The problem of finding large cliques in random graphs and its ``planted" variant, where one wants to recover a clique of size $\omega \gg \log{(n)}$ added to an \Erdos-\Renyi graph $G \sim G(n,\frac{1}{2})$, have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size $\omega = \Omega(\sqrt{n})$. By contrast, information theoretically, one can recover planted cliques so long as $\omega \gg \log{(n)}$. In this work, we continue the investigation of algorithms from the sum of squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson \cite{MPW15} and Deshpande and Montanari \cite{DM15}. Our main result is that degree four SoS does not recover the planted clique unless $\omega \gg \sqrt n / \polylog n$, improving upon the bound $\omega \gg n^{1/3}$ due to \cite{DM15}. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by ``correcting" the certificate of \cite{MPW15,DM15,FeigeK03}.

The Firefighter problem and a variant of it, known as Resource Minimization for Fire Containment (RMFC), are natural models for optimal inhibition of harmful spreading processes. Despite considerable progress on several fronts, the approximability of these problems is still badly understood. This is the case even when the underlying graph is a tree, which is one of the most-studied graph structures in this context and the focus of this paper. In their simplest version, a fire spreads from one fixed vertex step by step from burning to adjacent non-burning vertices, and at each time step B many non-burning vertices can be protected from catching fire. The Firefighter problem asks, for a given B, to maximize the number of vertices that will not catch fire, whereas RMFC (on a tree) asks to find the smallest B that allows for saving all leaves of the tree. Prior to this work, the best known approximation ratios were an O(1)-approximation for the Firefighter problem and an O(log^* n)-approximation for RMFC, both being LP-based and essentially matching the integrality gaps of two natural LP relaxations. We improve on both approximations by presenting a PTAS for the Firefighter problem and an O(1)-approximation for RMFC, both qualitatively matching the known hardness results. Our results are obtained through a combination of the known LPs with several new techniques, which allow for efficiently enumerating over super-constant size sets of constraints to strengthen the natural LPs.

There are two errors in our paper ``Approximating minimum-cost connectivity problems via uncrossable bifamilies'' (ACM Transactions on Algorithms (TALG), 9(1), Article No. 1, 2012). In that paper we consider the (undirected) {\sc Survivable Network} problem. The input consists of a graph $G=(V,E)$ with edge-costs, a set $T \subs V$ of terminals, and connectivity demands $\{r_{st}>0:st \in D \subs T \times T\}$. The goal is to find a minimum cost subgraph $H$ of $G$ that for all $st \in D$ contains $r_{st}$ pairwise internally-disjoint $st$-paths. We claimed ratios $O(k \ln k)$ for rooted demands when the set $D$ of demand pairs forms a star, where $k=\max_{st \in D} r_{st}$ is the maximum demand. This ratio is correct when the requirements are $r_{st}=k$ for all $t \in T \sem \{s\}$, but for general rooted demands our paper implies only ratio $O(k^2)$ (which however is still the currently best known ratio for the problem). We also obtained various ratios for the node-weighted version of the problem. These results are valid, but the proof needs a correction described here.

We consider the classical selection and sorting problems in a model where the initial permutation of the input has to be restored after completing the computation. While the requirement of the restoration is stringent compared to the classical versions of the problems, this model is more relaxed than a read-only memory (ROM) where the input elements are not allowed to be moved within the input array. We fi rst show that for a sequence of n integers, selection ( finding the median or more generally the k-th smallest element for a given k) can be done in O(n) time using O(lg n) words. In contrast, no linear-time selection algorithm is known which uses polylogarithmic space in ROM. For sorting n integers in this model, we fi rst present an O(n lg n)-time algorithm using O(lg n) words. When the universe size U is polynomial in n, we give a faster O(n)-time algorithm (analogous to radix sort) which uses O(n^eps) words of extra space for an arbitrarily small constant eps > 0. More generally, we show how to match the time bound of any word-RAM integer-sorting algorithms using O(n^eps) words of extra space. In sharp contrast, there is an (n^2/S)-time lower bound for integer sorting using O(S) space in ROM. For indivisible input elements, we prove the same lower bound for sorting in our model. En route, we develop linear-time in-place algorithms to extract leading bits of the input and to compress and decompress strings with low entropy.

Chazelle and Matou\v sek [\emph{J. Algorithms}, 1996] presented a derandomization of Clarkson's sampling-based algorithm [\emph{J. ACM}, 1995] for solving linear programs with $n$ constraints and $d$ variables in $d^{(7+o(1))d}n$ deterministic time. The time bound can be improved to $d^{(5+o(1))d}n$ with subsequent work by Br\"onnimann, Chazelle, and Matou\v sek [\emph{SIAM J. Comput.}, 1999]. We first point out a much simpler derandomization of Clarkson's algorithm that avoids $\eps$-approximations and runs in $d^{(3+o(1))d}n$ time. We then describe a few additional ideas that eventually improve the deterministic time bound to $d^{(1/2+o(1))d}n$.

We consider a new construction of locality-sensitive hash functions for Hamming space that is covering in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius r. The construction is efficient in the sense that the expected number of hash collisions between vectors at distance cr, for a given c>1, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (STOC '98). The efficiency of the new construction essentially matches their bound when the search radius is not too large --- e.g., when cr = o(log(n)/log log n), where n is the number of points in the data set, and when cr = log(n)/k where k is an integer constant. In general, it differs by at most a factor ln(4) in the exponent of the time bounds. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.

We consider the minimum-load k-facility location (MLKFL) problem: give a set F of facilities, a set C of clients, and an integer k\geq 0, and a distance function d(f,j). The goal is to open a set F'\subseteq F of k facilities, and assign each client j to a facility f(j)\in F so as to minimize \max_{f\in F}\sum_{j\in C:f(j)=f}d(f,j). This problem was studied under the name of min-max star cover in {EGK+03,AHL06}, who gave bicriteria approximation algorithms for when F=C. MLKFL is rather poorly understood, and only an $O(k)$-approximation is currently known even for line metrics. Our main result is a PTAS for MLKFL on line metrics. Complementing this, we prove that MLKFL is strongly NP-hard on line metrics. We also devise a QPTAS for it on tree metrics. MLKFL turns out to be surprisingly challenging even on line metrics; we show that: (a) even a configuration-style LP has a bad integrality gap; and (b) a multi-swap local-search heuristic has a bad locality gap. Our PTAS for line metrics consists of two main ingredients. First, we prove existence of a near-optimal solution possessing some nice properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP), and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to ``transfer'' this structure to the rounded integer solution. We then show how to find a solution having these structural properties using DP.

This paper shows the weighted matching problem on general graphs
can be solved in time $O(n(m + n\log n))$ for $n$ and $m$ the
number of vertices and edges, respectively. This was previously known
only for bipartite graphs. The crux is a data structure for blossom creation. It uses a dynamic nearest-common-ancestor algorithm
to simplify blossom steps, so they involve only back edges rather than arbitrary nontree edges.
The rest of the paper presents direct extensions of Edmonds'
blossom algorithm to weighted $b$-matching and $f$-factors. Again
the time bound is the one previously known for
bipartite graphs: for $b$-matching the time is $O(\min\{b(V),n\log
n\}(m + n\log n))$ and for $f$-factors the time is $O(\min {f(V),m\log n\}( m + n\log n) )$,
where $b(V)$ and $f(V)$ denote the sum of all degree constraints.
Several immediate applications of the $f$-factor algorithm
are given: The generalized shortest path structure of \cite{GS13}, i.e., the analog of the shortest path tree for
conservative undirected graphs, is shown to be a version of
the blossom structure for $f$-factors. This structure is found in time $O(|N|(m+n\log n))$ for $N$ the set of negative edges ($0<|N|

