ABSTRACT The Closest String problem asks to find a string s which is not too far from each string... more ABSTRACT The Closest String problem asks to find a string s which is not too far from each string in a set of m input strings, where the distance is taken as the Hamming distance. This well-studied problem has various applications in computational biology and drug design. In this paper, we introduce a new variant of Closest String where the input strings can contain wildcards that can match any letter in the alphabet, and the goal is to find a solution string without wildcards. We call this problem the Closest String with Wildcards problem, and we analyze it in the framework of parameterized complexity. Our study determines for each natural parameterization whether this parameterization yields a fixed-parameter algorithm, or whether such an algorithm is highly unlikely to exist. More specifically, let m denote the number of input strings, each of length n, and let d be the given distance bound for the solution string. Furthermore, let k denote the minimum number of wildcards in any input string. We present fixed-parameter algorithms for the parameters m, n, and k + d, respectively. On the other hand, we then show that such results are unlikely to exist when k and d are taken as single parameters. This is done by showing that the problem is NP-hard already for k = 0 and d ≥ 2. Finally, to complement the latter result, we present a polynomial-time algorithm for the case of d = 1. Apart from this last result, all other results hold even when the strings are over a general alphabet.
ABSTRACT The minimum linear arrangement (MLA) problem asks to embed a given graph on the integer ... more ABSTRACT The minimum linear arrangement (MLA) problem asks to embed a given graph on the integer line so that the sum of the edge lengths of the embedded graph is minimized. Most layout problems are either intractable, or not known to be tractable, parameterized by the treewidth of the input graphs. We investigate MLA with respect to three parameters that provide more structure than treewidth. In particular, we give a factor (1+ε)-approximation algorithm for MLA parameterized by (ε,k), where k is the vertex cover number of the input graph. By a similar approach, we describe two FPT algorithms that exactly solve MLA parameterized by, respectively, the max leaf and edge clique cover numbers of the input graph.
ABSTRACT The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt... more ABSTRACT The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). The algorithm relies on several new structural lemmas, which decompose the Steiner cut into important separators and minimal s-t cuts, and which only hold for the edge deletion version of the problem. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).
Proceedings of the Twenty-Third Annual ACM- …, Jan 17, 2012
In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bou... more In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d��� 2 be some constant and let L 1, L 2���{0, 1}* x N be two parameterized problems where the unparameterized version of L 1 is NP-hard. Assuming coNP ⊈ NP/poly, our framework essentially states that composing t L 1-instances each with parameter k, to an L 2-instance with parameter k'��� t 1/dk O (1), implies that L 2 does not have a kernel of size O ( ...
We give a novel characterization of W(1), the most important fixed-parameter intractability class... more We give a novel characterization of W(1), the most important fixed-parameter intractability class in the W-hierarchy, using Boolean circuits that consist solely of majority gates. Such gates have a Boolean value of 1 if and only if more than half of their inputs have value 1. Us- ing majority circuits, we define an analog of the W-hierarchy which we call the f W-hierarchy, and show that W(1) = f W(1) and W(t) � f W(t) for all t. This gives the first characterization of W(1) based on the weighted satisfiability problem for monotone Boolean circuits rather than anti- monotone. Our results are part of a wider program aimed at exploring the robustness of the notion of weft, showing that it remains a key param- eter governing the combinatorial nondeterministic computing strength of circuits, no matter what type of gates these circuits have.
We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements o... more We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded feedback-vertex-set are well-quasi-ordered by the topological-minor order, graphs with bounded vertex-covers are well-quasi-ordered by the subgraph order, and graphs with bounded circumference are well-quasi-ordered by the induced-minor order. Our results give an algorithm for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.
IEEE/ACM transactions on computational biology and bioinformatics / IEEE, ACM
The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set ... more The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is important in practical fields such as the investigation of diseases or other types of genetic mutations. In order to find the haplotypes which are as close as possible to the real set of haplotypes that comprise the genotypes, two models have been suggested which are by now well-studied: The perfect phylogeny model and the pure parsimony model. All known algorithms up till now for haplotype inference may find haplotypes that are not necessarily plausible, i.e., very rare haplotypes or haplotypes that were never observed in the population. In order to overcome this disadvantage, we study in this paper, a new constrained version of HIP under the above-mentioned models. In this new version, a pool of plausible haplotypes H is given together with the set of genotypes G, and the goal is to find a subset H ⊆ H that resolves G. For constrained perfect phlogeny hapl...
In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer q... more In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of size $i$ and has exactly $j$ 1-bits. The problem naturally generalizes to node-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an ${n^2}/{2^{\Omega(\log n/\log \log n)^{1/2}}}$ time solution for both strings and trees. This odd-looking time complexity improves the state of the art $O(n^2/\log^2 n)$ solutions by more than any poly-logarithmic factor. We obtain it by showing (1) A reduction from the string problem to computing min-plus matrix products using the recent seminal algorithm of Williams. (2) A black box reduction showing that any $O(n^{2-\varepsilon})$ algorithm for strings implies an $O(n^{2-\varepsilon'})$ algorithm for trees. Since currently $\varepsilon$ is non-constant, we now have $\varepsilon'=\varepsilon$.
ABSTRACT The Closest String problem asks to find a string s which is not too far from each string... more ABSTRACT The Closest String problem asks to find a string s which is not too far from each string in a set of m input strings, where the distance is taken as the Hamming distance. This well-studied problem has various applications in computational biology and drug design. In this paper, we introduce a new variant of Closest String where the input strings can contain wildcards that can match any letter in the alphabet, and the goal is to find a solution string without wildcards. We call this problem the Closest String with Wildcards problem, and we analyze it in the framework of parameterized complexity. Our study determines for each natural parameterization whether this parameterization yields a fixed-parameter algorithm, or whether such an algorithm is highly unlikely to exist. More specifically, let m denote the number of input strings, each of length n, and let d be the given distance bound for the solution string. Furthermore, let k denote the minimum number of wildcards in any input string. We present fixed-parameter algorithms for the parameters m, n, and k + d, respectively. On the other hand, we then show that such results are unlikely to exist when k and d are taken as single parameters. This is done by showing that the problem is NP-hard already for k = 0 and d ≥ 2. Finally, to complement the latter result, we present a polynomial-time algorithm for the case of d = 1. Apart from this last result, all other results hold even when the strings are over a general alphabet.
ABSTRACT The minimum linear arrangement (MLA) problem asks to embed a given graph on the integer ... more ABSTRACT The minimum linear arrangement (MLA) problem asks to embed a given graph on the integer line so that the sum of the edge lengths of the embedded graph is minimized. Most layout problems are either intractable, or not known to be tractable, parameterized by the treewidth of the input graphs. We investigate MLA with respect to three parameters that provide more structure than treewidth. In particular, we give a factor (1+ε)-approximation algorithm for MLA parameterized by (ε,k), where k is the vertex cover number of the input graph. By a similar approach, we describe two FPT algorithms that exactly solve MLA parameterized by, respectively, the max leaf and edge clique cover numbers of the input graph.
ABSTRACT The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt... more ABSTRACT The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). The algorithm relies on several new structural lemmas, which decompose the Steiner cut into important separators and minimal s-t cuts, and which only hold for the edge deletion version of the problem. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).
Proceedings of the Twenty-Third Annual ACM- …, Jan 17, 2012
In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bou... more In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d��� 2 be some constant and let L 1, L 2���{0, 1}* x N be two parameterized problems where the unparameterized version of L 1 is NP-hard. Assuming coNP ⊈ NP/poly, our framework essentially states that composing t L 1-instances each with parameter k, to an L 2-instance with parameter k'��� t 1/dk O (1), implies that L 2 does not have a kernel of size O ( ...
We give a novel characterization of W(1), the most important fixed-parameter intractability class... more We give a novel characterization of W(1), the most important fixed-parameter intractability class in the W-hierarchy, using Boolean circuits that consist solely of majority gates. Such gates have a Boolean value of 1 if and only if more than half of their inputs have value 1. Us- ing majority circuits, we define an analog of the W-hierarchy which we call the f W-hierarchy, and show that W(1) = f W(1) and W(t) � f W(t) for all t. This gives the first characterization of W(1) based on the weighted satisfiability problem for monotone Boolean circuits rather than anti- monotone. Our results are part of a wider program aimed at exploring the robustness of the notion of weft, showing that it remains a key param- eter governing the combinatorial nondeterministic computing strength of circuits, no matter what type of gates these circuits have.
We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements o... more We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded feedback-vertex-set are well-quasi-ordered by the topological-minor order, graphs with bounded vertex-covers are well-quasi-ordered by the subgraph order, and graphs with bounded circumference are well-quasi-ordered by the induced-minor order. Our results give an algorithm for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.
IEEE/ACM transactions on computational biology and bioinformatics / IEEE, ACM
The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set ... more The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is important in practical fields such as the investigation of diseases or other types of genetic mutations. In order to find the haplotypes which are as close as possible to the real set of haplotypes that comprise the genotypes, two models have been suggested which are by now well-studied: The perfect phylogeny model and the pure parsimony model. All known algorithms up till now for haplotype inference may find haplotypes that are not necessarily plausible, i.e., very rare haplotypes or haplotypes that were never observed in the population. In order to overcome this disadvantage, we study in this paper, a new constrained version of HIP under the above-mentioned models. In this new version, a pool of plausible haplotypes H is given together with the set of genotypes G, and the goal is to find a subset H ⊆ H that resolves G. For constrained perfect phlogeny hapl...
In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer q... more In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of size $i$ and has exactly $j$ 1-bits. The problem naturally generalizes to node-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an ${n^2}/{2^{\Omega(\log n/\log \log n)^{1/2}}}$ time solution for both strings and trees. This odd-looking time complexity improves the state of the art $O(n^2/\log^2 n)$ solutions by more than any poly-logarithmic factor. We obtain it by showing (1) A reduction from the string problem to computing min-plus matrix products using the recent seminal algorithm of Williams. (2) A black box reduction showing that any $O(n^{2-\varepsilon})$ algorithm for strings implies an $O(n^{2-\varepsilon'})$ algorithm for trees. Since currently $\varepsilon$ is non-constant, we now have $\varepsilon'=\varepsilon$.
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Papers by Danny Hermelin