We show that the Simplex Method, the Network Simplex Method both with Dantzigs original pivot rule and the Successive Shortest Path Algorithm are NP-mighty. That is, each of these algorithms can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithms execution. This result casts a more favorable light on these algorithms exponential worst-case running times. Furthermore, as a consequence of our approach, we obtain several novel hardness results. For example, for a given input to the Simplex Algorithm, deciding whether a given variable ever enters the basis during the algorithms execution and determining the number of iterations needed are both NP-hard problems. Finally, we close a long-standing open problem in the area of network flows over time by showing that earliest arrival flows are NP-hard to obtain.

We consider the problem of deterministic load balancing of tokens in the discrete model. A set of $n$ processors is connected into a $d$-regular undirected network. In every time step, each processor exchanges some of its tokens with each of its neighbors in the network. The goal is to minimize the discrepancy between the number of tokens on the most-loaded and the least-loaded processor as quickly as possible. Rabani {\it et al.}\ (1998) present a general technique for the analysis of a wide class of discrete load balancing algorithms. Their approach is to characterize the deviation between the actual loads of a discrete balancing algorithm with the distribution generated by a related Markov chain. The Markov chain can also be regarded as the underlying model of a continuous diffusion algorithm. Rabani {\it et al.} showed that after time $T = \bigo(\log (Kn)/\mu)$, any algorithm of their class achieves a discrepancy of $\bigo(d\log n/\mu)$, where $\mu$ is the spectral gap of the transition matrix of the graph, and $K$ is the initial load discrepancy in the system. In this work, we identify some natural additional conditions on deterministic balancing algorithms, resulting in a class of algorithms reaching a smaller discrepancy. This class contains fundamental algorithms. Specifically, we introduce the notion of \emph{cumulatively fair} load-balancing algorithms where in any interval of consecutive time steps, the total number of tokens sent out over an edge by a node is the same (up to constants) for all adjacent edges.

We show how to compute for n-vertex planar graphs in O(n^{11/6} polylog(n)) expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In O(n^{15/8} polylog(n)) expected time we can also compute the number of pairs of vertices at distance smaller than a given threshold, These are the first algorithms for these problems using time O(n^c) for some constant c < 2, even when restricted to undirected, unweighted planar graphs.

We present a simple deterministic single-pass (2+µ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5+µ).
Our algorithm uses O(n log^{2} n) space for constant values of µ. It relies on a variation of the local-ratio theorem, which may be of use for other algorithms in the semi-streaming model as well.

*O(*log

*k)*-competitive algorithm for Generalized Caching Anna Adamaszek (University of Copenhagen); Artur Czumaj (University of Warwick Faculty of Science); Matthias Englert (University of Warwick Faculty of Science); Harald Räcke (Technische Universitat Munchen Fakultat fur Elektrotechnik und Informationstechnik)

In the generalized caching problem, we have a set of pages and a cache of size *k*. Each page *p* has a size *w _{p}e1* and fetching cost

*c*for loading the page into the cache. At any point in time, the sum of the sizes of the pages stored in the cache cannot exceed

_{p}*k*. The input consists of a sequence of page requests. If a page is not present in the cache at the time it is requested, it has to be loaded into the cache incurring a cost of

*c*. We give a randomized

_{p}*O(*log

*k)*-competitive online algorithm for the generalized caching problem, improving the previous bound of

*O(*log

^{2}

*k)*by Bansal, Buchbinder, and Naor (STOC'08). This improved bound is tight and of the same order as the known bounds for the classic problem with uniform weights and sizes. We use the same LP based techniques as Bansal et al. but provide improved and slightly simplified methods for rounding fractional solutions online.

Given a graph $G$ and a parameter $k$, the {\sc Chordal Vertex Deletion (CVD)} problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for {\sc CVD} was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for {\sc CVD} of size $\OO(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $\OO(k^{10})$ [Jansen an Pilipczuk, SODA 2017]. While we do not completely resolve this question, we design a significantly smaller kernel of size $\OO(k^{12}\log^{10} k)$, inspired by the $\OO(k^2)$-size kernel for {\sc Feedback Vertex Set} [Thomass\'{e}, TALG 2010]. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.

Max-Cut, Edge Dominating Set, Graph Coloring, and Hamiltonian Cycle on graphs of bounded clique-width have received significant attention. Each of these problems can be solved in time g(k)n^f(k) on graphs of clique-width k. Fomin et al. showed that the running times cannot be improved to g(k)n^O(1) assuming W[1]\=FPT. However, this does not rule out non-trivial improvements to the exponent f(k) in the running times. In a follow-up paper, the same authors improved the running times for Edge Dominating Set and Max-Cut to n^O(k), and proved that these problems cannot be solved in time g(k)n^o(k) unless ETH fails. Thus, prior to this work, Edge Dominating Set and Max-Cut were known to have tight n^\Theta(k) algorithmic upper and lower bounds. In this paper we provide lower bounds for Hamiltonian Cycle and Graph Coloring. For Hamiltonian Cycle our lower bound g(k)n^o(k) matches asymptotically the recent upper bound n^O(k) due to Bergougnoux, Kante and Kwon [WADS 2017]. As opposed to the asymptotically tight n^\Theta(k) bounds for Edge Dominating Set, Max-Cut and Hamiltonian Cycle, the Graph Coloring problem has an upper bound of n^O(2^k) and a lower bound of merely n^o(\sqrt[4]{k}) (implicit from the W[1]-hardness proof). In this paper, we close the gap for Graph Coloring by proving a lower bound of n^{2^{o(k)}}. This shows that Graph Coloring behaves qualitatively different from the other three problems. To the best of our knowledge, Graph Coloring is the first natural problem known to require exponential dependence on the parameter in the exponent of n.

