Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P

Approximation algorithms for a graph-cut problem with applications to a clustering problem in bioinformatics

xiii, 71 leaves : ill. ; 29 cm.Clusters in protein interaction networks can potentially help identify functional relationships among proteins. We study the clustering problem by modeling it as graph cut problems. Given an edge weighted graph, the goal is to partition the graph into a prescribed number of subsets obeying some capacity constraints, so as to maximize the total weight of the edges that are within a subset. Identification of a dense subset might shed some light on the biological function of all the proteins in the subset. We study integer programming formulations and exhibit large integrality gaps for various formulations. This is indicative of the difficulty in obtaining constant factor approximation algorithms using the primal-dual schema. We propose three approximation algorithms for the problem. We evaluate the algorithms on the database of interacting proteins and on randomly generated graphs. Our experiments show that the algorithms are fast and have good performance ratio in practice

Linear-Time FPT Algorithms via Network Flow

In the area of parameterized complexity, to cope with NP-Hard problems, we introduce a parameter k besides the input size n, and we aim to design algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some function f(k) and constant d. Though FPT algorithms have been successfully designed for many problems, typically they are not sufficiently fast because of huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d for many important problems including Odd Cycle Transversal and Almost 2-SAT. More specifically, we can choose f(k) as a single exponential (4^k) and d as one, that is, linear in the input size. To the best of our knowledge, our algorithms achieve linear time complexity for the first time for these problems. To obtain our algorithms for these problems, we consider a large class of integer programs, called BIP2. Then we show that, in linear time, we can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can compute an optimal LP solution for Vertex Cover Above LP using network flow. Then, we perform an exhaustive search by fixing half-integral values in the optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we need to recompute an LP optimal solution after branching. To address this issue, we exploit network flow to update the optimal LP solution in linear time.Comment: 20 page

Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time \poly(n) 2^{O(k)} due to Chlamt\'a\v{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a $\rho$-approximation for series-parallel graphs (where $\rho \geq 1$), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to $1/\rho$. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than $17/16 - \epsilon$ for $\epsilon > 0$; assuming the Unique Games Conjecture the hardness becomes $1/\alpha_{GW} - \epsilon$. For graphs with large (but constant) treewidth, we show a hardness result of $2 - \epsilon$ assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation

Inference of single-cell phylogenies from lineage tracing data using Cassiopeia.

The pairing of CRISPR/Cas9-based gene editing with massively parallel single-cell readouts now enables large-scale lineage tracing. However, the rapid growth in complexity of data from these assays has outpaced our ability to accurately infer phylogenetic relationships. First, we introduce Cassiopeia-a suite of scalable maximum parsimony approaches for tree reconstruction. Second, we provide a simulation framework for evaluating algorithms and exploring lineage tracer design principles. Finally, we generate the most complex experimental lineage tracing dataset to date, 34,557 human cells continuously traced over 15 generations, and use it for benchmarking phylogenetic inference approaches. We show that Cassiopeia outperforms traditional methods by several metrics and under a wide variety of parameter regimes, and provide insight into the principles for the design of improved Cas9-enabled recorders. Together, these should broadly enable large-scale mammalian lineage tracing efforts. Cassiopeia and its benchmarking resources are publicly available at www.github.com/YosefLab/Cassiopeia

Constant Factor Approximation for Balanced Cut in the PIE model

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation-invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters $L$ and $R$ of equal size. Let $G$ be an arbitrary graph on $V$ with no edges between $L$ and $R$. Let $E_{random}$ be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in $L$ and in $R$). Then we say that $G + E_{random}$ is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost $O(|E_{random}|) + n \text{polylog}(n)$ in this model. In the regime when $|E_{random}| = \Omega(n \text{polylog}(n))$, this is a constant factor approximation with respect to the cost of the planted cut.Comment: Full version of the paper at the 46th ACM Symposium on the Theory of Computing (STOC 2014). 32 page

Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.Comment: Approx 201

Balanced Partitions of Trees and Applications

We study the k-BALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ceil(n/k) while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it cannot be approximated within any factor in polynomial time. However, little is known about restricted graph classes. We show that for trees k-BALANCED PARTITIONING remains surprisingly hard. In particular, approximating the cut size is APX-hard even if the maximum degree of the tree is constant. If instead the diameter of the tree is bounded by a constant, we show that it is NP-hard to approximate the cut size within n^c, for any constant c<1. In the face of the hardness results, we show that allowing near-balanced solutions, in which there are at most (1+eps)ceil(n/k) vertices in any of the k sets, admits a PTAS for trees. Remarkably, the computed cut size is no larger than that of an optimal balanced solution. In the final section of our paper, we harness results on embedding graph metrics into tree metrics to extend our PTAS for trees to general graphs. In addition to being conceptually simpler and easier to analyse, our scheme improves the best factor known on the cut size of near-balanced solutions from O(log^{1.5}(n)/eps^2) [Andreev and Räcke TCS 2006] to 0(log n), for weighted graphs. This also settles a question posed by Andreev and Räcke of whether an algorithm with approximation guarantees on the cut size independent from eps exists.ISSN:1868-896

Minimum Makespan Multi-vehicle Dial-a-Ride

Dial a ride problems consist of a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs, where each object requires to be moved from its source to destination vertex. We consider the multi-vehicle Dial a ride problem, with each vehicle having capacity k and its own depot-vertex, where the objective is to minimize the maximum completion time (makespan) of the vehicles. We study the "preemptive" version of the problem, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log^3 n)-approximation algorithm for preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint. We also show that the approximation ratios improve by a log-factor when the underlying metric is induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200