Andreas Emil Feldmann

Fast balanced partitioning is hard even on grids and trees

Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfactory approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is unavoidable. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a general reduction which identifies some sufficient conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite two-dimensional grid without holes. We apply the reduction to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfactory ratio. We also consider solutions in which the sets may deviate from being equal-sized. Our reduction is applied to grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase when the limit on the set sizes decreases. These are the first bicriteria inapproximability results for the k-BALANCED PARTITIONING problem.

An O(n 4 ) time algorithm to compute the bisection width of solid grid graphs

The bisection problem asks for a partition of the n vertices of a graph into two sets of size at most ⌈n/2⌉, so that the number of edges connecting the two sets is minimised. A grid graph is a finite connected subgraph of the infinite two-dimensional grid. It is called solid if it has no holes. Papadimitriou and Sideri [8] gave an O(n5) time algorithm to solve the bisection problem on solid grid graphs. We propose a novel approach that exploits structural properties of optimal cuts within a dynamic program. We show that our new technique leads to an O(n4) time algorithm.

Restricted cuts for bisections in solid grids: a proof via polygons

The graph bisection problem asks to partition the n vertices of a graph into two sets of equal size so that the number of edges across the cut is minimum. We study finite, connected subgraphs of the infinite two-dimensional grid that do not have holes. Since bisection is an intricate problem, our interest is in the tradeoff between runtime and solution quality that we get by limiting ourselves to a special type of cut, namely cuts with at most one bend each (corner cuts). We prove that optimum corner cuts get us arbitrarily close to equal sized parts, and that this limitation makes us lose only a constant factor in the quality of the solution. We obtain our result by a thorough study of cuts in polygons and the effect of limiting these to corner cuts.

Computing Approximate Nash Equilibria in Network Congestion Games

We consider the problem of computing i¾?-approximate Nash equilibria in network congestion games. The general problem is known to be PLS -complete for every i¾?> 0, but the reductions are based on artificial and steep delay functions with the property that already two players using the same resource cause a delay that is significantly larger than the delay for a single player. We consider network congestion games with delay functions such as polynomials, exponential functions, and functions from queuing theory. We analyse which approximation guarantees can be achieved for such congestion games by the method of randomised rounding. Our results show that the success of this method depends on different criteria depending on the class of functions considered. For example, queuing theoretical functions admit good approximations if the equilibrium load of every resource is bounded away appropriately from its capacity.

Computing approximate Nash equilibria in network congestion games

We consider the problem of computing e -approximate Nash equilibria in network congestion games. The general problem is known to be PLS-complete for every e > 0, but the reductions are based on artificial and steep delay functions with the property that already two players using the same resource cause a delay that is significantly larger than the delay for a single player. We consider network congestion games with delay functions such as polynomials, exponential functions, and functions from queuing theory. We analyse which approximation guarantees can be achieved for such congestion games by the method of randomized rounding. Our results show that the success of this method depends on different criteria depending on the class of functions considered. For example, queuing theoretical functions admit good approximations if the equilibrium load of every resource is bounded away appropriately from its capacity.

Simple cuts are fast and good: optimum right-angled cuts in solid grids

We consider the problem of bisecting a graph, i.e. cutting it into two equally sized parts while minimising the number of cut edges. In its most general form the problem is known to be NP-hard. Several papers study the complexity of the problem when restricting the set of considered graphs. We attempt to study the effects of restricting the allowed cuts. We present an algorithm that bisects a solid grid, i.e. a connected subgraph of the infinite two-dimensional grid without holes, using only cuts that correspond to a straight line or a right angled corner. It was shown in [13] that an optimal bisection for solid grids with n vertices can be computed in O(n5) time. Restricting the cuts in the proposed way we are able to improve the running time to O(n4). We prove that these restricted cuts still yield good solutions to the original problem: The best restricted cut is a bicriteria approximation to an optimal bisection w.r.t. both the differences in the sizes of the partitions and the number of edges that are cut.

The Parameterized Hardness of the k-Center Problem in Transportation Networks

In this paper we study the hardness of the

Balanced Partitions of Trees and Applications

k

An O(n^4) time algorithm to compute the bisection width of solid grid graphs

-Center problem on inputs that model transportation networks. For the problem, a graph

Balanced partitions of trees and applications

G=(V,E)