28 research outputs found
Stochastic Contextual Bandits with Graph-based Contexts
We naturally generalize the on-line graph prediction problem to a version of
stochastic contextual bandit problems where contexts are vertices in a graph
and the structure of the graph provides information on the similarity of
contexts. More specifically, we are given a graph , whose vertex set
represents contexts with {\em unknown} vertex label . In our stochastic
contextual bandit setting, vertices with the same label share the same reward
distribution. The standard notion of instance difficulties in graph label
prediction is the cutsize defined to be the number of edges whose end
points having different labels. For line graphs and trees we present an
algorithm with regret bound of where is
the number of arms. Our algorithm relies on the optimal stochastic bandit
algorithm by Zimmert and Seldin~[AISTAT'19, JMLR'21]. When the best arm
outperforms the other arms, the regret improves to . The regret bound in the later case is comparable to other optimal
contextual bandit results in more general cases, but our algorithm is easy to
analyze, runs very efficiently, and does not require an i.i.d. assumption on
the input context sequence. The algorithm also works with general graphs using
a standard random spanning tree reduction
Folding Every Point on a Polygon Boundary to a Point
We consider a problem in computational origami. Given a piece of paper as a
convex polygon and a point located within, fold every point on a
boundary of to and compute a region that is safe from folding, i.e.,
the region with no creases. This problem is an extended version of a problem by
Akitaya, Ballinger, Demaine, Hull, and Schmidt~[CCCG'21] that only folds
corners of the polygon. To find the region, we prove structural properties of
intersections of parabola-bounded regions and use them to devise a linear-time
algorithm. We also prove a structural result regarding the complexity of the
safe region as a variable of the location of point , i.e., the number of
arcs of the safe region can be determined using the straight skeleton of the
polygon .Comment: Preliminary results appeared in JCDCGGG'2
Faster Algorithms for Semi-Matching Problems
We consider the problem of finding \textit{semi-matching} in bipartite graphs
which is also extensively studied under various names in the scheduling
literature. We give faster algorithms for both weighted and unweighted case.
For the weighted case, we give an -time algorithm, where is
the number of vertices and is the number of edges, by exploiting the
geometric structure of the problem. This improves the classical
algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi
[Communications of the ACM 1974].
For the unweighted case, the bound could be improved even further. We give a
simple divide-and-conquer algorithm which runs in time,
improving two previous -time algorithms by Abraham [MSc thesis,
University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003
and Journal of Algorithms 2006]. We also extend this algorithm to solve the
\textit{Balance Edge Cover} problem in time, improving the
previous -time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC
2008].Comment: ICALP 201
A tight bound on approximating arbitrary metrics by tree metrics
In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion β¦(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems