Search for the end of a path in the d-dimensional grid and in other graphs

We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a vertex $s \in V(G)$. We denote the directed graph obtained from $G$ by directing all edges in both directions by $G'$. $D$ is a directed subgraph of $G'$ which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in $s$. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex $v\in V(G)$, and the answer is the set of the edges of $D$ incident to $v$, together with their directions. We also show lower bounds for the special case when $D$ consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph $G$ is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as fixed. In order to do this, we prove a separator theorem about grid graphs, which is interesting on its own right

S2: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S2 for this task. At each step, S2 selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S2 queries for the label of the vertex that bisects the *shortest shortest* path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S2 needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S2 on both real and synthetic data. While other graph-based active learning algorithms have shown promise in practice, our algorithm is the first with both good performance and theoretical guarantees. Finally, we demonstrate the implications of the S2 algorithm to the theory of nonparametric active learning. In particular, we show that S2 achieves near minimax optimal excess risk for an important class of nonparametric classification problems.Comment: A version of this paper appears in the Conference on Learning Theory (COLT) 201

(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube \B^n, the randomized query complexity of Local Search is $\Theta(2^{n/2}n^{1/2})$ and the quantum query complexity is $\Theta(2^{n/3}n^{1/6})$. We also show that for the constant dimensional grid $[N^{1/d}]^d$, the randomized query complexity is $\Theta(N^{1/2})$ for $d \geq 4$ and the quantum query complexity is $\Theta(N^{1/3})$ for $d \geq 6$. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for $[N^{1/2}]^2$ a new upper bound of $O(N^{1/4}(\log\log N)^{3/2})$ on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3: a line for grant added. v4: upper bound section rewritte

Collaborative search on the plane without communication

We generalize the classical cow-path problem [7, 14, 38, 39] into a question that is relevant for collective foraging in animal groups. Specifically, we consider a setting in which k identical (probabilistic) agents, initially placed at some central location, collectively search for a treasure in the two-dimensional plane. The treasure is placed at a target location by an adversary and the goal is to find it as fast as possible as a function of both k and D, where D is the distance between the central location and the target. This is biologically motivated by cooperative, central place foraging such as performed by ants around their nest. In this type of search there is a strong preference to locate nearby food sources before those that are further away. Our focus is on trying to find what can be achieved if communication is limited or altogether absent. Indeed, to avoid overlaps agents must be highly dispersed making communication difficult. Furthermore, if agents do not commence the search in synchrony then even initial communication is problematic. This holds, in particular, with respect to the question of whether the agents can communicate and conclude their total number, k. It turns out that the knowledge of k by the individual agents is crucial for performance. Indeed, it is a straightforward observation that the time required for finding the treasure is $\Omega$(D + D 2 /k), and we show in this paper that this bound can be matched if the agents have knowledge of k up to some constant approximation. We present an almost tight bound for the competitive penalty that must be paid, in the running time, if agents have no information about k. Specifically, on the negative side, we show that in such a case, there is no algorithm whose competitiveness is O(log k). On the other hand, we show that for every constant \epsilon \textgreater{} 0, there exists a rather simple uniform search algorithm which is $O( \log^{1+\epsilon} k)$-competitive. In addition, we give a lower bound for the setting in which agents are given some estimation of k. As a special case, this lower bound implies that for any constant \epsilon \textgreater{} 0, if each agent is given a (one-sided) $k^\epsilon$-approximation to k, then the competitiveness is $\Omega$(log k). Informally, our results imply that the agents can potentially perform well without any knowledge of their total number k, however, to further improve, they must be given a relatively good approximation of k. Finally, we propose a uniform algorithm that is both efficient and extremely simple suggesting its relevance for actual biological scenarios

The Lazy Flipper: MAP Inference in Higher-Order Graphical Models by Depth-limited Exhaustive Search

This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.Comment: C++ Source Code available from http://hci.iwr.uni-heidelberg.de/software.ph

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by (Allender et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple