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22
Faster algebraic algorithms for path and packing problems
, 2008
"... We study the problem of deciding whether an nvariate polynomial, presented as an arithmetic circuit G, contains a degree k squarefree term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2 k+1 in time t and space s, the problem can be decided with constant prob ..."
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Cited by 47 (2 self)
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We study the problem of deciding whether an nvariate polynomial, presented as an arithmetic circuit G, contains a degree k squarefree term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2 k+1 in time t and space s, the problem can be decided with constant probability in O((kn + t)2 k) time and O(kn + s) space. Based on this, we present new and faster algorithms for two well studied problems: (i) an O ∗ (2 mk) algorithm for the mset kpacking problem and (ii) an O ∗ (2 3k/2) algorithm for the simple kpath problem, or an O ∗ (2 k) algorithm if the graph has an induced ksubgraph with an odd number of Hamiltonian paths. Our algorithms use poly(n) random bits, comparing to the 2 O(k) random bits required in prior algorithms, while having similar low space requirements. 1
Improved algorithms for path, matching, and packing problems
 IN PROCEEDINGS OF THE 18TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2007
"... We develop new and improved randomized and deterministic algorithmic techniques for path, matching, and packing problems. Our randomized algorithms are based on a new divideandconquer technique, which leads to improved algorithms for these problems. For example, for the kpath problem, our randomi ..."
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Cited by 29 (0 self)
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We develop new and improved randomized and deterministic algorithmic techniques for path, matching, and packing problems. Our randomized algorithms are based on a new divideandconquer technique, which leads to improved algorithms for these problems. For example, for the kpath problem, our randomized algorithm runs in time O(4 k mk 3.42) and space O(nk log k + m), improving the previous best randomized algorithm for the problem that runs in time O(5.44 k km) and space O(2 k kn + m). To achieve improved deterministic algorithms, we develop a lower bound Ω(2.718 k) and an improved upper bound O(6.4 k n) on the number of kcolorings in a kcolor coding scheme. This leads directly to a deterministic algorithm of time O(12.8 k nm) for the kpath problem, improving the previous best deterministic algorithm for the problem that runs in time O(c k nm), where c> 8000. Our techniques also lead to similar or more significant improvements on randomized and deterministic algorithms for matching and packing problems, such as 3d matching, 3set packing, and triangle packing.
Kernelization of Packing Problems
, 2011
"... Kernelization algorithms are polynomialtime reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, dSet Matching for integers d ≥ 3 is the problem of nding a matching of size at least k in a given duniform hypergraph and has kernels w ..."
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Cited by 20 (2 self)
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Kernelization algorithms are polynomialtime reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, dSet Matching for integers d ≥ 3 is the problem of nding a matching of size at least k in a given duniform hypergraph and has kernels with O(k d) edges. Recently, Bodlaender et al. [ICALP 2008], Fortnow and Santhanam [STOC 2008], Dell and Van Melkebeek [STOC 2010] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexitytheoretic hypothesis that coNP is not contained in NP/poly. Under the same hypothesis, we show lower bounds for the kernelization of dSet Matching and other packing problems. Our bounds are tight for dSet Matching: It does not have kernels with O(k d−ɛ) edges for any ɛ> 0 unless the hypothesis fails. By reduction, this transfers to a bound of O(k d−1−ɛ) for the problem of nding k vertexdisjoint cliques of size d in standard graphs. It is natural to ask for tight bounds on the kernel sizes of such graph packing problems. We make rst progress in that direction by showing nontrivial kernels with O(k 2.5) edges for the problem of nding k vertexdisjoint paths of three edges each. This does not quite match the best lower bound of O(k 2−ɛ) that we can prove. Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(k d−ɛ) for a problem in duniform hypergraphs, one should reduce from a carefully chosen dpartite problem that is still NPhard. As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the numbertheoretical construction by Dell and Van Melkebeek [STOC 2010] with shorter elementary arguments. 1
Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization
"... Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized versi ..."
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Cited by 18 (2 self)
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Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized version of L1 is NPhard. Assuming coNP ̸ ⊆ NP/poly, our framework essentially states that composing t L1instances each with parameter k, to an L2instance with parameter k ′ ≤ t 1/d k O(1) , implies that L2 does not have a kernel of size O(k d−ε) for any ε> 0. We show two examples of weak composition and derive polynomial kernelization lower bounds for dBipartite Regular Perfect Code and dDimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lowerbounds for kernel sizes when the parameter is the solution size k (assuming coNP ̸ ⊆ NP/poly): – dSet Packing, dSet Cover, dExact Set Cover, Hitting Set with dBounded Occurrences, and Exact Hitting Set with dBounded Occurrences have no kernels of size O(k d−3−ε) for any ε> 0. – Kd Packing and Induced K1,d Packing have no kernels of size O(k d−4−ε) for any ε> 0. – dRedBlue Dominating Set and dSteiner Tree have no kernels of sizes O(k d−3−ε) and
Greedy Localization, Iterative Compression, and Modeled Crown Reductions: New FPT Techniques, an Improved Algorithm for Set Splitting, and a Novel 2k Kernelization for Vertex Cover
, 2004
"... The two objectives of this paper are: (1) to articulate three new general techniques for designing FPT algorithms, and (2) to apply these to obtain new FPT algorithms for Set Splitting and Vertex Cover. In the case of Set Splitting, we improve the best previous O ∗ (72 k) FPT algorithm due to Dehn ..."
