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We introduce and study a novel generalization of the classical Knapsack Problem (KP), called the Colored Knapsack Problem (CKP). In this problem, the items are partitioned into classes of colors and the packed items need to be ordered such…
We present a polynomial-space algorithm that computes the number independent sets of any input graph in time $O(1.1387^n)$ for graphs with maximum degree 3 and in time $O(1.2355^n)$ for general graphs, where n is the number of vertices.…
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this…
A probabilistic version of the Weisfeiler-Leman algorithm for computing the coherent closure of a colored graph is suggested. The algorithm is Monte Carlo and runs in time $ O(n^{1+\omega}\log^2 n) $, where $ n $ is the number of vertices…
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised…
Consider an n-vertex graph G = (V,E) of maximum degree Delta, and suppose that each vertex v \in V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it…
For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in \mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the probability that a point $q$ is contained inside a random closed hyper-ball obtained from a pair…
Monocular depth estimation is often described as an ill-posed and inherently ambiguous problem. Estimating depth from 2D images is a crucial step in scene reconstruction, 3Dobject recognition, segmentation, and detection. The problem can be…
The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree…
We consider the problem of finding a subgraph of a given graph which maximizes a given function evaluated at its degree sequence. While the problem is intractable already for convex functions, we show that it can be solved in polynomial…
The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$,…
We study dynamic conflict-free colorings in the plane, where the goal is to maintain a conflict-free coloring (CF-coloring for short) under insertions and deletions. - First we consider CF-colorings of a set $S$ of unit squares with respect…
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient…
Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves…
Depth measures quantify central tendency in the analysis of statistical and geometric data. Selecting a depth measure that is simple and efficiently computable is often important, e.g., when calculating depth for multiple query points or…
Time-of-flight cameras provide depth information, which is complementary to the photometric appearance of the scene in ordinary images. It is desirable to merge the depth and colour information, in order to obtain a coherent scene…
Any graph with maximum degree $\Delta$ admits a proper vertex coloring with $\Delta + 1$ colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm?…
We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with…
In this article, we consider colorable variations of the Unit Disk Cover ({\it UDC}) problem as follows. {\it $k$-Colorable Discrete Unit Disk Cover ({\it $k$-CDUDC})}: Given a set $P$ of $n$ points, and a set $D$ of $m$ unit disks (of…
We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum…