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The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem (CAP) is arguably one of the most basic problems in this area: given a k(-edge)-connected graph G and a set of extra edges (links), select a minimum cardinality subset A of links such that adding A to G increases its edge connectivity to k+1. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and J\'aj\' a, SIAM J. Comput., 10 (1981), pp. 270-283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290-306] that CAP can be reduced to the case k = 1, also known as the tree augmentation problem (TAP) for odd k, and to the case k = 2, also known as the cactus augmentation problem (CacAP) for even k. Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC'20, ACM, New York, 2020, pp. 815-825], several better than 2 approximation algorithms were known for TAP, culminating with a recent 1.458 approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC'18, ACM, New York, 1918, pp. 632-645]. However, for CacAP the best known approximation was 2. In this paper we breach the 2 approximation barrier for CacAP, hence, for CAP, by presenting a polynomial-time 2 ln(4) - 1120 967 + \varepsilon < 1.91 approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for TAP either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which wa

We study the resource augmented version of the k-server problem, also known as the k-server problem against weak adversaries or the (h,k)-server problem. In this setting, an online algorithm using k servers is compared to an offline algorithm using h servers, where h ≤ k. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any ">0, the competitive ratio drops to a constant if k=(1+") · h. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least ω(loglogh), even as k→∞. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics.

A binary tanglegram is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n^3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation. Keywords: Binary tanglegram – Crossing minimization – NP-hardness – Approximation algorithm – Fixed-parameter tractability

We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location (UFL) problem, which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye, and Zhang. Note that the approximability lower bound by Guha and Khuller is $1.463\dots$. An algorithm is a ($\lambda_f$,$\lambda_c$)-approximation algorithm if the solution it produces has total cost at most $\lambda_f\cdot F^*+\lambda_c\cdot C^*$, where $F^*$ and $C^*$ are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the $(1+2/e)$-approximation algorithm of Chudak and Shmoys, is a $(1.6774,1.3738)$-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve $(\gamma_f,1+2e^{-\gamma_f})$ established by Jain, Mahdian, and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a $(1.11,1.7764)$-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL.

A \emph{binary tanglegram} is a pair $$ of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossing and that the inter-tree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for general binary trees. We show that the problem is hard even if both trees are complete binary trees. For this case we give an $O(n^3)$-time 2-approximation and a new and simple fixed-parameter algorithm. We show that the maximization version of the dual problem for general binary trees can be reduced to a version of \textsc{MaxCut} for which the algorithm of Goemans and Williamson yields a 0.878-approximation.

We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location (UFL) problem, which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye, and Zhang. Note that the approximability lower bound by Guha and Khuller is 1.463 . . . . An algorithm is a (?f ,?c)-approximation algorithm if the solution it produces has total cost at most ?f ·F?+?c ·C?, where F? and C? are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the (1 + 2/e)-approximation algorithm of Chudak and Shmoys, is a (1.6774, 1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve (?f ,1+2e??f ) established by Jain, Mahdian, and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11, 1.7764)-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL., Delft Institute of Applied Mathematics, Electrical Engineering, Mathematics and Computer Science

The study of phylogenetic networks is of great interest to computational evolutionary biology and numerous different types of such structures are known. This article addresses the following question concerning rooted versions of phylogenetic networks. What is the maximum value of pset membership, variant[0,1] such that for every input set T of rooted triplets, there exists some network View the MathML source such that at least p|T| of the triplets are consistent with View the MathML source? We call an algorithm that computes such a network (where p is maximum) worst-case optimal. Here we prove that the set containing all triplets (the full triplet set) in some sense defines p. Moreover, given a network View the MathML source that obtains a fraction p′ for the full triplet set (for any p′), we show how to efficiently modify View the MathML source to obtain a fraction greater-or-equal, slantedp′ for any given triplet set T. We demonstrate the power of this insight by presenting a worst-case optimal result for level-1 phylogenetic networks improving considerably upon the 5/12 fraction obtained recently by Jansson, Nguyen and Sung. For level-2 phylogenetic networks we show that pgreater-or-equal, slanted0.61. We emphasize that, because we are taking |T| as a (trivial) upper bound on the size of an optimal solution for each specific input T, the results in this article do not exclude the existence of approximation algorithms that achieve approximation ratio better than p. Finally, we note that all the results in this article also apply to weighted triplet sets.

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem ($\textsc{MaxRTC}$) and the minimum rooted triplets inconsistency problem ($\textsc{MinRTI}$) in which the input is a set $\mathcal{R}$ of rooted triplets, and where the objectives are to find a largest cardinality subset of $\mathcal{R}$ which is consistent and a smallest cardinality subset of $\mathcal{R}$ whose removal from $\mathcal{R}$ results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic [25] results in a bottom-up-based 3-approximation for $\textsc{MaxRTC}$. We then demonstrate how any approximation algorithm for $\textsc{MinRTI}$ could be used to approximate $\textsc{MaxRTC}$, and thus obtain the first polynomial-time approximation algorithm for $\textsc{MaxRTC}$ with approximation ratio smaller than 3. Next, we prove that fo

A \emph{binary tanglegram} is a pair $$ of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge

We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197¿+¿e)-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point. The first author was supported by NWO. The second and third author were supported by the EU Marie Curie Research Training Network, contract numbers MRTN-CT-2003-504438-ADONET and MRTN-CT-2004-504438-ADONET respectively.

We present a simple two-person Bucket Game, based on throwing balls into buckets, and we discuss possible players’ strategies. We use these strategies to create an approximation algorithm for a generalization of the well known Set Cover problem, where we need to cover each element by at least k sets. Furthermore, we apply these strategies to construct a randomized algorithm for Dynamic Page Migration problem achieving the optimal competitive ratio against an oblivious adversary.