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Cited by 16 (2 self)
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The two objectives of this paper are: (1) to articulate three new general techniques for designing FPT algorithms, and (2) to apply these to obtain new FPT algorithms for Set Splitting and Vertex Cover. In the case of Set Splitting, we improve the best previous O ∗ (72 k) FPT algorithm due to Dehne, Fellows and Rosamond [DFR03], to O ∗ (8 k) by an approach based on greedy localization in conjunction with modeled crown reduction. In the case of Vertex Cover, wedescribe a new approach to 2k kernelization based on iterative compression and crown reduction, providing a potentially useful alternative to the NemhauserTrotter 2k kernelization.
Packing Edge Disjoint Triangles: A Parameterized View
 In Proc. 1st Int. Workshop on Parameterized and Exact Computation (IWPEC’04), volume 3162 of LNCS
, 2004
"... The problem of packing k edgedisjoint triangles in a graph has been thoroughly studied both in the classical complexity and the approximation fields and it has a wide range of applications in many areas, especially computational biology [BP96]. In this paper we present an analysis of the proble ..."
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Cited by 10 (1 self)
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The problem of packing k edgedisjoint triangles in a graph has been thoroughly studied both in the classical complexity and the approximation fields and it has a wide range of applications in many areas, especially computational biology [BP96]. In this paper we present an analysis of the problem from a parameterized complexity viewpoint.
Algorithm Engineering for ColorCoding with Applications to Signaling Pathway Detection
, 2007
"... Colorcoding is a technique to design fixedparameter algorithms for several NPcomplete subgraph isomorphism problems. Somewhat surprisingly, not much work has so far been spent on the actual implementation of algorithms that are based on colorcoding, despite the elegance of this technique and its ..."
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Cited by 9 (0 self)
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Colorcoding is a technique to design fixedparameter algorithms for several NPcomplete subgraph isomorphism problems. Somewhat surprisingly, not much work has so far been spent on the actual implementation of algorithms that are based on colorcoding, despite the elegance of this technique and its wide range of applicability to practically important problems. This work gives various novel algorithmic improvements for colorcoding, both from a worstcase perspective as well as under practical considerations. We apply the resulting implementation to the identification of signaling pathways in protein interaction networks, demonstrating that our improvements speed up the colorcoding algorithm by orders of magnitude over previous implementations. This allows more complex and larger structures to be identified in reasonable time; many biologically relevant instances can even be solved in seconds where, previously, hours were required.
A Problem Kernelization for Graph Packing
, 2009
"... For a fixed connected graph H, we consider the NPcomplete Hpacking problem, where, given an undirected graph G and an integer k ≥ 0, one has to decide whether there exist k vertexdisjoint copies of H in G. We give a problem kernel of O(k V (H)−1) vertices, that is, we provide a polynomialtime ..."
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Cited by 8 (1 self)
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For a fixed connected graph H, we consider the NPcomplete Hpacking problem, where, given an undirected graph G and an integer k ≥ 0, one has to decide whether there exist k vertexdisjoint copies of H in G. We give a problem kernel of O(k V (H)−1) vertices, that is, we provide a polynomialtime algorithm that reduces a given instance of Hpacking to an equivalent instance with at most O(k V (H)−1) vertices. In particular, this result specialized to H being a triangle improves a problem kernel for Triangle Packing from O(k³) vertices by Fellows et al. [WG 2004] to O(k²) vertices.
On the effective enumerability of NP problems
 Proc. 2nd Int. Workshop on Parameterized and Exact Computations (IWPEC 2006), Lecture Notes in Computer Science
, 2006
"... In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few ”best ” solutions to the instance. Motivated by the recent research performed in these fields, we propose in this paper a new framework to measure the ..."
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Cited by 5 (3 self)
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In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few ”best ” solutions to the instance. Motivated by the recent research performed in these fields, we propose in this paper a new framework to measure the effective enumerability of NP optimization problems. More specifically, given an instance of an NP problem, we consider the problem of enumerating a given number of best solutions for the instance, and study its average complexity in terms of the number of solutions. Our framework is different from the previouslyproposed ones, which studied the counting complexity of a problem, or the complexity of enumerating all solutions to a given instance of the problem. For example, even though it was shown by Flum and Grohe that counting the number of kpaths in a graph is fixedparameter intractable, we present a fixedparameter enumeration algorithm for the problem. The developed enumeration framework consists of two phases: the structuregeneration phase and the solutionenumeration phase. We show that most algorithmdesign techniques for fixedparameter tractable problems, such as search trees, color coding, and bounded treewidth, can be transformed into techniques for the structuregeneration phase. We design elegant enumeration techniques, and combine them with the use of effective data structures, to show how to generate smallsize structures and enumerate them efficiently. 1