is an upper bound on each absolute value of an entry in $A$. This improves upon the longstanding best bound of Papadimitriou (1981) of $(mÅ)^{O(m^2)}$, where in addition, the absolute values of the entries of $b$ also need to be bounded by $$. % and addresses an open problem raised by Fomin. Our result relies on a lemma of Steinitz that states that a set of vectors in $^m$ that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by $m$. We also use the Steinitz lemma to show that the $\ell_1$-distance of an optimal integer and fractional solution, %of the integer %program, also under the presence of upper bounds on the variables, is bounded by $m Å (2\,mÅ+1)^m$. Here $$ is again an upper bound on the absolute values of the entries of $A$. The novel strength of our bound is that it is independent of $n$. We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.

Energy is often the most constrained resource for battery-powered wireless devices and the lion's share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the energy complexity of LeaderElection and ApproximateCounting in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions. We consider four collision-detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD and Receiver-CD (in which only transmitters or only listeners detect collisions), and No-CD (in which no one detects collisions.) The take-away message of our results is quite surprising. For randomized LeaderElection algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD, and for deterministic LeaderElection algorithms there is another exponential gap, but in the reverse direction. In particular, the randomized energy complexity of LeaderElection is $\Theta(\log^* n)$ in Sender-CD but $\Theta(\log(\log^* n))$ in Receiver-CD, where $n$ is the (unknown) number of devices. Its deterministic complexity is $\Theta(\log N)$ in Receiver-CD but $\Theta(\log\log N)$ in Sender-CD, where $N$ is the (known) size of the devices' ID space. There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized LeaderElection and ApproximateCounting. A critical component of this algorithm is a new deterministic LeaderElection algorithm for dense instances, when $n=\Theta(N)$, with inverse-Ackermann-type ($O(\alpha(N))$) energy complexity.

Often in a scheduling problem, there is uncertainty about the jobs to be processed. The issue of uncertainty regarding the machines has been much less studied. In this paper, we study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. In order to evaluate algorithms in such an environment, we introduce the idea of an ±-robust algorithm, one which is guaranteed to return a schedule on any number m of machines that is within an ± factor of the optimal schedule on m machine, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a 5/3-robust algorithm for scheduling on parallel machines to minimize makespan, and show a lower bound 4/3. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. We also study a case of fair allocation, where the objective is to minimize the difference between the maximum and minimum machine load.

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go ``half the remaining distance" to optimal for a major integrality-gap conjecture of F\"uredi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).

We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Ordered Level Planarity is a problem with high centrality in the area of graph drawing: we give polynomial-time reductions to several variants of Level Planarity and Cluster Planarity, as well as embeddability problems on point sets. Together with our complexity results for Ordered Level Planarity, these reductions answer multiple open questions posed by the graph drawing community. We expect that our results may serve as a suitable basis for more reductions in the future.

We provide a tight analysis which settles the round complexity of the well-studied \emph{parallel randomized greedy MIS} algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA'12]. The parallel/distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS'87], and the previously best known bounds were $O(\log n)$ rounds in expectation for Erd\H{o}s-R\'{e}nyi random graphs by Calkin and Frieze [Random Struc. \& Alg. '90] and $O(\log^2 n)$ rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA'12]. We prove a high probability upper bound of $O(\log n)$ on the round complexity of this algorithm in general graphs, and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC'85, JALG'86].

