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We review the task of Sentence Pair Scoring, popular in the literature in various forms - viewed as Answer Sentence Selection, Semantic Text Scoring, Next Utterance Ranking, Recognizing Textual Entailment, Paraphrasing or e.g. a component of Memory Networks. We argue that all such tasks are similar from the model perspective and propose new baselines by comparing the performance of common IR metrics and popular convolutional, recurrent and attention-based neural models across many Sentence Pair Scoring tasks and datasets. We discuss the problem of evaluating randomized models, propose a statistically grounded methodology, and attempt to improve comparisons by releasing new datasets that are much harder than some of the currently used well explored benchmarks. We introduce a unified open source software framework with easily pluggable models and tasks, which enables us to experiment with multi-task reusability of trained sentence model. We set a new state-of-art in performance on the Ubuntu Dialogue dataset., Comment: submitted as paper to CoNLL 2016

An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set of a graph $G$ is a subset $S$ of vertices of $G$ such that the irreversible $k$-threshold process starting with $S$ black eventually changes all vertices of $G$ to black. We show that deciding the existence of an irreversible 2-conversion set of a given size is NP-complete, even for graphs of maximum degree 4, which answers a question of Dreyer and Roberts. Conversely, we show that for graphs of maximum degree 3, the minimum size of an irreversible 2-conversion set can be computed in polynomial time. Moreover, we find an optimal irreversible 3-conversion set for the toroidal grid, simplifying constructions of Pike and Zou., Comment: 18 pages, 12 figures; journal version

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph $G$ can be represented by closed intervals, i.e., whether $G$ is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker [J. Comput. System Sci., 13 (1976)] based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph $G$ with a partial representation $\cal R'$ fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation $\cal R$ of the entire graph $G$ extending $\cal R'$. We generalize the characterization of interval graphs by Fulkerson and Gross [Pac. J. Math., 15 (1965)] to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.

The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations. We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. This includes the case of partial representation extension. The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs [Balko et al. ISAAC'13]. So unless $\text{P} = \text{NP}$, proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.

In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B_k-VPG. We show here that for every fixed k, B_k-VPG is a proper subset of B_{k+1}-VPG and that recognition of graphs from B_k-VPG is NP-complete even when the input graph is given by a B_{k+1}-VPG representation. We also show that the class B_k-VPG (for k>0) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane., Comment: 17 pages, 14 figures, to appear in the proceedings of WG 2012

We study the computational complexity of graph planarization via edge contraction. The problem CONTRACT asks whether there exists a set $S$ of at most $k$ edges that when contracted produces a planar graph. We work with a more general problem called $P$-RESTRICTEDCONTRACT in which $S$, in addition, is required to satisfy a fixed MSOL formula $P(S,G)$. We give an FPT algorithm in time $O(n^2 f(k))$ which solves $P$-RESTRICTEDCONTRACT, where $P(S,G)$ is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion minimal solution $S$). As a specific example, we can solve the $\ell$-subgraph contractibility problem in which the edges of a set $S$ are required to form disjoint connected subgraphs of size at most $\ell$. This problem can be solved in time $O(n^2 f'(k,\ell))$ using the general algorithm. We also show that for $\ell \ge 2$ the problem is NP-complete.

A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na [``On intersection representations of co-planar graphs'', Discrete Mathematics, 178(1-3):251-255, 1998] asked the question of whether the complements of planar graphs are segment graphs. We show here that the complements of all partial 2-trees are segment graphs., Comment: 5 pages, 4 figures, preliminary version