Reducing graph transversals via edge contractions

For a graph invariant $\pi$, the Contraction($\pi$) problem consists in, given a graph $G$ and two positive integers $k,d$, deciding whether one can contract at most $k$ edges of $G$ to obtain a graph in which $\pi$ has dropped by at least $d$. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where $\pi$ is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ${\cal H}$ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ${\cal H}$, which in particular imply that Contraction($\pi$) is co-NP-hard even for fixed $k=d=1$ when $\pi$ is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when $\pi$ is the size of a minimum vertex cover, the problem is in XP parameterized by $d$.Comment: 19 pages, 2 figure

The traveling salesman problem for lines and rays in the plane

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of $n$ regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length $L$.Comment: 10 pages, 5 figure

Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings

In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is solved by polynomial number of solutions of subproblems, each being a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D.~Gijswijt and G.~Pap, An algorithm for weighted fractional matroid matching, J.\ Combin.\ Theory, Ser.~B 103 (2013), 509--520]

Helly's Theorem: New Variations and Applications

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.Comment: 40 pages, 1 figure

Three problems on well-partitioned chordal graphs

In this work, we solve three problems on well-partitioned chordal graphs. First, we show that every connected (resp., 2-connected) well-partitioned chordal graph has a vertex that intersects all longest paths (resp., longest cycles). It is an open problem [Balister et al., Comb. Probab. Comput. 2004] whether the same holds for chordal graphs. Similarly, we show that every connected well-partitioned chordal graph admits a (polynomial-time constructible) tree 3-spanner, while the complexity status of the Tree 3-Spanner problem remains open on chordal graphs [Brandstädt et al., Theor. Comput. Sci. 2004]. Finally, we show that the problem of finding a minimum-size geodetic set is polynomial-time solvable on well-partitioned chordal graphs. This is the first example of a problem that is NP -hard on chordal graphs and polynomial-time solvable on well-partitioned chordal graphs. Altogether, these results reinforce the significance of this recently defined graph class as a tool to tackle problems that are hard or unsolved on chordal graphs.acceptedVersio

Direct and Binary Direct Bases for One-set Updates of a Closure System

We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the $D$-basis introduced in Adaricheva, Nation, Rand [Disc.Appl.Math. 2013]. Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, when the closure system is given by the canonical direct or binary-direct basis. In this problem, a new closed set is added to or removed from the closure system forcing the re-write of a given basis. Our goal is to obtain the same type of implicational basis for the new closure system as was given for original closure system and to make the basis update an optimal process.Comment: 17 pages, 1 table, 1 figure, poster session presentation ICFCA-201

Reconstructing fully-resolved trees from triplet cover distances

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset \cl of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in \cl. It is known that any set \cl with this property for a tree in which all interior vertices have degree 3 must form a {\em cover} for $T$ -- that is, for each interior vertex $v$ of $T$, \cl must contain a pair of leaves from each pair of the three components of $T-v$. Here we provide a partial converse of this result by showing that if a set \cl of leaf pairs forms a cover of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in \cl. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning `triplet covers', and is relevant to a problem arising in evolutionary genomics.Comment: 10 pages, 2 figure

Reconstructing fully-resolved trees from triplet cover distances

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset L of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in L. It is known that any set L with this property for a tree in which all interior vertices have degree 3 must form a cover for T {that is, for each interior vertex v of T, L must contain a pair of leaves from each pair of the three components of T ̶ v. Here we provide a partial converse of this result by showing that if a set L of leaf pairs forms a cover of a certain type for such a tree T then T and its edge weights can be uniquely determined from the distances between the pairs of leaves in L. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning `triplet covers', and is relevant to a problem arising in evolutionary genomics

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

We devise a method to enumerate the inclusion-wise minimal hitting sets of a hypergraph. The algorithm has delay $O( m^{k^*+1} \, n^2)$ on $n$-vertex, $m$-edge hypergraphs, where $k^*$ is the rank of the transversal hypergraph, i.e., the cardinality of the largest minimal solution. In particular, on classes of hypergraphs for which $k^*$ is bounded, the delay is polynomial. The algorithm uses space linear in the input size only. The enumeration methods solves the extension problem for minimal hitting sets as a subroutine. We show that this problem, parameterised by the cardinality of the set which is to be extended, is one of the first natural W[3]-complete problems. We give an algorithm for the subroutine that is optimal under the assumption that $W[2] \neq \mathrm{FPT}$ or the exponential time hypothesis (ETH), respectively. Despite the hardness of the extension problem, we provide empirical evidence indicating that the enumeration outperforms its theoretical worst-case guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations. Our analysis suggest that these hypergraphs exhibit structure that allows the subroutine to be fast on average.Comment: 26 pages, 8 PDF figure

Stabbers of line segments in the plane

The problem of computing a representation of the stabbing lines of a set S of segments in the plane was solved by Edelsbrunner et al. We provide efficient algorithms for the following problems: computing the stabbing wedges for S, finding a stabbing wedge for a set of parallel segments with equal length, and computing other stabbers for S such as a double-wedge and a zigzag. The time and space complexities of the algorithms depend on the number of combinatorially different extreme lines, critical lines, and the number of different slopes that appear in S.Preprin