Related papers: Optimal FPT-Approximability for Modular Linear Equ…
A property $\Pi$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $\Pi$, every superset $Y \subseteq U$ of $X$ also satisfies $\Pi$. Many combinatorial properties can be seen as monotone properties. The problem…
In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable)…
The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{\Delta,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $\Delta$ and…
The problem of maximizing the $p$-th power of a $p$-norm over a halfspace-presented polytope in $\R^d$ is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in 1986 that this problem…
Consider a real line equipped with a (not necessarily intrinsic) distance. We deal with the minimum-weight perfect matching problem for a complete graph whose points are located on the line and whose edges have weights equal to distances…
We consider the problem of approximating a multiple-input multiple-output (MIMO) $p\times m$ rational transfer function $H(s)$ of high degree by another $p\times m$ rational transfer function $\hat H(s)$ of much smaller degree, so that the…
We consider integer programming problems in standard form $\max \{c^Tx : Ax = b, \, x\geq 0, \, x \in Z^n\}$ where $A \in Z^{m \times n}$, $b \in Z^m$ and $c \in Z^n$. We show that such an integer program can be solved in time $(m…
Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in…
Submodular function minimization (SFM) is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint…
Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…
We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as $\mathbb{N}$, where the relations are defined via first-order formulas whose only predicate is $=$.…
Nowadays, frequent pattern mining (FPM) on large graphs receives increasing attention, since it is crucial to a variety of applications, e.g., social analysis. Informally, the FPM problem is defined as finding all the patterns in a large…
In this paper, we investigate a class of submodular problems which in general are very hard. These include minimizing a submodular cost function under combinatorial constraints, which include cuts, matchings, paths, etc., optimizing a…
This work, for the first time, introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular maximization over a ground set of size $n$ subject to a knapsack constraint, $\mathsf{DLA}$ and…
We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let $a,n,m$ be positive integers and suppose $m$ factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters $t_i\in [1,…
Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…
For many optimization problems in machine learning, finding an optimal solution is computationally intractable and we seek algorithms that perform well in practice. Since computational intractability often results from pathological…
We study a two-dimensional generalization of the classical Bin Packing problem, denoted as 2D Demand Bin Packing. In this context, each bin is a horizontal timeline, and rectangular tasks (representing electric appliances or computational…
We show how to reduce a general, strictly-feasible LP problem, into a min-max problem, which can be solved by the algorithm from the third section of my thesis.
We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. While it is widely known that greedy methods work…