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Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
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Cited by 32 (6 self)
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Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
Finding small separators in linear time via treewidth reduction
"... We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an ind ..."
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Cited by 17 (1 self)
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We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)and (H,C,≤K)coloring problems as well, which are cardinality constrained variants of the classical Hcoloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.
Static Analysis and Compiler Design for Idempotent Processing
"... Recovery functionality has many applications in computing systems, from speculation recovery in modern microprocessors to fault recovery in highreliability systems. Modern systems commonly recover using checkpoints. However, checkpoints introduce overheads, add complexity, and often save more state ..."
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Cited by 5 (3 self)
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Recovery functionality has many applications in computing systems, from speculation recovery in modern microprocessors to fault recovery in highreliability systems. Modern systems commonly recover using checkpoints. However, checkpoints introduce overheads, add complexity, and often save more state than necessary. This paper develops a novel compiler technique to recover program state without the overheads of explicit checkpoints. The technique breaks programs into idempotent regions—regions that can be freely reexecuted—which allows recovery without checkpointed state. Leveraging the property of idempotence, recovery can be obtained by simple reexecution. We develop static analysis techniques to construct these regions and demonstrate low overheads and large region sizes for an LLVMbased implementation. Across a set of diverse benchmark suites, we construct idempotent regions close in size to those that could be obtained with perfect runtime information. Although the resulting code runs more slowly, typical performance overheads are in the range of just 212%. The paradigm of executing entire programs as a series of idempotent regions we call idempotent processing, and it has many applications in computer systems. As a concrete example, we demonstrate it applied to the problem of compilerautomated hardware fault recovery. In comparison to two other stateoftheart techniques, redundant execution and checkpointlogging, our idempotent processing technique outperforms both by over 15%.
Multicut Algorithms via Tree Decompositions
, 2010
"... Various forms of multicut problems are of great importance in the area of network design. In general, these problems are intractable. However, several parameters have been identified which lead to fixedparameter tractability (FPT). Recently, Gottlob and Lee have proposed the treewidth of the struc ..."
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Cited by 4 (0 self)
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Various forms of multicut problems are of great importance in the area of network design. In general, these problems are intractable. However, several parameters have been identified which lead to fixedparameter tractability (FPT). Recently, Gottlob and Lee have proposed the treewidth of the structure representing the graph and the set of pairs of terminal vertices as one such parameter. In this work, we show how this theoretical FPT result can be turned into efficient algorithms for optimization, counting, and enumeration problems in this area.
On the Parameterized Complexity of Finding Separators with NonHereditary Properties?
"... Abstract. We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomialtime solvable (Tarjan 1985), while for example the problems of finding a minimum s–t separator that induces a connected g ..."
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Abstract. We study the problem of finding small s–t separators that induce graphs having certain properties. It is known that finding a minimum clique s–t separator is polynomialtime solvable (Tarjan 1985), while for example the problems of finding a minimum s–t separator that induces a connected graph or forms an independent set are fixedparameter tractable when parameterized by the size of the separator (Marx, O’Sullivan and Razgon, ACM Trans. Algor., to appear). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on boundeddegree graphs. Our results are as follows: (1) Finding a minimum cconnected s–t separator is FPT for c = 2 and W [1]hard for any c ≥ 3. (2) Finding a minimum s–t separator with diameter at most d is W [1]hard for any d ≥ 2. (3) Finding a minimum rregular s–t separator is W [1]hard for any r ≥ 1. (4) For any decidable graph property, finding a minimum s–t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree. (5) Finding a connected s–t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless NP ⊆ coNP/poly. In order to prove (1), we show that the natural cconnected generalization of the wellknown Steiner Tree problem is FPT for c = 2 and W [1]hard for any c ≥ 3. 1
Computer Sciences Department Compiler Construction of Idempotent Regions
"... Recovery functionality has many applications in computing systems, from speculation recovery in modern microprocessors to fault recovery in highreliability systems. Modern systems commonly recover using checkpoints. However, checkpoints introduce overheads, add complexity, and often conservatively ..."
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Recovery functionality has many applications in computing systems, from speculation recovery in modern microprocessors to fault recovery in highreliability systems. Modern systems commonly recover using checkpoints. However, checkpoints introduce overheads, add complexity, and often conservatively save more state than necessary. This paper develops a compiler technique to recover program state without the overheads of explicit checkpoints. Our technique breaks programs into idempotent regions—regions that can be freely reexecuted—which allows recovery without checkpointed state. Leveraging the property of idempotence, recovery can be obtained by simple reexecution. We develop static analysis techniques to construct these regions in a compiler, and demonstrate low overheads and large region sizes using an LLVMbased implementation. Across a set of diverse benchmark suites, we construct idempotent regions almost as large as those that could be obtained with perfect runtime information. Although the resulting code runs slower, typical execution time overheads are in the range of just 212%. 1
How to Cut a Graph into Many Pieces
, 2011
"... In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We fi ..."
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In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We first provide an overview of the hardness of approximation of this problem, which stems mainly from its close relation to the INDEPENDENT SET and to the MAXIMUM CLIQUE problem. Next, we show that the problem is solvable in polynomial time for interval graphs and graphs of bounded treewidth. We also show that MAXINUM COMPONENTS is fixedparameter tractable on planar graphs with the size of the separator as the parameter. Our main contribution is the derivation of an efficient polynomialtime approximation scheme for the problem on planar graphs.
Restricted vertex multicut on permutation graphs
"... Abstract Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NPcomplete for trees and polynomialtime solvable for interval graphs. In thi ..."
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Abstract Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NPcomplete for trees and polynomialtime solvable for interval graphs. In this paper we give a polynomialtime algorithm for the problem on permutation graphs. Furthermore we show that the problem remains NPcomplete on split graphs whereas it becomes polynomialtime solvable for the class of cobipartite graphs.