arXiv subjects

Manish Purohit

Publications and source records attributed to Manish Purohit.

28 records · Page 2Linked to original sources

arXiv2020

Online Learning with Imperfect Hints

We consider a variant of the classical online linear optimization problem in which at every step, the online player receives a "hint" vector before choosing the action for that round. Rather surprisingly, it was shown that if the hint vector is guaranteed to have a positive correlation with the cost vector, then the online player can achieve a regret of $O(\log T)$, thus significantly improving over the $O(\sqrt{T})$ regret in the general setting. However, the result and analysis require the correlation property at \emph{all} time steps, thus raising the natural question: can we design online learning algorithms that are resilient to bad hints? In this paper we develop algorithms and nearly matching lower bounds for online learning with imperfect directional hints. Our algorithms are oblivious to the quality of the hints, and the regret bounds interpolate between the always-correlated hints case and the no-hints case. Our results also generalize, simplify, and improve upon previous results on optimistic regret bounds, which can be viewed as an additive version of hints.

cs.LG↗

arXiv2019

Semi-Online Bipartite Matching

In this paper we introduce the \emph{semi-online} model that generalizes the classical online computational model. The semi-online model postulates that the unknown future has a predictable part and an adversarial part; these parts can be arbitrarily interleaved. An algorithm in this model operates as in the standard online model, i.e., makes an irrevocable decision at each step. We consider bipartite matching in the semi-online model, for both integral and fractional cases. Our main contributions are competitive algorithms for this problem that are close to or match a hardness bound. The competitive ratio of the algorithms nicely interpolates between the truly offline setting (no adversarial part) and the truly online setting (no predictable part).

cs.DS↗

arXiv2019

Near Optimal Coflow Scheduling in Networks

The coflow scheduling problem has emerged as a popular abstraction in the last few years to study data communication problems within a data center. In this basic framework, each coflow has a set of communication demands and the goal is to schedule many coflows in a manner that minimizes the total weighted completion time. A coflow is said to complete when all its communication needs are met. This problem has been extremely well studied for the case of complete bipartite graphs that model a data center with full bisection bandwidth and several approximation algorithms and effective heuristics have been proposed recently. In this work, we study a slightly different model of coflow scheduling in general graphs (to capture traffic between data centers) and develop practical and efficient approximation algorithms for it. Our main result is a randomized 2 approximation algorithm for the single path and free path model, significantly improving prior work. In addition, we demonstrate via extensive experiments that the algorithm is practical, easy to implement and performs well in practice.

cs.DC↗

arXiv2019

Hiring Under Uncertainty

In this paper we introduce the hiring under uncertainty problem to model the questions faced by hiring committees in large enterprises and universities alike. Given a set of $n$ eligible candidates, the decision maker needs to choose the sequence of candidates to make offers so as to hire the $k$ best candidates. However, candidates may choose to reject an offer (for instance, due to a competing offer) and the decision maker has a time limit by which all positions must be filled. Given an estimate of the probabilities of acceptance for each candidate, the hiring under uncertainty problem is to design a strategy of making offers so that the total expected value of all candidates hired by the time limit is maximized. We provide a 2-approximation algorithm for the setting where offers must be made in sequence, an 8-approximation when offers may be made in parallel, and a 10-approximation for the more general stochastic knapsack setting with finite probes.

cs.DS↗

arXiv2018

A Tight Approximation for Co-flow Scheduling for Minimizing Total Weighted Completion Time

Co-flows model a modern scheduling setting that is commonly found in a variety of applications in distributed and cloud computing. In co-flow scheduling, there are $m$ input ports and $m$ output ports. Each co-flow $j \in J$ can be represented by a bipartite graph between the input and output ports, where each edge $(i,o)$ with demand $d_{i,o}^j$ means that $d_{i,o}^j$ units of packets must be delivered from port $i$ to port $o$. To complete co-flow $j$, we must satisfy all of its demands. Due to capacity constraints, a port can only transmit (or receive) one unit of data in unit time. A feasible schedule at each time $t$ must therefore be a bipartite matching. We consider co-flow scheduling and seek to optimize the popular objective of total weighted completion time. Our main result is a $(2+ε)$-approximation for this problem, which is essentially tight, as the problem is hard to approximate within a factor of $(2 - ε)$. This improves upon the previous best known 4-approximation. Further, our result holds even when jobs have release times without any loss in the approximation guarantee. The key idea of our approach is to construct a continuous-time schedule using a configuration linear program and interpret each job's completion time therein as the job's deadline. The continuous-time schedule serves as a witness schedule meeting the discovered deadlines, which allows us to reduce the problem to a deadline-constrained scheduling problem. * This result is flawed; see the first page for the details.

cs.DS↗

arXiv2015

On Correcting Inputs: Inverse Optimization for Online Structured Prediction

Algorithm designers typically assume that the input data is correct, and then proceed to find "optimal" or "sub-optimal" solutions using this input data. However this assumption of correct data does not always hold in practice, especially in the context of online learning systems where the objective is to learn appropriate feature weights given some training samples. Such scenarios necessitate the study of inverse optimization problems where one is given an input instance as well as a desired output and the task is to adjust the input data so that the given output is indeed optimal. Motivated by learning structured prediction models, in this paper we consider inverse optimization with a margin, i.e., we require the given output to be better than all other feasible outputs by a desired margin. We consider such inverse optimization problems for maximum weight matroid basis, matroid intersection, perfect matchings, minimum cost maximum flows, and shortest paths and derive the first known results for such problems with a non-zero margin. The effectiveness of these algorithmic approaches to online learning for structured prediction is also discussed.

cs.LG↗

arXiv2015

Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.

cs.DS↗

arXiv2015

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{Ω\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.

cs.DS↗

arXiv2013

Analyzing the Optimal Neighborhood: Algorithms for Budgeted and Partial Connected Dominating Set Problems

We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem, we are given an undirected graph G = (V,E) and an integer n', and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n' vertices. We obtain the first polynomial time algorithm with an O(\ln Δ) approximation factor for this problem, thereby significantly extending the results of Guha and Khuller (Algorithmica, Vol. 20(4), Pages 374-387, 1998) for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem, there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a (1/13)(1 - 1/e) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a monotone "special" submodular function. This generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies a O(\ln q) approximation (where q denotes the quota) and an O(1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function.

cs.DS↗

arXiv2013

Improved algorithms and analysis for the laminar matroid secretary problem

In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected solution weight is within a constant factor of the optimal) are known for many types of matroid secretary problems. We examine the laminar matroid and show an algorithm achieving provably 0.053 competitive ratio.

cs.GT↗