We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and matrices of low treewidth. Our goal is to construct algorithms with running time of the form poly(k) * n, where k is the width of a tree decomposition given on input. Such procedures would outperform the best known algorithms already for moderate values of the treewidth, like O(n^{1/c}) for some small constant c. Our results include: - an algorithm for computing the determinant and the rank of an n x n matrix using O(k^3 * n) time and arithmetic operations; - an algorithm for solving a system of linear equations using O(k^3 * n) time and arithmetic operations; - an O(k^3 * n log n)-time randomized algorithm for finding the cardinality of a maximum matching in a graph; - an O(k^4 * n log^2 n)-time randomized algorithm for constructing a maximum matching in a graph; - an O(k^2 * n log n)-time algorithm for finding a maximum vertex flow in a directed graph. Moreover, we give an approximation algorithm for treewidth with suitable time complexity: given a graph G and integer k, in time O(k^7 * n log n) it concludes that G has treewidth larger than k, or constructs a decomposition of width O(k^2). The results stand in contrast with the recent work of Abboud et al. [SODA 2016], which shows that the algorithms with similar running times are unlikely for the problems of finding the diameter and radius of a graph of low treewidth.

For any integer $n\geq 1$ a \emph{middle levels Gray code} is a cyclic listing of all bitstrings of length $2n+1$ that have either $n$ or $n+1$ entries equal to 1 such that any two consecutive bitstrings in the list differ in exactly one bit. The question whether such a Gray code exists for every $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T.~Mütze. Proof of the middle levels conjecture. \textit{arXiv:1404.4442}, 2014]. In this work we provide the first efficient algorithm to compute a middle levels Gray code. For a given bitstring, our algorithm computes the next $\ell$ bitstrings in the Gray code in time $\cO(n\ell(1+\frac{n}{\ell}))$, which is $\cO(n)$ on average per bitstring provided that $\ell=\Omega(n)$.

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic $1/2$-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.

Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p' in R^d by a voter v in V if dist(v,p) < dist(v,p'). A point p is called a plurality point if it is preferred by at least as many voters as any other point p'. We present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector

We study the problem of detecting {\em outlier pairs} of strongly correlated variables among a collection of $n$ variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of $n$ vectors with unit Euclidean norm and dimension $d$, and we are asked to find all the outlier pairs of vectors whose inner product is at least $\rho$ in absolute value, subject to the promise that all but at most $q$ pairs of vectors have inner product at most $\tau$ in absolute value for some constants $0<\tau<\rho<1$. Improving on an algorithm of G.~Valiant [FOCS~2012; J.\,ACM~2015], we present a randomized algorithm that for Boolean inputs ($\{-1,1\}$-valued data normalized to unit Euclidean length) runs in time \[ \tilde O\bigl(n^{\max\,\{1-\gamma+M(\Delta\gamma,\gamma),\,M(1-\gamma,2\Delta\gamma)\}}+qdn^{2\gamma}\bigr)\,, \] where $0<\gamma<1$ is a constant tradeoff parameter and $M(\mu,\nu)$ is the exponent to multiply an $\lfloor n^\mu\rfloor\times\lfloor n^\nu\rfloor$ matrix with an $\lfloor n^\nu\rfloor\times \lfloor n^\mu\rfloor$ matrix and $\Delta=1/(1-\log_\tau\rho)$. As corollaries we obtain randomized algorithms that run in time \[ \tilde O\bigl(n^{\frac{2\omega}{3-\log_\tau\rho}}+qdn^{\frac{2(1-\log_\tau\rho)}{3-\log_\tau\rho}}\bigr) \] and in time \[ \tilde O\bigl(n^{\frac{4}{2+\alpha(1-\log_\tau\rho)}}+qdn^{\frac{2\alpha(1-\log_\tau\rho)}{2+\alpha(1-\log_\tau\rho)}}\bigr)\,, \] where $2\leq\omega<2.38$ is the exponent for square matrix multiplication and $0.3<\alpha\leq 1$ is the exponent for\, rectangular matrix multiplication. We present further corollaries for the light bulb problem and for learning sparse Boolean functions. (The notation {$\tilde O(\cdot)$} hides polylogarithmic factors in $n$ and $d$ whose degree may depend on $\rho$ and $\tau$.)