We consider the fundamental task of graph exploration. An $n$-node graph has unlabeled nodes, and all ports at any node of degree $d$ are arbitrarily numbered $0,\dots, d-1$. A mobile agent, initially situated at some starting node $v$, has to visit all nodes and stop. The time of the exploration is the number of edge traversals. We consider the problem of how much knowledge the agent has to have a priori, in order to explore the graph in a given time, using a deterministic algorithm. Following the paradigm of algorithms with advice, this a priori information (advice) is provided to the agent by an oracle, in the form of a binary string, whose length is called the size of advice. We consider two types of oracles. The instance oracle knows the entire instance of the exploration problem, i.e., the port-numbered map of the graph and the starting node of the agent in this map. The map oracle knows the port-numbered map of the graph but does not know the starting node of the agent. The central question of the paper is the following. What is the minimum size of advice that must be given to the agent by each of these oracles, so that the agent explores the graph in a given time?

We provide a remarkably simple algorithm to compute all (at most four) common tangents of two disjoint simple polygons. Given each polygon as a read-only array of its corners in cyclic order, the algorithm runs in linear time and constant workspace and is the first to achieve the two complexity bounds simultaneously. The set of common tangents provides basic information about the convex hulls of the polygonswhether they are nested, overlapping, or disjointand our algorithm thus also decides this relationship.

We settle the complexity of the Independent Set Reconfiguration problem on bipartite graphs under all three commonly studied reconfiguration models. We show that under the token jumping or token addition/removal model the problem is NP-complete. For the token sliding model, we show that the problem remains PSPACE-complete.

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this paper, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed value of r a polynomial kernel for the distance-r dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class C of graphs which is closed under subgraphs, the distance-r dominating set problem admits any kernel on C for every value of r if, and only if, it already admits a polynomial kernel for every value of r (assuming FPT \neq W[2]).

Miller et al. \cite{MPVX15} devised a distributed\footnote{They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in \cite{MPVX15}.} algorithm in the CONGEST model, that given a parameter $k = 1,2,\ldots$, constructs an $O(k)$-spanner of an input unweighted $n$-vertex graph with $O(n^{1+1/k})$ expected edges in $O(k)$ rounds of communication. In this paper we improve the result of \cite{MPVX15}, by showing a $k$-round distributed algorithm in the same model, that constructs a $(2k-1)$-spanner with $O(n^{1+1/k}/\epsilon)$ edges, with probability $1- \epsilon$, for any $\epsilon>0$. Moreover, when $k = \omega(\log n)$, our algorithm produces (still in $k$ rounds) {\em ultra-sparse} spanners, i.e., spanners of size $n(1+ o(1))$, with probability $1- o(1)$. To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large $k$, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for $(1+\epsilon,\beta)$-spanners and emulators. In particular, we show that for any unweighted $n$-vertex graph and any $\epsilon > 0$, there exists a $(1+ \epsilon, ({{\log\log n} \over \epsilon})^{\log\log n})$-emulator with $O(n)$ edges. All previous constructions of $(1+\epsilon,\beta)$-spanners and emulators employ a superlinear number of edges, for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs.

We give the first optimal bounds for returning the $\ell_1$-heavy hitters in a data stream of insertions, together with their approximate frequencies, closing a long line of work on this problem. For a stream of $m$ items in $\{1, 2, \ldots, n\}$ and parameters $0 < \epsilon < \phi \leq 1$, let $f_i$ denote the frequency of item $i$, i.e., the number of times item $i$ occurs in the stream. With arbitrarily large constant probability, our algorithm returns all items $i$ for which $f_i \geq \phi m$, returns no items $j$ for which $f_j \leq (\phi -\epsilon)m$, and returns approximations $\tilde{f}_i$ with $|\tilde{f}_i - f_i| \leq \epsilon m$ for each item $i$ that it returns. Our algorithm uses $O(\epsilon^{-1} \log\phi^{-1} + \phi^{-1} \log n + \log \log m)$ bits of space, processes each stream update in $O(1)$ worst-case time, and can report its output in time linear in the output size. We also prove a lower bound, which implies that our algorithm is optimal up to a constant factor in its space complexity. A modification of our algorithm can be used to estimate the maximum frequency up to an additive $\epsilon m$ error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of $\ell_1$-heavy hitters. We also introduce several variants of the heavy hitters problem, inspired by rank aggregation and voting schemes, and show how our techniques can be applied in such settings.

The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even -matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by DvoYák and Kupec. Knowing that edge CSP is tractable for even -matroid constraints allows us to extend the tractability result to a larger class of -matroids that includes many classes that were known to be tractable before, namely co-independent, compact, local and binary.