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Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
Planarization of Graphs Embedded on Surfaces
 in WG
, 1995
"... A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an nvertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar res ..."
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A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an nvertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genusg embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Deleting Vertices To Bound Path Length
 IEEE Transation on Computers
, 1998
"... We examine the vertex deletion problem for weighted directed acyclic graphs (wdags). The objective is to delete the fewest number of vertices so that the resulting wdag has no path of length > d. Several simplified versions of this problem are shown to be NPhard. However, the problem is solved in l ..."
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We examine the vertex deletion problem for weighted directed acyclic graphs (wdags). The objective is to delete the fewest number of vertices so that the resulting wdag has no path of length > d. Several simplified versions of this problem are shown to be NPhard. However, the problem is solved in linear time when the wdag is a rooted tree and in quadratic time when the wdag is a seriesparallel graph. Keywords And Phrases Vertex deletion, directed acyclic graphs, rooted trees, seriesparallel graphs, NPhard __________________ + Research supported, in part, by the National Science Foundation under grants DCR8420935 and MIPS8617374. ++ Research supported, in part, by SDIO/IST Contract No. N0001490J1793 managed by US Office of Naval Research.   2 1 Introduction A variety of vertex deletion problems formulated on graphs and digraphs are known to be NPhard [KRIS79]. In this paper, we propose a new formulation of the vertex deletion problem that is applicable to edge weighte...
The Complexity of Some Problems on Very Sparse Graphs
, 1997
"... We study the complexity of the problems Dominating Set, Max Cut, Vertex Feedback Set, Steiner Tree, Hamiltonian Circuit, and Chromatic Index on graphs G of bounded maximum degree and large girth. All results are essentially best possible. We also construct regular classone graphs of large girth and ..."
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We study the complexity of the problems Dominating Set, Max Cut, Vertex Feedback Set, Steiner Tree, Hamiltonian Circuit, and Chromatic Index on graphs G of bounded maximum degree and large girth. All results are essentially best possible. We also construct regular classone graphs of large girth and small order. Finally, we point out how vertex resp. edge feedback sets of size O(log n) can be used to solve Max Cut, Independent Set, Node Cover, Dominating Set, Vertex Feedback Set and Steiner Tree in polynomial time.
NPHard Network Upgrading Problems
"... Graphs with delays associated with their edges are often used to model communication and signal flow networks. Network performance can be improved by upgrading the network vertices. Such an improvement reduces the edge delays and comes at a cost. We study different formulations of this network perfo ..."
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Graphs with delays associated with their edges are often used to model communication and signal flow networks. Network performance can be improved by upgrading the network vertices. Such an improvement reduces the edge delays and comes at a cost. We study different formulations of this network performance improvement problem and show that these are NPhard. Keywords And Phrases Network performance, performance enhancement, vertex upgrades, NPhard __________________ + This research was supported, in part, by the National Science Foundation under grant MIP8617374.   2 1 Introduction A communication network can be modeled as an undirected connected graph in which the edge delays ( ³ 0) represent the time taken to communicate between a pair of vertices that are directly connected. Two vertices that are not directly connected can communicate by using a series of edges that form a path from one vertex to the other. The total delay along the communication path is the sum of the dela...
The NodeDeletion Problem for Hereditary . . .
, 1980
"... We consider the family of graph problems called nodedeletion problems, defined as follows: For a fixed graph property l7, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies l7? We show that if l7 is nontrivial and hereditary on indu ..."
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We consider the family of graph problems called nodedeletion problems, defined as follows: For a fixed graph property l7, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies l7? We show that if l7 is nontrivial and hereditary on induced subgraphs, then the nodedeletion problem for n is NPcomplete for both undirected and directed graphs.