The approximate nearest neighbor problem ($\epsilon$-ANN) in high dimensional Euclidean space has been mainly addressed by Locality Sensitive Hashing (LSH), which has polynomial dependence in the dimension, sublinear query time, but subquadratic space requirement. In this paper, we introduce a new definition of ``low-quality'' embeddings for metric spaces. It requires that, for some query point $q$, there exists an approximate nearest neighbor among the pre-images of the $k>1$ approximate nearest neighbors in the target space. Focusing on Euclidean spaces, we employ random projections in order to reduce the original problem to one in a space of dimension inversely proportional to $k$. The $k$ approximate nearest neighbors can be efficiently retrieved by a data structure such as BBD-trees. The same approach is applied to the problem of computing an approximate near neighbor, where we obtain a data structure requiring linear space, and query time in $O(d n^{\rho})$, for $\rho\approx 1-\epsilon^2/\log(1/\epsilon)$. This directly implies a solution for $\epsilon$-ANN, while achieving a better exponent in the query time than the method based on BBD-trees. Better bounds are obtained in the case of doubling subsets of $\ell_2$, by combining our method with $r$-nets. We implement our method in C++, and present experimental results in dimension up to $500$ and $10^6$ points, which show that performance is better than predicted by the analysis. In addition, we compare our ANN approach to E2LSH, which implements LSH, and we show that the theoretical advantages of each method are reflected on their actual performance.

A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3-separators. This seminal result generalizes Tutte's theorem that every 4-connected planar graph is Hamiltonian as well as Barnette's theorem that every 3-connected planar graph has a spanning tree with maximum degree at most 3. The algorithmic challenge of finding such a closed 2-walk is to overcome big overlapping subgraphs in the decomposition, which are also inherent in Tutte's and Thomassen's decompositions. We solve this problem by extending the decomposition of Gao, Richter and Yu in such a way that all pieces, in which the graph is decomposed into, are edge-disjoint. This implies the first polynomial-time algorithm that computes the closed 2-walk mentioned above.

We consider the problems of online and stochastic packet queuing in a distributed system of n nodes with queues, where the communication between the nodes is done via a multiple access channel. In the online setting, in each round, an arbitrary number of packets can be injected into the system, each to an arbitrary node's queue. Two measures of performance are considered: the total number of packets in the system, called the total load, and the maximum queue size, called the maximum load. We develop a deterministic distributed algorithm that is asymptotically optimal with respect to both complexity measures, in a competitive way. More precisely, the total load of our algorithm is bigger than the total load of any other algorithm, including centralized online solutions, by only an additive term of O(n^2), while the maximum queue size of our algorithm is at most n times bigger than the maximum queue size of any other algorithm, with an extra additive O(n). The optimality for both measures is justified by proving the corresponding lower bounds, which also separates nearly exponentially distributed solutions from the centralized ones. Next, we show that our algorithm is also stochastically stable for any expected injection rate smaller or equal to 1. This is the first solution to the stochastic queuing problem on a multiple access channel that achieves such stability for the (highest possible) rate equal to 1.

Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k+1 quantifiers takes time $O(m^k)$ and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require $O(m^{k-\epsilon})$ time for any $\epsilon > 0$. (Here, m represents the size of the input structure, i.e. the number of tuples in all relations.) We give algorithms for every first-order property that improves this upper bound to $m^k/2^{\Theta(\sqrt{\log n})}$, i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [AWY15,CW16]. While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurovi et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurovi et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including: (i) a strengthening of the heuristic by Hujdurovi et al. via a new min-max result in digraphs generalizing Dilworth's theorem, which may be of independent interest, (ii) APX-hardness results for both problems, (iii) approximation algorithms, and (iv) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

A simple digraph is semi-complete if for any two of its vertices u and v, at least one of the arcs (u,v) and (v,u) is present. We study the complexity of computing two layout parameters of semi-complete digraphs: cutwidth and optimal linear arrangement (Ola). We prove that: " Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis. " The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polyno- mial kernel unless NP coNP/poly. By contrast, Ola admits a linear kernel. These results essentially complete the complexity analysis of computing cutwidth and Ola on semi- complete digraphs. Our techniques can be also used to analyze the sizes of minimal obstructions for having small cutwidth under the induced subdigraph relation.

