The strong thin tree conjecture states that every k-edge-connected graph G contains an O(1/k)-thin spanning tree, meaning a spanning tree which contains at most an O(1/k) fraction of the edges across each cut in G. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an O(polylog log n/k)-thin tree. In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of η-near minimum cuts of the graph for η = 1/40, in other words, for the set of cuts with fewer than (1+1/40)k edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Benczúr and Goemans, to reduce to the previously solved problem of finding a spanning tree that is O(1/k)-thin for all sets in a laminar family.

The Weighted Tree Augmentation Problem (WTAP) is a fundamental network design problem where the goal is to find a minimum-cost set of additional edges (links) to make an input tree 2-edge-connected. While a 2-approximation is standard and the integrality gap of the classic Cut LP relaxation is known to be at least 1.5, achieving approximation factors significantly below 2 has proven challenging. Recent advances of Traub and Zenklusen using local search culminated in a ratio of 1.5+є, establishing the state-of-the-art. In this work, we present a randomized approximation algorithm for WTAP with an approximation ratio below 1.49. Our approach is based on designing and rounding a strong linear programming relaxation for WTAP which incorporates variables that represent subsets of edges and the links used to cover them, inspired by lift-and-project methods like Sherali-Adams.

In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter k, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length k. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a k-approximation and showed that it is UGC-Hard to approximate better than ⌊0.5k⌋ for any constant k ≥ 4 using a work of Svensson from 2012. The approximation ratio was improved to 2/3(k+1) by Klein and Wexler in 2016. In this work, we introduce a randomized rounding framework based on distributions over vertex labels in [0,1]. The most natural distribution is to sample labels independently from the uniform distribution over [0,1]. We show this leads to a (2-√2)(k+1) ≈ 0.585(k+1)-approximation. By using a modified (but still independent) label distribution, we obtain a 0.549(k+1)-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below 0.542(k+1). Finally, we show a 0.5(k+1)-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.

We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

We give a poly-time algorithm for the k-edge-connected spanning subgraph (k-ECSS) problem that returns a solution of cost no greater than the cheapest (k+10)-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of 2 for k-ECSS whenever the optimal value of (k+10)-ECSS is close to that of k-ECSS. This is a property that holds for the closely related problem k-edge-connected spanning multi-subgraph (k-ECSM), which is identical to k-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a 1+O(1/k)-approximation algorithm for k-ECSM, which resolves a conjecture of Pritchard and improves upon a recent 1+O(1/√k)-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for k-ECSM, showing that our approximation ratio is tight up to the constant factor in O(1/k), unless P=NP.

In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied degree-bounded spanning tree problem, as well as a previously studied setting where a chain of cuts is given. We give the first constant-factor approximation algorithm; in particular we show how to obtain a multiplicative violation of the crossing bounds of less than 22 while losing less than a factor of 5 in terms of cost. Our result compares to the natural $L P$ relaxation. As a consequence, our results show that given a k-edge-connected graph and a laminar family $\mathcal{L} \subseteq 2^{V}$ of cuts, there exists a spanning tree which contains only an $O(1 / k)$ fraction of the edges across every cut in $\mathcal{L}$. This can be viewed as progress towards the Thin Tree Conjecture, which (in a strong form) states that this guarantee can be obtained for all cuts simultaneously.

A matroid $\mathcal{M}$ on a set $E$ of elements has the $α$-partition property, for some $α>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $α$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $α$-partition property for any constant $α>0$ (independent of $d$). Furthermore, we refute a recent conjecture of Bérczi, Schwarcz, and Yamaguchi by showing the same matroid is $2^d/d$-colorable but cannot be reduced to an $α2^d/d$-colorable partition matroid for any $α$ that is sublinear in $d$.

In this extended abstract, we show that for some $\epsilon>10^{-36}$ and any metric TSP instance, the max entropy algorithm studied by [1] returns a solution of expected cost at most $\frac{3}{2}-\epsilon$ times the cost of the optimal solution to the subtour elimination LP. This implies that the integrality gap of the subtour LP is at most $\frac{3}{2}-\epsilon$. This analysis also shows that there is a randomized $\frac{3}{2}-\epsilon$ approximation for the 2-edge-connected multi-subgraph problem, improving upon Christofides’ algorithm.

We give a randomized 1+5.06/√k-approximation algorithm for the minimum k-edge connected spanning multi-subgraph problem, k-ECSM.

For some > 10−36 we give a randomized 3/2− approximation algorithm for metric TSP.

We design a 1.49993-approximation algorithm for the metric traveling salesperson problem (TSP) for instances in which an optimal solution to the subtour linear programming relaxation is half-integral. These instances received significant attention over the last decade due to a conjecture of Schalekamp, Williamson and van Zuylen stating that half-integral LP solutions have the largest integrality gap over all fractional solutions. So, if the conjecture of Schalekamp et al. holds true, our result shows that the integrality gap of the subtour polytope is bounded away from 3/2.

Issue tracking software of large software projects receive a large volume of issue reports each day. Each of these issues is typically triaged by hand, a time consuming and error prone task. Additionally, issue reporters lack the necessary understanding to know whether their issue has previously been reported. This leads to issue trackers containing a lot of duplicate reports, adding complexity to the triaging task.