#### Solving the Sigma-Tau Problem

JOE SAWADA (University of Guelph); AARON WILLIAMS (Williams College)We present a new locally differentially private algorithm for the heavy hitters problem which achieves optimal worst-case error as a function of all standardly considered parameters. Prior work obtained error rates which depend optimally on the number of users, the size of the domain, and the privacy parameter, but depend sub-optimally on the failure probability. We strengthen existing lower bounds on the error to incorporate the failure probability, and show that our new upper bound is tight with respect to this parameter as well. Our lower bound is based on a new understanding of the structure of locally private protocols. We further develop these ideas to obtain the following general results beyond heavy hitters. 1. Advanced Grouposition: In the local model, group privacy for k users degrades proportionally to root k, instead of linearly in k as in the central model. Stronger group privacy yields improved max-information guarantees, as well as stronger lower bounds (via "packing arguments"), over the central model. 2. Building on a transformation of Bassily and Smith (STOC 2015), we give a generic transformation from any non-interactive approximate-private local protocol into a pure-private local protocol. Again in contrast with the central model, this shows that we cannot obtain more accurate algorithms by moving from pure to approximate local privacy.

Given a traveling salesman problem (TSP) tour $H$ in graph $G$ a $k$-move is an operation which removes $k$ edges from $H$, and adds $k$ edges of $G$ so that a new tour $H'$ is formed. The popular $k$-OPT heuristic for TSP finds a local optimum by starting from an arbitrary tour $H$ and then improving it by a sequence of $k$-moves. Until 2016, the only known algorithm to find an improving $k$-move for a given tour was the naive solution in time $O(n^k)$. At ICALP'16 de~Berg, Buchin, Jansen and Woeginger showed an $O(n^{\floor{2/3k}+1})$-time algorithm. We show an algorithm which runs in $O(n^{(1/4+\epsilon_k)k})$ time, where $\lim_{k\rightarrow\infty} \epsilon_k = 0$. It improves over the state of the art for every $k\ge 5$. For the most practically relevant case $k=5$ we provide a slightly refined algorithm running in $O(n^{3.4})$ time. We also show that for the $k=4$ case, improving over the $O(n^3)$-time algorithm of de~Berg et al. would be a major breakthrough: an $O(n^{3-\epsilon})$-time algorithm for any $\epsilon>0$ would imply an $O(n^{3-\delta})$-time algorithm for the \probAPSP problem, for some $\delta>0$.

We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An \textbf{embedding} $\varphi:G\rightarrow M$ of a graph $G$ into a 2-manifold $M$ maps the vertices in $V(G)$ to distinct points and the edges in $E(G)$ to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs, due to data compression or low resolution. This raises the computational problem of deciding whether a given map $\varphi:G\rightarrow M$ comes from an embedding. A map $\varphi:G\rightarrow M$ is a \textbf{weak embedding} if it can be perturbed into an embedding $\psi_\eps:G\rightarrow M$ with $\|\varphi-\psi_\eps\|<\eps$ for every $\eps>0$. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyn\v{c}l~\cite{FK18_ht}, which reduces to solving a system of linear equations over $\mathbb{Z}_2$. It runs in $O(n^{2\omega})\leq O(n^{4.75})$ time, where $\omega\in [2,2.373)$ is the matrix multiplication exponent and $n$ is the number of vertices and edges of $G$. We improve the running time to $O(n\log n)$. Our algorithm is also conceptually simpler than~\cite{FK18_ht}: We perform a sequence of \emph{local operations} that gradually ``untangles'' the image $\varphi(G)$ into an embedding $\psi(G)$, or reports that $\varphi$ is not a weak embedding. It generalizes a recent technique developed for the case that $G$ is a cycle and the embedding is a simple polygon~\cite{AAET17}, and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid containing a set T of given terminal elements. For graphic matroids this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gavenciak, Kral', and Oum [ICALP'12], who proved that the case of the problem with |T|=2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gavenciak, Kral', and Oum by showing that for regular matroids - the Minimum Spanning Circuit problem, deciding whether there is a circuit with at most \ell elements containing T, is FPT parameterized by k=\ell-|T|; - the Spanning Circuit problem, deciding whether there is a circuit containing T, is FPT parameterized by |T|. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: Minimum Spanning Circuit parameterized by \ell is W[1]-hard on binary matroids even when |T | = 1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that Minimum Spanning Circuit parameterized by |T| is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.