Given a 1.5-dimensional terrain $T$, also known as an $x$-monotone polygonal chain, the {\sc Terrain Guarding} problem seeks a set of points of minimum size on $T$ that {\em guards} all of the points on $T$. Here, we say that a point $p$ guards a point $q$ if no point of the line segment $\overline{pq}$ is strictly below $T$. The {\sc Terrain Guarding} problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that {\sc Terrain Guarding} is NP-hard. In spite of the remarkable developments in approximation algorithms for {\sc Terrain Guarding}, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an $n^{O(\sqrt{k})}$-time algorithm for both {\sc Discrete Terrain Guarding} and {\sc Continuous Terrain Guarding}. We also make non-trivial progress with respect to the second question: we show that {\sc Discrete Orthogonal Terrain Guarding}, a well-studied special case of {\sc Terrain Guarding}, is fixed-parameter tractable.

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2-\epsilon)^{n}m^{O(1)} time, we show that for any e > 0; {\sc Independent Set} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Dominating Set} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Max Cut} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Odd Cycle Transversal} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, For any qe 3, q-{\sc Coloring} cannot be solved in (q-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Partition Into Triangles} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time. Our lower bounds match the running times for the best known algorithms for the problems, up to the e in the base.

In the Interval Completion problem we are given an n-vertex graph G and an integer k, and the task is to transform G by making use of at most k edge additions into an interval graph. This is a fundamental graph modifi cation problem with applications in sparse matrix multiplication and molecular biology. The question about fi xed-parameter tractability of Interval Completion was asked by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999] and was answered a rmatively more than a decade later by Villanger at el. [STOC 2007; SIAM J. Comput. 2009], who presented an algorithm with running time O(k^{2k} n^3 m). We give the first subexponential parameterized algorithm solving Interval Completion in time k^O(sqrt(k)) n^O(1). This adds Interval Completion to a very small list of parameterized graph modi fication problems solvable in subexponential time.

The Weighted Tree Augmentation Problem (WTAP) is a fundamental well-studied problem in the field of network design. Given an undirected tree $G=(V,E)$, an additional set of edges $L \subseteq V\times V$ disjoint from $E$ called \textit{links} and a cost vector $c\in \mathbb{R}_{\geq 0}^L$, WTAP asks to find a minimum-cost set $F\subseteq L$ with the property that $(V,E\cup F)$ is $2$-edge connected. The special case where $c_\ell = 1$ for all $\ell\in L$ is called the Tree Augmentation Problem (TAP). For the class of bounded cost vectors, we present a first improved approximation algorithm for WTAP since more than three decades. Concretely, for any $M\in \mathbb{R}_{\geq 1}$ and $\epsilon > 0$, we present an LP based $(\delta+\epsilon)$-approximation for WTAP restricted to cost vectors $c$ in $[1,M]^L$ for $\delta \approx 1.96417$. More generally, our result is a $(\delta+\epsilon)$-approximation algorithm with running time $n^{r^{O(1)}}$, where $r = c_{\max}/c_{\min}$ is the ratio between the largest and the smallest cost of any link. For the special case of TAP we improve this factor to $\frac{5}{3}+\epsilon$. Our results rely on several new ideas, including a new LP relaxation of WTAP and a two-phase rounding algorithm.

We prove the first non-trivial performance ratio strictly above 0.5 for the weighted Ranking algorithm on the oblivious matching problem where nodes in a general graph can have arbitrary weights. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the weighted and the unweighted versions of the problem. Using a new class of continuous LP, we prove that the ratio for the weighted case is at least 0.501512, and improve the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). Unlike previous continuous LP in which the primal solution must be continuous everywhere, our new continuous LP framework allows the monotone component of the primal function to have jump discontinuities, and the other primal components to take non-conventional forms such as the Dirac delta function.

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefixes of the ordering are non-negative, and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. referred to this problem as the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms. We consider a related problem, which we call the alternating stock size problem, where the numbers of positive and negative integers in the input set S are equal. The problem is the same as above, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79. Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovasz, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is NP-hard to approximate the Gromov-Hausdorff distance better than a factor of 3 for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time O(min{n, (rn)})-approximation algorithm for computing the GH distance between a pair of metric trees, where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an O(n)-approximation algorithm.