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Faster Parameterized Algorithms using Linear Programming
 CoRR
"... We investigate the parameterized complexity ofVertex Cover parameterized by the di↵erence between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining ..."
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We investigate the parameterized complexity ofVertex Cover parameterized by the di↵erence between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O⇤(2.618k) algorithm for the problem. Here k is the excess of the vertex cover size over the LP optimum, and we write O⇤(f(k)) for a time complexity of the form O(f(k)nO(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of O⇤(2.3146k). Following this, using previously known as well as new reductions, we give O⇤(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion andAlmost 2SAT, andO⇤(1.5214k) algorithms forKönig Vertex Deletion andVertex Cover parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal this is the first algorithm which improves upon the dependence on k of the seminal O⇤(3k) algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k c log k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.
Constraint satisfaction problems parameterized above or below tight bounds: A survey
 The Multivariate Algorithmic Revolution and Beyond, volume 7370 of Lecture Notes in Computer Science
, 2012
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Note on Maximal Bisection above Tight Lower Bound
"... In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not har ..."
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Cited by 1 (0 self)
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In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not hard to see that ⌈E/2 ⌉ is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (MaxBisecATLB): decide whether a graph G = (V, E) has a bisection of size at least ⌈E/2 ⌉ + k, where k is the parameter. We show that this parameterized problem has a kernel with O(k²) vertices and O(k³) edges, i.e., every instance of MaxBisecATLB is equivalent to an instance of MaxBisecATLB on a graph with at most O(k²) vertices and O(k³) edges.
Finding small stabilizers for unstable graphs
"... An undirected graph G = (V,E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edgedeletion question ..."
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An undirected graph G = (V,E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edgedeletion question: given a graph G = (V,E), can we find a minimumcardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik [19] we are given an undirected graph G = (V,E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is wellknown to be nonempty iff graph G is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is nonempty. We show that this problem is vertexcover hard. We then prove that there is a minimumcardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.
Transversal, Split Vertex Deletion and Almost 2SAT, and an O∗(1.5214k)
"... We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combini ..."
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We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O∗((2.618)k) algorithm for the problem. Here k is the excess of the vertex cover size over the LP optimum, and we write O∗(f(k)) for a time complexity of the form O(f(k)nO(1)), where f(k) grows exponentially with k. We proceed to show that a more sophisticated branching algorithm achieves a runtime of O∗(2.3146k). Following this, using known and new reductions, we give O∗(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle