KAIST Discrete Math Seminar

Approximation algorithms and fixed-parameter tractable (FPT) algorithms have been two major ways to deal with NP-hardness of combinatorial optimization problems. The notion of FPT approximation can be naturally defined, and it is getting significant attention recently. Starting from gentle introductions to approximation algorithms and FPT algorithms, I will talk about my three recent results on FPT approximability.
– Given a graph G = (V, E) and an integer k, we study k-Vertex Separator, where the goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most k vertices. We give an O(log k)-FPT approximation algorithm for k-Vertex Separator. Our result improves the best previous graph partitioning algorithms.
– We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. We present an O(log k)-FPT approximation algorithm for k-Path Transversal. There was no nontrivial approximation algorithm for k > 4 before this work.
– Finally, k-cut is the problem where we want to remove the minimum number of edges such that the graph has at least k connected components. We give a (2 – ε)-FPT approximation algorithm for some epsilon > 0, improving upon a (non-FPT) 2-approximation.
The third result is joint work with Anupam Gupta and Jason Li.

When network users are satisficing decision-makers, the resulting traffic pattern attains a satisficing user equilibrium, which may deviate from the (perfectly rational) user equilibrium. In a satisficing user equilibrium traffic pattern, the total system travel time can be worse than in the case of the PRUE. We show how bad the worst-case satisficing user equilibrium traffic pattern can be, compared to the perfectly rational user equilibrium. We call the ratio between the total system travel times of the two traffic patterns the price of satisficing, for which we provide an analytical bound. Using the sensitivity analysis for variational inequalities, we propose a numerical method to quantify the price of satisficing for any given network instance.

We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC 2008]. We say that a class ? of graphs admits low rank-width colorings if there exist functions N:ℕ→ℕ and Q:ℕ→ℕ such that for all p∈ℕ, every graph G∈? can be vertex colored with at most N(p) colors such that the union of any i≤p color classes induces a subgraph of rank-width at most Q(i).
Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class ? of bounded expansion and every positive integer r, the class {G^r: G∈?} of r-th powers of graphs from ?, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. In this talk, we provide the color refinement technique necessary to show the first result. This is joint work with Sebastian Sierbertz and Michał Pilipczuk.