Search

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy is to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable, in particular we design a 2^O(k²)⋅n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by [Dereniowski, Osula, and Rzążewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a "global" demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versat

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

Front Matter, Table of Contents, Preface, Conference Organization

LIPIcs, Vol. 154, STACS 2020, Complete Volume

Given a string S of length n, the classic string indexing problem is to preprocess S into a compact data structure that supports efficient subsequent pattern queries. In this paper we consider the basic variant where the pattern is given in compressed form and the goal is to achieve query time that is fast in terms of the compressed size of the pattern. This captures the common client-server scenario, where a client submits a query and communicates it in compressed form to a server. Instead of the server decompressing the query before processing it, we consider how to efficiently process the compressed query directly. Our main result is a novel linear space data structure that achieves near-optimal query time for patterns compressed with the classic Lempel-Ziv 1977 (LZ77) compression scheme. Along the way we develop several data structural techniques of independent interest, including a novel data structure that compactly encodes all LZ77 compressed suffixes of a string in linear space and a general decomposition of tries that reduces the search time from logarithmic in the size of the trie to logarithmic in the length of the pattern.

The node search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above two games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivty to search for an agile robber is bounded by 2, that is the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. In this paper, we investigate the connected search game against a lazy robber. A lazy robber moves only when the searchers' strategy threatens the location that he currently occupies. We introduce two alternative graph-theoretic formulations of this game, one in terms of connected tree decompositions, and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that connected treewidth parameter is closed under contractions and prove that for every k >= 2, the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k=2. One may observe that, so far, only a few complete obstruction sets are explicitly known for contraction closed graph classes. We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.

The node search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above two games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivty to search for an agile robber is bounded by 2, that is the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. In this paper, we investigate the connected search game against a lazy robber. A lazy robber moves only when the searchers' strategy threatens the location that he currently occupies. We introduce two alternative graph-theoretic formulations of this game, one in terms of connected tree decompositions, and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that connected treewidth parameter is closed under contractions and prove that for every k >= 2, the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k=2. One may observe that, so far, only a few complete obstruction sets are explicitly known for contraction closed graph classes. We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.

LIPIcs, Volume 126, STACS'19, Complete Volume

Front Matter, Table of Contents, Preface, Conference Organization

Front Matter, Table of Contents, Preface, Conference Organization

LIPIcs, Volume 115, IPEC'18, Complete Volume