Related papers: 3SUM, 3XOR, Triangles
Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural "Axial" and "Planar" versions,…
The 3-domatic number problem asks whether a given graph can be partitioned intothree dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695^n (up to polynomial factors). This result improves the…
The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized…
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in…
We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from…
We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested Clique model, where $n$ fully connected nodes communicate synchronously using $O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of…
We study the problem of finding large cuts in $d$-regular triangle-free graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a cut of expected size $(1/2 + 0.177/\sqrt{d})m$, where $m$ is the number of edges. We…
We present a deterministic algorithm which, given a graph G with n vertices and an integer 1<m < n, computes in n^{O(ln m)} time the sum of weights w(S) over all m-subsets S of the set of vertices of G, where w(S)=exp{gamma t m +O(1/m)}…
A simplicial vertex of a graph is a vertex whose neighborhood is a clique. It is known that listing all simplicial vertices can be done in $O(nm)$ time or $O(n^{\omega})$ time, where $O(n^{\omega})$ is the time needed to perform a fast…
Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of…
We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although it is known that this can be achieved in polynomial time for any graph, the algorithm…
We show that in the quantum query model the complexity of detecting a triangle in an undirected graph on $n$ nodes can be done using $O(n^{1+{3\over 7}}\log^{2}n)$ quantum queries. The same complexity bound applies for outputting the…
Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of…
We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1. Is the 3SUM problem on $n$…
In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and…
We provide an algorithm that, given an $n$-vertex $m$-edge Eulerian graph with polynomially bounded weights, computes an $\breve{O}(n\log^{2} n \cdot \varepsilon^{-2})$-edge $\varepsilon$-approximate Eulerian sparsifier with high…
We consider the problem of finding all allowed edges in a bipartite graph $G=(V,E)$, i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this…
It has been shown that for a certain special type of quantum graphs the random-matrix form factor can be recovered to at least third order in the scaled time \tau using periodic-orbit theory. Two types of contributing pairs of orbits were…
In the 3SUM-Indexing problem the goal is to preprocess two lists of elements from $U$, $A=(a_1,a_2,\ldots,a_n)$ and $B=(b_1,b_2,...,b_n)$, such that given an element $c\in U$ one can quickly determine whether there exists a pair $(a,b)\in A…
We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a…