Maximum Edge-Disjoint Paths in kk-sums of Graphs

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be $\Omega(\sqrt{n})$ even for planar graphs due to a simple topological obstruction and a major focus, following earlier work, has been understanding the gap if some constant congestion is allowed. In this context, it is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is nor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to $k$-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of $k$-sums of bounded genus graphs

The Complexity of Scheduling for p-norms of Flow and Stretch

We consider computing optimal k-norm preemptive schedules of jobs that arrive over time. In particular, we show that computing the optimal k-norm of flow schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity). Further we show that computing the optimal k-norm of stretch schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity).Comment: Conference version accepted to IPCO 201

Імунологічний скринінг вагітних на наявність гострого токсоплазмозу і ЦМВ-інфекції в Донецькій області

Захворюваність вагітних і новонароджених TORCH-інфекціями в останні роки набула епідемічного характеру і зберігає тенденцію до зростання. Це зумовлено підвищенням інфікованості жінок фертильного віку збудниками, здатними викликати внутрішньоутробні інфекції плода. За статистикою, під час вагітності токсоплазмозом заражається не більше 1% жінок, з яких 20% передають токсоплазмоз плоду. Нажаль, існуючі профілактичні та протиепідемічні заходи в сучасних екологічних умовах не забезпечують стабілізації і зниження захворюваності вагітних і новонароджених

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles with possibly different speeds, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot

Approximating Node-Weighted k-MST on Planar Graphs

We study the problem of finding a minimum weight connected subgraph spanning at least $k$ vertices on planar, node-weighted graphs. We give a (4+\eps)-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual $3$-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math.\ Prog.\ '06). More generally, our approach readily gives a (\nicefrac{4}{3}\cdot r+\eps)-approximation on any graph class where the algorithm of Byrka et al.\ for the prize-collecting version gives an $r$-approximation. We argue that this can be interpreted as a generalization of an analogous result by K\"onemann et al. (Algorithmica~'11) for partial cover problems. Together with a lower bound construction by Mestre (STACS'08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al.\ in a \emph{non-black-box} fashion could not beat the factor \nicefrac{4}{3}\cdot r when the tree merging step relies only on the solutions output by the LMP algorithm

Four Soviets walk the dog, with an application to Alt's conjecture

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n^2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard.In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n^2 \sqrt log n (log log n)^{3/2}) on a pointer machine and in time O(n^2 (log log n)^2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n^{2¿}), for some ¿ > 0. This provides evidence that the decision problem may not be 3SUM-hard after all and reveals an intriguing new aspect of this well-studied problem

Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach

Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.Comment: 17 pages, Conference version to appear in ESA 201

Scheduling over Scenarios on Two Machines

We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all possible scenarios. Each scenario is a subset of jobs that must be executed in that scenario and all scenarios are given explicitly. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and lower bounds on their approximability. With this research into optimization problems over scenarios, we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at link.springer.co