Balancing Relevance and Diversity in Online Bipartite Matching via Submodularity

In bipartite matching problems, vertices on one side of a bipartite graph are paired with those on the other. In its online variant, one side of the graph is available offline, while the vertices on the other side arrive online. When a vertex arrives, an irrevocable and immediate decision should be made by the algorithm; either match it to an available vertex or drop it. Examples of such problems include matching workers to firms, advertisers to keywords, organs to patients, and so on. Much of the literature focuses on maximizing the total relevance---modeled via total weight---of the matching. However, in many real-world problems, it is also important to consider contributions of diversity: hiring a diverse pool of candidates, displaying a relevant but diverse set of ads, and so on. In this paper, we propose the Online Submodular Bipartite Matching (\osbm) problem, where the goal is to maximize a submodular function $f$ over the set of matched edges. This objective is general enough to capture the notion of both diversity (\emph{e.g.,} a weighted coverage function) and relevance (\emph{e.g.,} the traditional linear function)---as well as many other natural objective functions occurring in practice (\emph{e.g.,} limited total budget in advertising settings). We propose novel algorithms that have provable guarantees and are essentially optimal when restricted to various special cases. We also run experiments on real-world and synthetic datasets to validate our algorithms.Comment: To appear in AAAI 201

Buyback Problem - Approximate matroid intersection with cancellation costs

In the buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to some constraints on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost that is a fixed fraction of the bid value. Previous to our work, deterministic and randomized algorithms were known when the constraint is a matroid constraint. We extend this and give a deterministic algorithm for the case when the constraint is an intersection of $k$ matroid constraints. We further prove a matching lower bound on the competitive ratio for this problem and extend our results to arbitrary downward closed set systems. This problem has applications to banner advertisement, semi-streaming, routing, load balancing and other problems where preemption or cancellation of previous allocations is allowed

Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $\Omega(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes value oracle access to $f$ and membership oracles for the matroids defining the $p$-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201

Submodular Secretary Problem with Shortlists under General Constraints

In submodular k-secretary problem, the goal is to select k items in a randomly ordered input so as to maximize the expected value of a given monotone submodular function on the set of selected items. In this paper, we introduce a relaxation of this problem, which we refer to as submodular k-secretary problem with shortlists. In the proposed problem setting, the algorithm is allowed to choose more than k items as part of a shortlist. Then, after seeing the entire input, the algorithm can choose a subset of size k from the bigger set of items in the shortlist. We are interested in understanding to what extent this relaxation can improve the achievable competitive ratio for the submodular k-secretary problem. In particular, using an O(k) shortlist, can an online algorithm achieve a competitive ratio close to the best achievable online approximation factor for this problem? We answer this question affirmatively by giving a polynomial time algorithm that achieves a 1 - 1/e - epsilon -O(k^{-1}) competitive ratio for any constant epsilon>0, using a shortlist of size eta {epsilon}(k)=O(k). Also, for the special case of m-submodular functions, we demonstrate an algorithm that achieves a 1 - epsilon competitive ratio for any constant epsilon > 0, using an O(1) shortlist. Finally, we show that our algorithm can be implemented in the streaming setting using a memory buffer of size eta{epsilon}(k)=O(k) to achieve a 1 - 1/e - epsilon - O(k^{-1}) approximation for submodular function maximization in the random order streaming model. This substantially improves upon the previously best known approximation factor of 1/2 + 8*10^{-14} [Norouzi-Fard et al. 2018] that used a memory buffer of size O(k log k). We further generalize our results to the case of matroid constraints. We design an algorithm that achieves a 1/2(1 - 1/e^2 - epsilon - O(1/k)) competitive ratio for any constant epsilon>0, using a shortlist of size O(k). This is especially surprising considering that the best known competitive ratio for the matroid secretary problem is O(log log k). An important application of our algorithm is for the random order streaming of submodular functions. We show that our algorithm can be implemented in the streaming setting using O(k) memory. It achieves a 1/2 (1 - 1/e^2 - epsilon - O(1/k)) approximation. The previously best known approximation ratio for streaming submodular maximization under matroid constraint is 0.25 (adversarial order) due to [Feldman et al.], [Chekuri et al.], and [Chakrabarti et al.]. Moreover, we generalize our results to the case of p-matchoid constraints and give a frac{1}{p+1}(1 - 1/e^{p+1} - epsilon - O(1/k)) approximation using O(k) memory, which asymptotically approaches the best known offline guarantee frac{1}{p+1} [Nemhauser et al.]. Finally we empirically evaluate our results on real world data sets such as YouTube video and Twitter stream