Improved Hardness for Cut, Interdiction, and Firefighter Problems

We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). * For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction. * For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut} is hard to approximate within a factor 2 - epsilon, matching the trivial 2-approximation algorithm. * Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm. Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Register Loading via Linear Programming

We study the following optimization problem. The input is a number k and a directed graph with a specified “start ” vertex, each of whose vertices may have one “memory bank requirement”, an integer. There are k “registers”, labeled 1...k. A valid solution associates to the vertices with no bank requirement one or more “load instructions ” L[b,j], for bank b and register j, such that every directed trail from the start vertex to some vertex with bank requirement c contains a vertex u that has been associated L[c,i] (for some register i ≤ k) and no vertex following u in the trail has been associated an L[b,i], for any other bank b. The objective is to minimize the total number of associated load instructions. We give a k(k +1)-approximation algorithm based on linear programming rounding, with (k+1) being the best possible unless Vertex Cover has approximation 2−ǫ for ǫ> 0. We also present a O(klogn) approximation, with n being the number of vertices in the input directed graph. Based on the same linear program, another rounding method outputs a valid solution with objective at most 2k times the optimum for k registers, using 2k−1 registers. This version of the paper corrects some minor errors that made it in the final Algorithmica paper.

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

Approximating Unique Games Using Low Diameter Graph Decomposition

We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games