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41
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms | a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
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Cited by 1213 (77 self)
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the rapidly developing systematic connections between FPT and useful heuristic algorithms | a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
On problems without polynomial kernels
- LECT. NOTES COMPUT. SCI
, 2007
"... Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parame ..."
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Cited by 143 (17 self)
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Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. nonparametric complexity), and evolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which might be of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine which allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth or cliquewidth.
On the parameterized complexity of multiple-interval graph problems
- Theor. Comput. Sci
"... Abstract. Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specifi ..."
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Cited by 50 (8 self)
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Abstract. Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k-Independent Set, k-Dominating Set, and k-Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k-Clique is in FPT, while k-Independent Set and k-Dominating Set are both W[1]-hard. We also prove that k-Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k-Multicolored Clique problem, a variant of k-Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.
Faster algebraic algorithms for path and packing problems
, 2008
"... We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free 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 n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free 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 m-set k-packing problem and (ii) an O ∗ (2 3k/2) algorithm for the simple k-path problem, or an O ∗ (2 k) algorithm if the graph has an induced k-subgraph 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
Determinant sums for undirected hamiltonicity
- in Prof. of FOCS’10, 2010
"... We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O ∗ (1.657 n) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O ∗ (2 n) bound established for TSP almost fif ..."
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Cited by 46 (1 self)
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We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O ∗ (1.657 n) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O ∗ (2 n) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger’s 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to O ∗ (1.414 n) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for k-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which weareset tocount weightedarclabeled cycle coversoverafinite field ofcharacteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Björklund STACS 2010) to evaluate it. 1
Limits and applications of group algebras for parameterized problems
- In Automata, Languages and Programming: Thirty-Sixth International Colloquium (ICALP
, 2009
"... The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem req ..."
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Cited by 30 (1 self)
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The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem requires only O ∗ (2 k) time and oracle access to an arithmetic circuit, i.e. the ability to evaluate the circuit on elements from a suitable group algebra. This algorithm has been used to obtain the best known algorithms for several parameterized problems. In this paper we use communication complexity to show that the O ∗ (2 k) algorithm is essentially optimal within this evaluation oracle framework. On the positive side, we give new applications of the method: finding a copy of a given tree on k nodes, a spanning tree with at least k leaves, a minimum set of nodes that dominate at least t nodes, and an m-dimensional k-matching. In each case we achieve a faster algorithm than what was known. We also apply the algebraic method to problems in exact counting. Among other results, we show that a combination of dynamic programming and a variation of the algebraic method can break the trivial upper bounds for exact parameterized counting in fairly general settings. 1
On Limited Versus Polynomial Nondeterminism
- THE CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE
, 1997
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Topology-free querying of protein interaction networks
- In Proceedings of 13th RECOMB
, 2009
"... Abstract. In the network querying problem, one is given a protein complex or pathway of species A and a protein–protein interaction network of species B; the goal is to identify subnetworks of B that are similar to the query. Existing approaches mostly depend on knowledge of the interaction topology ..."
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Cited by 27 (2 self)
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Abstract. In the network querying problem, one is given a protein complex or pathway of species A and a protein–protein interaction network of species B; the goal is to identify subnetworks of B that are similar to the query. Existing approaches mostly depend on knowledge of the interaction topology of the query in the network of species A; however, in practice, this topology is often not known. To combat this problem, we develop a topology-free querying algorithm, which we call Torque. Given a query, represented as a set of proteins, Torque seeks a matching set of proteins that are sequence-similar to the query proteins and span a connected region of the network, while allowing both insertions and deletions. The algorithm uses alternatively dynamic programming and integer linear programming for the search task. We test Torque with queries from yeast, fly, and human, where we compare it to the QNet topology-based approach, and with queries from less studied species, where only topology-free algorithms apply. Torque detects many more matches than QNet, while in both cases giving results that are highly functionally coherent. 1
On problems without polynomial kernels (Extended abstract).
- ICALP 2008, Part I. LNCS,
, 2008
"... Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NP-hard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexity-theoretic as ..."
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Cited by 21 (3 self)
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Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NP-hard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexity-theoretic assumptions. These problems include k- k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth or cliquewidth.
Sharp tractability borderlines for finding connected motifs in vertex-colored graphs.
- In Proc. ICALP’07, LNCS 4596,
, 2007
"... Abstract. We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem has applications in metabolic network analysis, an ..."
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Cited by 19 (8 self)
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Abstract. We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem has applications in metabolic network analysis, an important area in bioinformatics. We give two positive results and three negative results that together draw sharp borderlines between tractable and intractable instances of the problem.
