Abstract. We present a [Formula: see text]-approximation algorithm for the matching augmentation problem (MAP): given a multigraph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A [Formula: see text]-approximation algorithm for the same problem was presented recently; see Cheriyan et al. [ Math. Program., 182 (2020), pp. 315–354]. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

Given a connected undirected graph $\overline{G}$ on $n$ vertices and nonnegative edge costs $c$, the $\ensuremath{{2ECM}}$ problem is that of finding a 2-edge connected spanning multisubgraph of $\overline{G}$ of minimum cost. The natural linear program (LP) for $\ensuremath{{2ECM}}$, which coincides with the subtour LP for the traveling salesman problem on the metric closure of $\overline{G}$, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution $x$, Carr and Ravi (1998) showed that the integrality gap is at most $\frac43$: they show that the vector $\frac43 x$ dominates a convex combination of incidence vectors of 2-edge connected spanning multisubgraphs of $\overline{G}$. We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lovász's splitting-off theorem. Our proof naturally leads to a $\frac43$-approximation algorithm for half-integral instances. Given a half-integral solution $x$ to the LP for $\ensuremath{{2ECM}}$, we give an $O(n^2)$-time algorithm to obtain a 2-edge connected spanning multisubgraph of $\overline{G}$ with cost at most $\frac43 c^T x$. We also consider a related problem of finding a cheap 2-edge connected spanning subgraph of a 3-regular, 3-edge connected graph $G = (V,E)$ with arbitrary edge costs $c$. We give a polynomial-time Las Vegas algorithm that finds a random 2-edge connected spanning subgraph $H$ of $G$ whose expected cost, $\mathbb{E}\left[{c(H)}\right]$, is at most $\frac45 c(E)$.

We present a $\frac53$-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A $\frac74$-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a $\frac{7}{4}$-approximation algorithm," {\em Math. Program.}, 182(1):315--354, 2020; arXiv:1810.07816. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in O(nm) time, which improves on the best known time complexity of O(n3) and offers practical improvements for sparse systems with small values of m. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of , where m+ is the number of unique fill edges and original edges that the algorithm encounters and Δ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in O(nm1 – ∊) time, for any ∊ > 0, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called U-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.

Given a connected undirected graph $\bar{G}$ on $n$ vertices, and non-negative edge costs $c$, the 2ECM problem is that of finding a $2$-edge~connected spanning multisubgraph of $\bar{G}$ of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of $\bar{G}$, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution $x$, Carr and Ravi (1998) showed that the integrality gap is at most $\frac43$: they show that the vector $\frac43 x$ dominates a convex combination of incidence vectors of $2$-edge connected spanning multisubgraphs of $\bar{G}$. We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lov\'{a}sz's splitting-off theorem. Our proof naturally leads to a $\frac43$-approximation algorithm for half-integral instances. Given a half-integral solution $x$ to the LP for 2ECM, we give an $O(n^2)$-time algorithm to obtain a $2$-edge connected spanning multisubgraph of $\bar{G}$ whose cost is at most $\frac43 c^T x$.

This Research Full Paper presents a qualitative interview and descriptive study on computing resilient identity development of African Americans who have earned a Ph.D. in a computing field. Low sense of belonging and self-efficacy contributes to low participation and performance of African Americans have lower participation and performance in computing as compared to their White and Asian counterparts. Computing identity including sense of belonging and self-efficacy contributes to this deficit. To increase African American successful representation in computing, resilience is explored to identify the support systems, challenges, and coping processes of African Americans who have earned a Ph.D. in computing. In depth, semi-structured interviews of African American post-docs, faculty, and industry researchers in computing fields were implemented. Interviews were audio recorded and transcribed. Transcriptions were analyzed with a hybrid inductive-deductive qualitative content analysis. Surveys were employed to document participants' work resilience and personality to supplement the qualitative data. Results include the resilient identity development of participants by reciting background information, challenges and support systems in their respective employment, and how they react to such stressors and supports, and the productive they are while persevering. Findings from this work can be used to improve academia and industry conditions for African American professionals and to identify resources that were suggested to be pivotal in participants' resilient identity development within the computing field. This paper is in conjunction with other papers in an extended case study on resilient identity development in African American computer scientists.

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence---increasing or decreasing---has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as an $x$-monotone polygonal path for which every maximal subpath, with positive- or negative-slope edges, has at least three vertices. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length (not necessarily contiguous) subsequence that is a rollercoaster. It was conjectured that every sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, while the best known lower bound is $\Omega(n/\log n)$. In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the $O(n\log n)$-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of $\{1,\dots,n\}$ can be computed in $O(n\log\log n)$ time. The search for rollercoasters was motivated by the orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is an embedded caterpillar where every vertex has degree either 4 or 1 and such that the two leaves adjacent to each spine vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every $n$-vertex top-view caterpillar on every set of $\frac{25}{3}(n+4)$ points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is $O(n\log n)$. We also show that such a drawing can be obtained in linear time when the points are given in sorted order.

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three points. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length subsequence that is a rollercoaster. It was conjectured that every sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, while the best known lower bound is $Ω(n/\log n)$. In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the $O(n\log n)$-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of $\{1,\dots,n\}$ can be computed in $O(n \log \log n)$ time. The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every $n$-node top-view caterpillar on every set of $\frac{25}{3}n$ points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is $O(n \log n)$. We also show that such a drawing can be obtained in linear time, provided that the points are given in sorted order.