308 research outputs found
String Matching: Communication, Circuits, and Learning
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
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
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 [Alon, Yuster, Zwick'95], where is the number of vertices of the host graph . While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of or even faster (e.g. for -cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time 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 that require time . 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 : For any there exists and a pattern graph of treewidth such that Subgraph Isomorphism on pattern has no algorithm running in time . Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth : For any there exists a pattern graph of treewidth such that for any Subgraph Isomorphism on pattern has no algorithm running in time . 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 , (3) Subgraph Isomorphism parameterized by pathwidth, and (4) a weighted problem variant
Bounds and algorithms for graph trusses
The -truss, introduced by Cohen (2005), is a graph where every edge is
incident to at least 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 -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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