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16
Finding small separators in linear time via treewidth reduction
"... We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an ind ..."
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We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)and (H,C,≤K)coloring problems as well, which are cardinality constrained variants of the classical Hcoloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.
Counting Perfect Matchings as Fast as Ryser
, 2012
"... We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5) ..."
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We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5)/2) n) ⊂ O(1.619 n) time algorithm by Koivisto, and for polynomial space, the O(1.942 n) time algorithm by Nederlof. Our new algorithm’s runtime matches up to polynomial factors that of Ryser’s 1963 algorithm for bipartite graphs. We present our algorithm in the more general setting of computing the hafnian over an arbitrary ring, analogously to Ryser’s algorithm for permanent computation. We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an n element ground set cannot be solved in O ∗ (2 (1−ɛ1)n) time for any ɛ1> 0 unless there are O ∗ (2 (1−ɛ2)n) time algorithms for computing an n × n 0/1 matrix permanent, for some ɛ2> 0 depending only on ɛ1.
A new direction for counting perfect matchings
 IEEE Symposium on Foundation of Computer Science (FOCS
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Sharp Separation and Applications to Exact and Parameterized Algorithms
"... Abstract. Many divideandconquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a tradeoff between the size of the separator a ..."
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Abstract. Many divideandconquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a tradeoff between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NPhard problems. We illustrate that by presenting two applications. Our first application is a parameterized algorithm with running time O(16 k+o(k) + n O(1) ) for the Maximum Internal Subtree problem in directed graphs. This is a significant improvement over the best previously known parameterized algorithm for the problem by [Cohen et al.’09], running in time O(49.4 k + n O(1)). The second application is a O(2 n+o(n) ) time algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some wellstudied problems, among them
Abusing the Tutte Matrix: An Algebraic Instance Compression for the Ksetcycle Problem
"... We give an algebraic, determinantbased algorithm for the KCycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the O ∗ (2 K  ) running time of the algorithm of Björklund et al. (SODA, 2012). F ..."
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We give an algebraic, determinantbased algorithm for the KCycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the O ∗ (2 K  ) running time of the algorithm of Björklund et al. (SODA, 2012). Furthermore, our approach is open for treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to a polynomial compression of the problem, i.e., a polynomialtime reduction of the KCycle problem into an algebraic problem with coding size O(K  3). This is surprising, as several related problems (e.g., kCycle and the Disjoint Paths problem) are known not to admit such a reduction unless the polynomial hierarchy collapses. Furthermore, despite the result, we are not aware of any witness for the KCycle problem of size polynomial in K  + log n, which seems (for now) to separate the notions of polynomial compression and polynomial kernelization (as a polynomial kernelization for a problem in NP necessarily implies a small witness).
Solving 3superstring in 3n/3 time
 In Proceedings of the 38th International Symposium on Mathematical Foundations of Computer Science (MFCS ’13), volume 8087 of LNCS
, 2013
"... Abstract. In the shortest common superstring problem (SCS) one is given a set s1,..., sn of n strings and the goal is to find a shortest string containing each si as a substring. While many approximation algorithms for this problem have been developed, it is still not known whether it can be solved ..."
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Abstract. In the shortest common superstring problem (SCS) one is given a set s1,..., sn of n strings and the goal is to find a shortest string containing each si as a substring. While many approximation algorithms for this problem have been developed, it is still not known whether it can be solved exactly in fewer than 2n steps. In this paper we present an algorithm that solves the special case when all of the input strings have length 3 in time 3n/3 and polynomial space. The algorithm generates a combination of a de Bruijn graph and an overlap graph, such that a SCS is then a shortest directed rural postman path (DRPP) on this graph. We show that there exists at least one optimal DRPP satisfying some natural properties. The algorithm works basically by exhaustive search, but on the reduced search space of such paths of size 3n/3. 1
Families with infants: a general approach to solve hard partition problems?
"... Abstract. We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NPhard problems can be naturally stated as such partition problems. We show that if one can find a ..."
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Abstract. We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NPhard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of socalled families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NPhard problems (the traveling salesman problem, the graph coloring problem, the problem of counting perfect matchings) on graphs of bounded average degree, as well as to simplify the proofs of several known results. 1
Sharp Separation and Applications to Exact and Parameterized Algorithms∗
"... Many divideandconquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a tradeoff between the size of the separator and the sha ..."
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Many divideandconquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a tradeoff between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NPhard problems. We illustrate that by presenting two applications. Our first application is a O(2n+o(n))time algorithm for the DEGREE CONSTRAINED SPANNING TREE problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some wellstudied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, and Maximum Internal Spanning Tree. The second application is a parameterized algorithm with running time O(16k+o(k) + nO(1)) for the kINTERNAL OUTBRANCHING problem: here the goal is to compute an outbranching of a digraph with at least k internal nodes. This is a significant improvement over the best previously known parameterized algorithm for the problem by [Cohen et al.’09], running in time O(49.4k + nO(1)). 1
Parameterized singleexponential time polynomial space algorithm for Steiner Tree
"... Abstract. In the Steiner tree problem, we are given as input a connected nvertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimumweight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.9 ..."
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Abstract. In the Steiner tree problem, we are given as input a connected nvertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimumweight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97k ·n4 · logW) using O(n3 · lognW · log k) space. This is the first singleexponential time, polynomialspace FPT algorithm for the weighted Steiner Tree problem. 1
Packing Resizable Items with Application to Video Delivery over Wireless Networks ∗
"... Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B> 0, and a set I of items. Each item j ∈ I is of size sj> 0. A packed item must stay in the bin for a fixed t ..."
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Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B> 0, and a set I of items. Each item j ∈ I is of size sj> 0. A packed item must stay in the bin for a fixed time interval. To accommodate more items in the bin, each item j can be compressed to a size pj ∈ [0, sj) for at most a fraction qj ∈ [0, 1) of the packing interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is strongly NPhard already for highly restricted instances. Our main result is an approximation algorithm that packs, for any instance I of PRI, at least 2 3OP T (I) − 3 items, where OP T (I) is the number of items packed in an optimal solution. Our algorithm yields better ratio for instances in which the maximum compression time of an item is qmax ∈ (0, 1 2). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OP T (I) − 2 items. Finally, we show that a nontrivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS). 1