308 research outputs found

    String Matching: Communication, Circuits, and Learning

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    String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings. - Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. - Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k. - Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem

    Characterization and computation of ancestors in reaction systems

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    AbstractIn reaction systems, preimages and nth ancestors are sets of reactants leading to the production of a target set of products in either 1 or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality nth ancestors, computing their size or counting them, are intractable. In this paper, we characterize all nth ancestors using a Boolean formula that can be computed in polynomial time. Once simplified, this formula can be exploited to easily solve all preimage and ancestor problems. This allows us to directly relate the difficulty of ancestor problems to the cost of the simplification so that new insights into computational complexity investigations can be achieved. In particular, we focus on two problems: (i) deciding whether a preimage/nth ancestor exists and (ii) finding a preimage/nth ancestor of minimal size. Our approach is constructive, it aims at finding classes of reactions systems for which the ancestor problems can be solved in polynomial time, in exact or approximate way

    Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

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    The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time O(ntw(H)+1)O(n^{tw(H)+1}) [Alon, Yuster, Zwick'95], where nn is the number of vertices of the host graph GG. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of O(ntw(H)+1ε)O(n^{tw(H)+1-\varepsilon}) or even faster (e.g. for kk-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time no(tw(H)/log(tw(H)))n^{o(tw(H) / \log(tw(H)))} for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx'07]. In this paper, we demonstrate the existence of maximally hard pattern graphs HH that require time ntw(H)+1o(1)n^{tw(H)+1-o(1)}. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth tt: For any ε>0\varepsilon > 0 there exists t3t \ge 3 and a pattern graph HH of treewidth tt such that Subgraph Isomorphism on pattern HH has no algorithm running in time O(nt+1ε)O(n^{t+1-\varepsilon}). Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth t3t \ge 3: For any t3t \ge 3 there exists a pattern graph HH of treewidth tt such that for any ε>0\varepsilon>0 Subgraph Isomorphism on pattern HH has no algorithm running in time O(nt+1ε)O(n^{t+1-\varepsilon}). In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for tw<3tw < 3, (3) Subgraph Isomorphism parameterized by pathwidth, and (4) a weighted problem variant

    Bounds and algorithms for graph trusses

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    The kk-truss, introduced by Cohen (2005), is a graph where every edge is incident to at least kk triangles. This is a relaxation of the clique. It has proved to be a useful tool in identifying cohesive subnetworks in a variety of real-world graphs. Despite its simplicity and its utility, the combinatorial and algorithmic aspects of trusses have not been thoroughly explored. We provide nearly-tight bounds on the edge counts of kk-trusses. We also give two improved algorithms for finding trusses in large-scale graphs. First, we present a simplified and faster algorithm, based on approach discussed in Wang & Cheng (2012). Second, we present a theoretical algorithm based on fast matrix multiplication; this converts a triangle-generation algorithm of Bjorklund et al. (2014) into a dynamic data structure
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