Minimum Entropy Orientations

We study graph orientations that minimize the entropy of the in-degree sequence. The problem of finding such an orientation is an interesting special case of the minimum entropy set cover problem previously studied by Halperin and Karp [Theoret. Comput. Sci., 2005] and by the current authors [Algorithmica, to appear]. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit. This improves on the only previously known algorithm which has an additive error guarantee of log_2 e bits (approx. 1.4427 bits).Comment: Referees' comments incorporate

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$

Nucleus accumbens core lesions retard instrumental learning and performance with delayed reinforcement in the rat.

BACKGROUND: Delays between actions and their outcomes severely hinder reinforcement learning systems, but little is known of the neural mechanism by which animals overcome this problem and bridge such delays. The nucleus accumbens core (AcbC), part of the ventral striatum, is required for normal preference for a large, delayed reward over a small, immediate reward (self-controlled choice) in rats, but the reason for this is unclear. We investigated the role of the AcbC in learning a free-operant instrumental response using delayed reinforcement, performance of a previously-learned response for delayed reinforcement, and assessment of the relative magnitudes of two different rewards. RESULTS: Groups of rats with excitotoxic or sham lesions of the AcbC acquired an instrumental response with different delays (0, 10, or 20 s) between the lever-press response and reinforcer delivery. A second (inactive) lever was also present, but responding on it was never reinforced. As expected, the delays retarded learning in normal rats. AcbC lesions did not hinder learning in the absence of delays, but AcbC-lesioned rats were impaired in learning when there was a delay, relative to sham-operated controls. All groups eventually acquired the response and discriminated the active lever from the inactive lever to some degree. Rats were subsequently trained to discriminate reinforcers of different magnitudes. AcbC-lesioned rats were more sensitive to differences in reinforcer magnitude than sham-operated controls, suggesting that the deficit in self-controlled choice previously observed in such rats was a consequence of reduced preference for delayed rewards relative to immediate rewards, not of reduced preference for large rewards relative to small rewards. AcbC lesions also impaired the performance of a previously-learned instrumental response in a delay-dependent fashion. CONCLUSIONS: These results demonstrate that the AcbC contributes to instrumental learning and performance by bridging delays between subjects' actions and the ensuing outcomes that reinforce behaviour

Information-theoretic lower bounds for quantum sorting

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04)

The Clique Problem in Ray Intersection Graphs

Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure

Ramsey-type theorems for lines in 3-space

We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove that: (1) The intersection graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}). (2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no 6-subset is stabbed by one line. (3) Every set of n lines in general position in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds -- such as the Guth-Katz bound on point-line incidences in R^3 -- combined with Tur\'an-type results on independent sets in sparse graphs and hypergraphs. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi