Related papers: 3SUM, 3XOR, Triangles
A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is…
Given a weighted graph $G$ and an error parameter $\epsilon > 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_{\epsilon}$ (containing…
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses…
We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the…
Let G be a simple connected plane graph and let C_1 and C_2 be cycles in G bounding distinct faces f_1 and f_2. For a positive integer l, let r(l) denote the number of integers n such that -l<=n<=l, n is divisible by 3, and n has the same…
In this paper, we show that every graph with $m$ edges admits a 3-partition such that \[ \max_{1 \leq i \leq 3} e(V_i) \leq \frac{m}{9} + \frac{1}{9}h(m) \quad \text{and} \quad e(V_1, V_2, V_3) \geq \frac{2}{3}m + \frac{1}{3}h(m), \] where…
We use here the results on the influence graph by Boissonnat et al. to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from…
We give efficient algorithms for finding power-sum decomposition of an input polynomial $P(x)= \sum_{i\leq m} p_i(x)^d$ with component $p_i$s. The case of linear $p_i$s is equivalent to the well-studied tensor decomposition problem while…
In this paper we consider the problem of reconstructing a hidden weighted hypergraph of constant rank using additive queries. We prove the following: Let $G$ be a weighted hidden hypergraph of constant rank with n vertices and $m$…
A triple of vertices in a graph is a \emph{frustrated triangle} if it induces an odd number of edges. We study the set $F_n\subset[0,\binom{n}{3}]$ of possible number of frustrated triangles $f(G)$ in a graph $G$ on $n$ vertices. We prove…
The 3SUM problem represents a class of problems conjectured to require $\Omega (n^2)$ time to solve, where $n$ is the size of the input. Given two polygons $P$ and $Q$ in the plane, we show that some variants of the decision problem,…
We improve on random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time construction that transforms any graph on n vertices into an O(n\log n)-edge graph on the same…
In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let $G$ be a digraph with $m$ edges and $n$ vertices, and let $G\setminus e$ be the digraph obtained after deleting edge $e$ from $G$. As a first…
Derandomization is the process of taking a randomized algorithm and turning it into a deterministic algorithm, which has attracted great attention in classical computing. In quantum computing, it is challenging and intriguing to derandomize…
Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing,…
The area of sublinear algorithms have recently received a lot of attention. In this setting, one has to choose specific access model for the input, as the algorithm does not have time to pre-process or even to see the whole input. A…
In this short note, we give a novel algorithm for $O(1)$ round triangle counting in bounded arboricity graphs. Counting triangles in $O(1)$ rounds (exactly) is listed as one of the interesting remaining open problems in the recent survey of…
The popular 3SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements $x_1, x_2, x_3$ such that $x_1+x_2=x_3$. A closely related problem is to check…
An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular…
We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017]. We show that the LP solution can be sparsified via cut-sparsification…