Max-Cut and Max-Bisection are NP-hard on unit disk graphs

We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that $\lambda$-precision graphs are planar for $\lambda$ > 1 / \sqrt{2}$

Linear orderings of random geometric graphs (extended abstract)

In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.Postprint (published version

Optimization in Geometric Graphs: Complexity and Approximation

We consider several related problems arising in geometric graphs. In particular, we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact and approximate solutions. In addition, we establish complexity-based theoretical justifications for several greedy heuristics. Unit ball graphs, which are defined in the three dimensional Euclidian space, have several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks involves several decision problems that can be reduced to well known optimization problems in graph theory. For instance, the notion of a \virtual backbone" in a wire- less network is strongly related to a minimum connected dominating set in its graph theoretic representation. Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition, max-cut and min-bisection. Although these problems have been widely studied in the context of unit disk graphs, which are the two dimensional version of unit ball graphs, there is no established result on the complexity and approximation status for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for several problems using geometrical properties of these graphs. We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through computational tests. We propose using minimum bottleneck connected dominating set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem is NP-hard in general graphs but solvable in polynomial time in unit disk and unit ball graphs. We also demonstrate work on theoretical foundations for simple greedy heuristics. Particularly, similar to the notion of "best" approximation algorithms with respect to their approximation ratios, we prove that several simple greedy heuristics are "best" in the sense that it is NP-hard to recognize the gap between the greedy solution and the optimal solution. We show results for several well known problems such as maximum clique, maximum independent set, minimum vertex coloring and discuss extensions of these results to a more general class of problems. In addition, we propose a "worst-out" heuristic based on edge contractions for the max-cut problem and provide analytical and experimental comparisons with a well known "best-in" approach and its modified versions

A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k=0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixed-parameter tractable algorithms. That is, we show that the problem admits a 2^{O(sqrt{k} log k)}n^{O(1)} time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n,k,k^2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments

Studies in Efficient Discrete Algorithms

This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity