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Bidimensionality and EPTAS
"... Bidimensionality theory appears to be a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] e ..."
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Bidimensionality theory appears to be a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al [SODA 2010] established a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. One of the important conditions required in the framework developed by Demaine and Hajiaghayi [SODA 2005] is that to obtain an EPTAS for a graph optimization problem Π, we have to know a constantfactor approximation algorithm for Π. Our approach eliminates this strong requirement, which makes it amenable to more problems. At the heart of our framework is a decomposition lemma which states that for “most ” bidimensional problems, there is a polynomial time algorithm which given an Hminorfree graph G as input and an ɛ> 0 outputs a vertex set X of size ɛ · OP T such that the treewidth of G \ X is O(1/ɛ). Here, OP T is the objective function value of the problem in question This allows us to obtain EPTASs on (apex)minorfree graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including Cycle Packing, VertexH
Representative Families: A Unified TradeoffBased Approach
"... Abstract. LetM=(E, I) be a matroid, and let S be a family of subsets of size p of E. A subfamily S ̂ ⊆ S represents S if for every pair of sets X ∈S and Y ⊆E\X such that X∪Y ∈ I, there is a set X ̂ ∈ S ̂ disjoint from Y such that X̂∪Y ∈ I. Fomin et al. (Proc. ACMSIAM Symposium on Discrete Algori ..."
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Abstract. LetM=(E, I) be a matroid, and let S be a family of subsets of size p of E. A subfamily S ̂ ⊆ S represents S if for every pair of sets X ∈S and Y ⊆E\X such that X∪Y ∈ I, there is a set X ̂ ∈ S ̂ disjoint from Y such that X̂∪Y ∈ I. Fomin et al. (Proc. ACMSIAM Symposium on Discrete Algorithms, 2014) introduced a powerful technique for fast computation of representative families for uniform matroids. In this paper, we show that this technique leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes, among others, kPartial Cover, kInternal OutBranching, and Long Directed Cycle. Our approach exploits an interesting tradeoff between running time and the size of the representative families. 1
Faster computation of representative families for uniform matroids with applications
 CoRR
"... Abstract. LetM=(E, I) be a matroid, and let S be a family of subsets of size p of E. A subfamily S ̂ ⊆ S represents S if for every pair of sets X ∈S and Y ⊆E\X such that X∪Y ∈ I, there is a set X ̂ ∈ S ̂ disjoint from Y such that X̂∪Y ∈I. In this paper, we present a fast computation of representat ..."
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Abstract. LetM=(E, I) be a matroid, and let S be a family of subsets of size p of E. A subfamily S ̂ ⊆ S represents S if for every pair of sets X ∈S and Y ⊆E\X such that X∪Y ∈ I, there is a set X ̂ ∈ S ̂ disjoint from Y such that X̂∪Y ∈I. In this paper, we present a fast computation of representative families for uniform matroids. We use our computation to develop deterministic algorithms that solve kPartial Cover and kInternal OutBranching in times O∗(2.619k) and O∗(6.855k), respectively. We thus significantly improve the best known randomized algorithm for kPartial Cover and deterministic algorithm for kInternal OutBranching, that run in times O∗(5.437k) and O∗(16k+o(k)), respectively. Finally, we improve the running times of several algorithms that rely on efficient computation of representative families. 1
Parameterized complexity of edge interdiction problems
 In Proc. 20th COCOON, volume 8591 of LNCS
, 2014
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Planar kPath in Subexponential Time and Polynomial Space
"... In the kPath problem we are given an nvertex graph G together with an integer k and asked whether G contains a path of length k as a subgraph. We give the first subexponential time, polynomial space parameterized algorithm for kPath on planar graphs, and more generally, on Hminorfree graphs. ..."
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In the kPath problem we are given an nvertex graph G together with an integer k and asked whether G contains a path of length k as a subgraph. We give the first subexponential time, polynomial space parameterized algorithm for kPath on planar graphs, and more generally, on Hminorfree graphs. The running time of our algorithm is O(2 O( √ k log 2 k) n O(1)
A FixedParameter Algorithm for Max Edge
"... Abstract. In a graph, an edge is said to dominate itself and its adjacent edges. Given an undirected and edgeweighted graph G = (V,E) and an integer k, Max Edge Domination problem (MaxED) is to find a subset K ⊆ E with cardinality at most k such that total weight of edges dominated by K is maximiz ..."
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Abstract. In a graph, an edge is said to dominate itself and its adjacent edges. Given an undirected and edgeweighted graph G = (V,E) and an integer k, Max Edge Domination problem (MaxED) is to find a subset K ⊆ E with cardinality at most k such that total weight of edges dominated by K is maximized. MaxED is NPhard due to the NPhardness of the minimum edge dominating set problem. In this paper, we present fixedparameter algorithms for MaxED with respect to treewidth ω. We first present an O(3ω ·k ·n ·(k+ω2))time algorithm. This algorithm enables us to design a subexponential fixedparameter algorithm of MaxED for apexminorfree graphs, which is a graph class that includes planar graphs.
Graph Minors and Parameterized Algorithm Design
"... Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic ..."
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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic consequences, we present the algorithmic applications of core theorems such as the gridexclusion theorem, and we give a brief description of the irrelevant vertex technique.