Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ∈ E}, and G has edge weights (w_{e})_{e ∈ E} or weights (w_u)_{u ∈ U}. Additionally, G has a downward-closed set of probing constraints (𝒞_{v})_{v ∈ V}, where 𝒞_v indicates which sequences of edges adjacent to an online v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each 𝒞_v. In the weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of 𝒞_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each 𝒞_v corresponds to a patience constraint 𝓁_v (i.e., 𝓁_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e. - When the offline vertices are unweighted. - When the online probabilities are vertex uniform; i.e., p_{u,v} = p_v for all (u,v) ∈ E. - When the patience constraint 𝓁_v satisfies 𝓁_v ∈ {[1,|U|} for every online vertex; i.e., every online either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem.

We study competitive online algorithms for EV (electrical vehicle) charging under the scenario of an aggregator serving a large number of EVs together with its background load, using both its own renewable energy (for free) and the energy procured from the external grid. The goal of the aggregator is to minimize its peak procurement from the grid, subject to the constraint that each EV has to be fully charged before its deadline. Further, the aggregator can predict the future demand and the renewable energy supply with some levels of uncertainty. The key challenge here is how to develop a model that captures the prior knowledge from such prediction, and how to best utilize this prior knowledge to reduce the peak under future uncertainty. In this paper, we first propose a 2-level increasing precision model (2-IPM), to capture the system uncertainty. We develop a powerful computation approach that can compute the optimal competitive ratio under 2-IPM over any online algorithm, and also online algorithms that can achieve the optimal competitive ratio. A dilemma for online algorithm design is that an online algorithm with good competitive ratio may exhibit poor average-case performance. We then propose a new Algorithm-Robustification procedure that can convert an online algorithm with reasonable average-case performance to one with both the optimal competitive ratio and good average-case performance. The robustified version of a well-known heuristic algorithm, Receding Horizon Control (RHC), is found to demonstrate superior performance via trace-based simulations.

We study the admission control problem in general networks. Communication requests arrive over time, and the online algorithm accepts or rejects each request while maintaining the capacity limitations of the network. The admission control problem has been usually analyzed as a benefit problem, where the goal is to devise an online algorithm that accepts the maximum number of requests possible. The problem with this objective function is that even algorithms with optimal competitive ratios may reject almost all of the requests, when it would have been possible to reject only a few. This could be inappropriate for settings in which rejections are intended to be rare events.In this paper, we consider preemptive online algorithms whose goal is to minimize the number of rejected requests. Each request arrives together with the path it should be routed on. We show an O(log2(mc))-competitive randomized algorithm for the weighted case, where m is the number of edges in the graph and c is the maximum edge capacity. For the unweighted case, we give an O(log m log c)-competitive randomized algorithm. This settles an open question of Blum, Kalai and Kleinberg raised in [10]. We note that allowing preemption and handling requests with given paths are essential for avoiding trivial lower bounds.The admission control problem is a generalization of the online set cover with repetitions problem, whose input is a family of m subsets of a ground set of n elements. Elements of the ground set are given to the online algorithm one by one, possibly requesting each element a multiple number of times. (If each element arrives at most once, this corresponds to the online set cover problem.) The algorithm must cover each element by different subsets, according to the number of times it has been requested.We give an O(log m log n)-competitive randomized algorithm for the online set cover with repetitions problem. This matches a recent lower bound of Ω(log m log n) given by Feige and Korman for the competitive ratio of any randomized polynomial time algorithm, under the BPP ≠ NP assumption. Given any constant e > 0, an O(log m logn)-competitive deterministic bicriteria algorithm is shown that covers each element by at least (1-e)k sets, where k is the number of times the element is covered by the optimal solution.

Cloud providers such as Amazon EC2 and Google GCE typically price their idle compute resources at a significant discount in the form of transient servers. Unlike regular servers, cloud transient servers have no reliability guarantee and can be revoked anytime. In order to enjoy the price premium of transient servers while mitigating the impact of indefinite server revocations, cloud users continuously checkpoint current computational states to stable storage. To determine the optimal checkpointing interval, prior works have largely built upon predicting future revocations. However, we argue that existing prediction models may not work well if providers employ different server revocation algorithms in the future, and the required historical information can be unavailable in practice. In this work, we propose two online algorithms, one competitive and the other heuristic, that determine the checkpointing intervals without predicting future revocation based on history. Our competitive online algorithm achieves the best possible competitive ratio against the optimal, yet impractical, offline solution. Our heuristic algorithm adaptively reacts to server revocations, achieving better average performance than the competitive algorithm, especially for short-running tasks. Extensive trace-driven simulations confirm our analytical results and demonstrate the efficacy of the two online algorithms in the context of Amazon EC2 Spot Instances.

We consider the problem of on-line call admission and routing on trees and meshes. Previous work gave randomized on-line algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probability. We investigate the question of devising for these problems on-line competitive algorithms that also guarantee a "good" solution with "good" probability. We give a new family of randomized algorithms with asymptotically optimal competitive ratios and "good" probability to get a profit close to the expectation. We complement these results by providing bounds on the probability of any optimally competitive randomized on-line algorithm for the problems we consider to get a profit close to the expectation. To the best of our knowledge, this is the first study of the relationship between the tail distribution and the competitive ratio of randomized on-line benefit algorithms.

Scheduling multicasts on unit-capacity trees and meshes

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.

We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless $NP = RP$. Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d.items and valuations satisfying diminishing returns is $(1-1/e)$-competitive, which is optimal even for coverage valuations, unless $NP=RP$. For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem.

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

Consider the following generalized notion of graph coloring: a coloring of the vertices of a graph G is valid w.r.t. some given graph F if there is no copy of F in G whose vertices all receive the same color. We study the problem of computing valid colorings of the binomial random graph Gn,p on n vertices with edge probability p = p(n) in the following online setting: the vertices of an initially hidden instance of Gn,p are revealed one by one (together with all edges leading to previously revealed vertices) and have to be colored immediately and irrevocably with one of r available colors. It is known that for any fixed graph F and any fixed integer r ≥ 2 this problem has a threshold p0 (F, r, n) in the following sense: For any function p(n) = o(p0) there is a strategy that a.a.s. (asymptotically almost surely, i.e., with probability tending to 1 as n tends to infinity) finds an r-coloring of Gn,p that is valid w.r.t. F online, and for any function p(n) = ω(p0) any online strategy will a.a.s. fail to do so. In this work we establish a general correspondence between this probabilistic problem and a deterministic two-player game in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. This characterization allows us to compute, for any F and r, a value γ = γ(F, r) such that the threshold of the probabilistic problem is given by p0(F, r, n) = n−γ. Our approach yields polynomial-time coloring algorithms that a.a.s. find valid colorings of Gn,p online in the entire regime below the respective thresholds, i.e., for any p(n) = o(n−γ).

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.

In this article, we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit , we present a (randomized) online algorithm with a competitive ratio of ln( p max − p min ) + 1, when bids are in a range [ p min , p max ], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. We also show that if the online algorithm is allowed to subsidize matches---match money-losing pairs if it has already collected enough money from previous pairs to pay for them---then it can actually be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, for maximizing the number of trades, the ability to subsidize is at least as valuable as knowing the future. We also consider objectives of maximizing buy or sell volume and social welfare. We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.We also consider the issue of incentive compatibility, and develop a nearly optimal incentive-compatible algorithm for maximizing social welfare. For maximizing profit , we show that no incentive-compatible algorithm can achieve a sublinear competitive ratio, even if only one buy bid and one sell bid are alive at a time. However, we provide an algorithm that, under certain mild assumptions on the bids, performs nearly as well as the best fixed pair of buy and sell prices, a weaker but still natural performance measure. This latter result uses online learning methods, and we also show how such methods can be used to improve our “optimal” algorithms to a broader notion of optimality. Finally, we show how some of our results can be generalized to settings in which the buyers and sellers themselves have online bidding strategies, rather than just each having individual bids.

Online algorithms are usually analyzed using the notion of competitive ratio which compares the solution obtained by the algorithm to that obtained by an online adversary for the worst possible input sequence. Often this measure turns out to be too pessimistic, and one popular approach especially for scheduling problems has been that of which was first proposed by Kalyanasundaram and Pruhs. Although resource augmentation has been very successful in dealing with a variety of objective functions, there are problems for which even a (arbitrary) constant speedup cannot lead to a constant competitive algorithm. In this paper we propose a model which requires no resource augmentation but which permits the online algorithm to not serve an epsilon-fraction of the requests.The problems considered in this paper are in the restricted assignment setting where each job can be assigned only to a subset of machines. For the load balancing problem where the objective is to minimize the maximum load on any machine, we give O(log2 1/e)-competitive algorithm which rejects at most an e-fraction of the jobs. For the problem of minimizing the maximum weighted flow-time, we give an O(1/e4)-competitive algorithm which can reject at most an e-fraction of the jobs by weight. We also extend this result to a more general setting where the weights of a job for measuring its weighted flow-time and its contribution towards total allowed rejection weight are different. This is useful, for instance, when we consider the objective of minimizing the maximum stretch. We obtain an O(1/e6)-competitive algorithm in this case.Our algorithms are immediate dispatch, though they may not be immediate reject. All these problems have strong lower bounds in speed augmentation model.

On-line scheduling of parallel jobs with runtime restrictions

The planted clique conjecture states that no polynomial-time algorithm can find a hidden clique of size k ≪ √n in an n-vertex Erdős–Rényi random graph with a k-clique planted. In this paper, we prove the equivalence among many (in fact, most) variants of planted clique conjectures, such as search versions with a success probability exponentially close to 1 and with a non-negligible success probability, a worst-case version (the k-clique problem on incompressible graphs), decision versions with small and large success probabilities, and decision versions with adversarially chosen k and binomially distributed k. In particular, we establish the equivalence between the planted clique problem introduced by Jerrum and Kučera and its decision version suggested by Saks in the 1990s. Moreover, the equivalence among decision versions identifies the optimality of a simple edge counting algorithm: By counting the number of edges, one can efficiently distinguish an n-vertex random graph from a random graph with a k-clique planted with probability Θ(k2/n) for any k ≤ √n. We show that for any k, no polynomial-time algorithm can distinguish these two random graphs with probability ≫ k2 / n if and only if the planted clique conjecture holds. The equivalence among search versions identifies the first one-way function that admits a polynomial-time security-preserving self-reduction from exponentially weak to strong one-way functions. These results reveal a detection-recovery gap in success probabilities for the planted clique problem. We also present another equivalence between the existence of a refutation algorithm for the planted clique problem and an average-case polynomial-time algorithm for the k-clique problem with respect to the Erdős–Rényi random graph.