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49
Dominating Sets in Planar Graphs: BranchWidth and Exponential Speedup
, 2002
"... Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance. ..."
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Cited by 69 (18 self)
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Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.
Subexponential parameterized algorithms on graphs of boundedgenus and Hminorfree Graphs
"... ... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossing ..."
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Cited by 63 (22 self)
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... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossingminorfree graphs, and anyclass of graphs that is closed under taking minors. Specifically, the running time is 2O(pk)nh, where h is a constant depending onlyon H, which is polynomial for k = O(log² n). We introducea general approach for developing algorithms on Hminorfreegraphs, based on structural results about Hminorfree graphs at the
Finding, minimizing, and counting weighted subgraphs
 In Proceedings of the FourtyFirst Annual ACM Symposium on the Theory of Computing
, 2009
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of cop ..."
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Cited by 31 (4 self)
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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) nodeweight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edgeweighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edgeweighted problem is much harder than its nodeweighted version. 1
Limits and applications of group algebras for parameterized problems
 In Automata, Languages and Programming: ThirtySixth 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 degreek 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 degreek 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 mdimensional kmatching. 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
Parameterized approximation problems
 In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006
, 2006
"... Parameterized complexity is fast becoming accepted as an important strand in the mainstream of algorithm design and analysis, alongside approximation, randomization, and the like. It is fair to say that most of the work in the area has focussed on exact algorithms for decision problems. On the other ..."
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Cited by 22 (1 self)
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Parameterized complexity is fast becoming accepted as an important strand in the mainstream of algorithm design and analysis, alongside approximation, randomization, and the like. It is fair to say that most of the work in the area has focussed on exact algorithms for decision problems. On the other hand it is clear that parameterized ideas have applications to many other questions of algorithmic design. For example, in [6]
Solving #SAT Using Vertex Covers
, 2006
"... We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of form ..."
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Cited by 21 (10 self)
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We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph ” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clusteringwidth. Our algorithm runs in uniform polynomial time on formulas with bounded clusteringwidth. It is known that the number of models of formulas with bounded cliquewidth, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clusteringwidth and the other parameters mentioned are incomparable: there are formulas with bounded clusteringwidth and arbitrarily large cliquewidth, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clusteringwidth and bounded cliquewidth, treewidth, and branchwidth.
Static Analysis and Optimization of Semantic Web Queries
 In Proceedings of the ACM Symposium on Principles of Database Systems
, 2012
"... Static analysis is a fundamental task in query optimization. In this paper we study static analysis and optimization techniques for SPARQL, which is the standard language for querying Semantic Web data. Of particular interest for us is the optionality feature in SPARQL. It is crucial in Semantic Web ..."
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Cited by 18 (6 self)
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Static analysis is a fundamental task in query optimization. In this paper we study static analysis and optimization techniques for SPARQL, which is the standard language for querying Semantic Web data. Of particular interest for us is the optionality feature in SPARQL. It is crucial in Semantic Web data management, where data sources are inherently incomplete and the user is usually interested in partial answers to queries. This feature is one of the most complicated constructors in SPARQL and also the one that makes this language depart from classical query languages such as relational conjunctive queries. We focus on the class of welldesigned SPARQL queries, which has been proposed in the literature as a fragment of the language with good properties regarding query evaluation. We first propose a tree representation for SPARQL queries, called pattern trees, which captures the class of welldesigned SPARQL graph patterns and which can be considered as a query execution plan. Among other results, we propose several transformation rules for pattern trees, a simple normal form, and study equivalence and containment. We also study the enumeration and counting problems for this class of queries.
Balanced families of perfect hash functions and their applications
 PROC. ICALP
, 2007
"... The construction of perfect hash functions is a wellstudied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to[k] is a δbalanced (n, k)family of perfect hash functions if for every S ⊆ [n], S  = k, the number of funct ..."
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Cited by 12 (3 self)
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The construction of perfect hash functions is a wellstudied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to[k] is a δbalanced (n, k)family of perfect hash functions if for every S ⊆ [n], S  = k, the number of functions that are 11 on S is between T/δ and δT for some constant T>0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 11 on S,for each S of size k. In the new notion of balanced families, we require the number of 11 functions to be almost the same (taking δ to be close to 1) for every such S. Our main result is that for any constant δ>1, a δbalanced (n, k)family of perfect hash functions of size 2 O(k log log k) log n can be constructed in time 2 O(k log log k) n log n. Using the technique of colorcoding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length k and the number of simple log n cycles of size k for any k ≤ O() in a graph with n vertices. The log log log n approximation is up to any fixed desirable relative error.
On parameterized path and chordless path problems
 In 22nd IEEE Conference on Computational Complexity
, 2007
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 11 (5 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Counting Stars and Other Small Subgraphs in Sublinear Time
"... Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting t ..."
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Cited by 10 (3 self)
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Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design sublineartime algorithms for approximating the number of copies of certain constantsize subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length2 paths and more generally on counting the number of stars of a certain size. Specifically, we design an algorithm that, given an approximation parameter 0 < ɛ < 1 and query access to a graph G, outputs an estimate ˆνs such that with high constant probability, (1−ɛ)νs(G) ≤ ˆνs ≤ (1+ɛ)νs(G), where νs(G) denotes the number of stars of size s + 1 in the graph. The expected query ( complexity and { running time of}) the algorithm are O