Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi, Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of metric relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an outerplanar metric.Comment: 27 pages, 4 figires, extended abstract to appear in the proceedings of APPROX-RANDOM 201

A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.Comment: 30 pages, 5 figure

Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators

Since 1997 there has been a steady stream of advances for the maximum disjoint paths problem. Achieving tractable results has usually required focusing on relaxations such as: (i) to allow some bounded edge congestion in solutions, (ii) to only consider the unit weight (cardinality) setting, (iii) to only require fractional routability of the selected demands (the all-or-nothing flow setting). For the general form (no congestion, general weights, integral routing) of edge-disjoint paths ({\sc edp}) even the case of unit capacity trees which are stars generalizes the maximum matching problem for which Edmonds provided an exact algorithm. For general capacitated trees, Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz, Shepherd provided a $4$-approximation. This is essentially the only setting where a constant approximation is known for the general form of \textsc{edp}. We extend their result by giving a constant-factor approximation algorithm for general-form \textsc{edp} in outerplanar graphs. A key component for the algorithm is to find a {\em single-tree} $O(1)$ cut approximator for outerplanar graphs. Previously $O(1)$ cut approximators were only known via distributions on trees and these were based implicitly on the results of Gupta, Newman, Rabinovich and Sinclair for distance tree embeddings combined with results of Anderson and Feige.Comment: 19 pages, 6 figure

A face cover perspective to ℓ1\ell_1 embeddings of planar graphs

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into $\ell_1$ with constant distortion. However, given an $n$-vertex weighted planar graph, the best upper bound on the distortion is only $O(\sqrt{\log n})$, by Rao [SoCG99]. In this paper we study the case where there is a set $K$ of terminals, and the goal is to embed only the terminals into $\ell_1$ with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into $\ell_1$. The more general case, where the set of terminals can be covered by $\gamma$ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of $O(\log \gamma)$ by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to $O(\sqrt{\log\gamma})$. Since every planar graph has at most $O(n)$ faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into $\ell_1$. Therefore, our result provides a polynomial time $O(\sqrt{\log \gamma})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $\gamma$ faces

Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable