Given a set of n non-overlapping geometric objects, can we separate a constant fraction of them using straight-line cuts that extend from edge to edge? In 1996, Urrutia posed this question for compact convex objects. Pach and Tardos later refuted it for general line segments by constructing a family where any separable subfamily has size at most O (n^{log₃ 2}). However, for axis-parallel rectangles, they provided positive evidence, showing that an Ω(1/log n)-fraction can be separated. This problem naturally arises in geometric approximation algorithms. In particular, when restricting cuts to only orthogonal straight lines, known as a guillotine cut sequence, any bound on the separability ratio directly translates into a clean and simple dynamic programming for computing a maximum independent set of geometric objects. This paper focuses on the case when the objects are squares. For squares of arbitrary sizes, an Ω(1)-fraction can be separated (Abed et al., APPROX 2015), recently improved to 1/40 (and 1/160 ≈ 0.62% for the weighted case) (Khan and Pittu, APPROX 2020). We further improve this bound, showing that a 9/256 ≈ 3.51% can be separated for the weighted case. This result significantly narrows the possible range for squares to [3.51%, 50%]. The key to our improvement is a refined analysis of the existing framework.

The fundamental Sparsest Cut problem takes as input a graph G together with edge capacities and demands and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For n -vertex graphs G of treewidth k , Chlamtáč, Krauthgamer, and Raghavendra (APPROX’10) presented an algorithm that yields a factor- \(2^{2^k}\) approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . Later, Gupta, Talwar, and Witmer (STOC’13) showed how to obtain a 2-approximation algorithm with a blown-up runtime of \(n^{O(k)}\) . An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . In this article, we make significant progress towards this goal via the following results: (i) A factor- \(O(k^2)\) approximation that runs in time \(2^{O(k)} \cdot n^{O(1)}\) , directly improving the work of Chlamtáč et al. while keeping the runtime single-exponential in k . (ii) For any \(\varepsilon \in (0,1]\) , a factor- \(O(1/\varepsilon ^2)\) approximation whose runtime is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)}\) , implying a constant-factor approximation whose runtime is nearly single-exponential in k and a factor- \(O(\log ^2 k)\) approximation in time \(k^{O(k)} \cdot n^{O(1)}\) . Key to these results is a new measure of a tree decomposition that we call combinatorial diameter , which may be of independent interest.

In this paper, we study a primal and dual relationship about triangles: For any graph G, let v(G) be the maximum number of edge-disjoint triangles in G, and τ(G) be the minimum subset F of edges such that G \ F is triangle-free. It is easy to see that v(G) ≤ τ(G) ≤ 3v(G), and in fact, this rather obvious inequality holds for a much more general primal-dual relation between k-hyper matching and covering in hypergraphs. Tuza conjectured in 1981 that τ(G) ≤ 2v(G), and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every k ≥ 2, there exist a (multi)-set F ⊆ E(G): |F| ≤ 2kv(G) such that each triangle in G overlaps at least k elements in F. Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in F based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.

Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $ε>0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-ε$ while violating only the covering constraints by a multiplicative factor of $1-ε$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraints as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach that can be applied to several special cases of the problem yielding not only {\em deterministic} but also considerably faster algorithms. For example, for the well studied special case of only packing constraints (Kulik {\em et. al.} [Math. Oper. Res. `13] and Chekuri {\em et. al.} [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.

A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph $G$ contains a cactus subgraph $C$ where $C$ contains at least a $\frac{1}{6}$ fraction of the triangular faces of $G$. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing "dense planar structures" inside any graph: (i) A $\frac{1}{6}$ approximation algorithm for, given any graph $G$, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous $\frac{1}{11}$-approximation; (ii) An alternate (and arguably more illustrative) proof of the $\frac{4}{9}$ approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind.