In the classical survivable network design problem (SNDP), we are given an undirected graph G=(V,E) with costs on edges and a connectivity requirement k(s,t) for each pair of vertices. The goal is to find a minimum-cost subgraph H⊆ G such that every pair (s,t) is connected by k(s,t) edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an O(k 3 log n )-approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput.’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This article concerns the generalization of EC-SNDP into the subset-to-subset setting, namely Group EC-SNDP. We develop a framework, which yields the first non-trivial (true) approximation algorithm for Group EC-SNDP. Previously, only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15], and a true approximation algorithm is known only for the single-source variant with connectivity requirement k(S,T) ∈ { 0,1,2} [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor. Comput. Sci.’12]. On the negative side, in terms of the number of connectivity demands q , we give an Ω (q /log q )-hardness result for large k , complementing the previous inapproximability results, e.g., hardness in terms of k : k 1/5-ɛ -hardness [Cheriyan et al., SODA’12; Laekhanukit, SODA’14; Chalermsook et al., SODA’15; Manurangsi, IPL’19]; hardness in terms of n : 2 log 1-ɛ n -hardness [Chalermsook et al., SODA’15].

We consider a generalized poset sorting problem (GPS), in which we are given a query graph $G = (V, E)$ and an unknown poset $\mathcal{P}(V, \prec)$ that is defined on the same vertex set $V$, and the goal is to make as few queries as possible to edges in $G$ in order to fully recover $\mathcal{P}$, where each query $(u, v)$ returns the relation between $u, v$, i.e., $u \prec v$, $v \prec u$ or $u \not \sim v$. This generalizes both the poset sorting problem [Faigle et al., SICOMP 88] and the generalized sorting problem [Huang et al., FOCS 11]. We give algorithms with $\tilde{O}(n\cdot \mathrm{poly}(k))$ query complexity when $G$ is a complete bipartite graph or $G$ is stochastic under the \ER model, where $k$ is the \emph{width} of the poset, and these generalize [Daskalakis et al., SICOMP 11] which only studies complete graph $G$. Both results are based on a unified framework that reduces the poset sorting to partitioning the vertices with respect to a given pivot element, which may be of independent interest. Our study of GPS also leads to a new $\tilde{O}(n^{1 - 1 / (2W)})$ competitive ratio for the so-called weighted generalized sorting problem where $W$ is the number of distinct weights in the query graph. This problem was considered as an open question in [Charikar et al., JCSS 02], and our result makes important progress as it yields the first nontrivial sublinear ratio for general weighted query graphs (for any bounded $W$). We obtain this via an $\tilde{O}(nk + n^{1.5})$ query complexity algorithm for the case where every edge in $G$ is guaranteed to be comparable in the poset, which generalizes a $\tilde{O}(n^{1.5})$ bound for generalized sorting [Huang et al., FOCS 11].

Online bipartite matching with one-sided arrival and its variants have been extensively studied since the seminal work of Karp, Vazirani, and Vazirani (STOC 1990). Motivated by real-life applications with dynamic market structures, e.g., ride-sharing, two generalizations of the classical one-sided arrival model are proposed to allow non-bipartite graphs and to allow all vertices to arrive online. Namely, online matching with general vertex arrival is introduced by Wang and Wong (ICALP 2015), and fully online matching is introduced by Huang et al. (JACM 2020).

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance.

We consider the Online Rent Minimization problem, where online jobs with release times, deadlines, and processing times must be scheduled on machines that can be rented for a fixed length period of $T$. The objective is to minimize the number of machine rents. This problem generalizes the Online Machine Minimization problem where machines can be rented for an infinite period, and both problems have an asymptotically optimal competitive ratio of $O(\log(p_{\max}/p_{\min}))$ for general processing times, where $p_{\max}$ and $p_{\min}$ are the maximum and minimum processing times respectively. However, for small values of $p_{\max}/p_{\min}$, a better competitive ratio can be achieved by assuming unit-size jobs. Under this assumption, Devanur et al. (2014) gave an optimal $e$-competitive algorithm for Online Machine Minimization, and Chen and Zhang (2022) gave a $(3e+7)\approx 15.16$-competitive algorithm for Online Rent Minimization. In this paper, we significantly improve the competitive ratio of the Online Rent Minimization problem under unit size to $6$, by using a clean oracle-based online algorithm framework.

There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze [ 9 ]. We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. [ 12 ]. We revisit theModified Randomized Greedy (MRG)algorithm by Aronson et al. [ 1 ] that is proved to achieve a (0.5+ε)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm namedRandom Decision Order (RDO)that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that theRDOalgorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio ofMRG. Furthermore, we generalize theRDOalgorithm to the edge-weighted case and prove that it achieves a 0.501-approximation ratio. This result solves the open question by Chan et al. [ 4 ] and Gamlath et al. [ 12 ] about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1-1/e)-approximation for edge-weighted bipartite graphs, which generalizes the (1-1/e)-approximation ratio of Gamlath et al. [ 12 ] for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.

In the classical survivable network design problem (SNDP), we are given an undirected graph $G-(V,E)$ with costs on edges and a connectivity requirement $k(5,t)$ for each pair of vertices. The goal is to find a minimum-cost subgraph $H\sqsubseteq G$ such that every pair $(s,t)$ are connected by $k(s,t)$ edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an $O(k^{3}\log n)-$ approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This paper concerns the generalization of SNDP into the subset-to-subset setting, namely Group EC-SNDR We develop the framework, which yields the first non-trivial (true) approximation algorithm for Group. EC-SNDE Previously only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15l, and a true approximation algorithm is known only for the single-source variant with connectivity requirement $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor Comput. Sci’12].

In the k-outconnected directed Steiner tree problem (k-DST), we are given an n-vertex directed graph G = (V,E) with edge costs, a connectivity requirement k, a root r ∈ V and a set of terminals T ⊆ V. The goal is to find a minimum-cost subgraph H ⊆ G that has k edge-disjoint paths from the root vertex r to every terminal t ∈ T. The problem is NP-hard, and inapproximability results are known in several parameters, e.g., hardness in terms of n: log^{2-ε}n-hardness for k = 1 [Halperin and Krauthgamer, STOC'03], 2^{log^{1-ε}n}-hardness for general case [Cheriyan, Laekhanukit, Naves and Vetta, SODA'12], hardness in terms of k [Cheriyan et al., SODA'12; Laekhanukit, SODA'14; Manurangsi, IPL'19] and hardness in terms of |T| [Laekhanukit, SODA'14]. In this paper, we show the approximation hardness of k-DST for various parameters. - Ω(|T|/log |T|)-approximation hardness, which holds under the standard complexity assumption NP≠ ZPP. The inapproximability ratio is tightened to Ω(|T|) under the Strongish Planted Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021]. The latter hardness result matches the approximation ratio of |T| obtained by a trivial approximation algorithm, thus closing the long-standing open problem. - Ω(2^{k/2} / k)-approximation hardness for the general case of k-DST under the assumption NP≠ZPP. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in k. - Ω((k/L)^{L/4})-approximation hardness for k-DST on L-layered graphs for L ≤ O(log n). This almost matches the approximation ratio of O(k^{L-1}⋅ L ⋅ log |T|) achieved in O(n^L)-time due to Laekhanukit [ICALP'16]. We further extend our hardness results in terms of |T| to the undirected cases of k-DST, namely the single-source k-vertex-connected Steiner tree and the k-edge-connected group Steiner tree problems. Thus, we obtain Ω(|T|/log |T|) and Ω(|T|) approximation hardness for both problems under the assumption NP≠ ZPP and the Strongish Planted Clique Hypothesis, respectively. This again matches the upper bound obtained by trivial algorithms.

We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order. We show that the problem is hard in both offline and online settings even when k = 1 and c = ∞: There is a hardness of approximation of Ω(n) for the offline problem, and a lower bound of Ω(n) on the competitive ratio of any online algorithm, where n is number of points in the metric. We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapacitated (i.e, c = ∞) food delivery problem on tree metrics; we also have negative result showing that the condition c = ∞ is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio.

We introduce the Online Food Delivery Problem (OFDP) to model the delivery problem commonly encountered in online food-ordering-and-delivery platforms. In the OFDP the requests (orders) are submitted online, and the depot (restaurant) needs to decide when to send out a server to serve the submitted requests. In addition, the server has to return to the depot (to pickup foods) before serving new requests. The objective is to minimize maximum flow time, i.e., the maximum time between the submission and completion of a request. This problem can also be viewed as a variant of the Online Dial-a-Ride problem, for which however the max flow time objective is inapproximable in general. We study the OFDP on star graphs, and give both algorithmic and hardness results. We analyze a natural greedy strategy and show that it achieves the optimal competitive ratio 3 among all myopic algorithms, which are algorithms that immediately send out the server whenever there are unserved requests. Then we prove that a far-sighted (i.e., non-myopic) algorithm with proper waiting strategy can achieve 8/3-competitive ratio. On the negative side, we give a simple lower bound example that excludes the possibility of any ( 2 − ϵ ) -competitive algorithms. • This paper introduces the Online Food Delivery Problem encountered in online food-ordering-and-delivery platforms. • We show that a natural greedy strategy achieves the optimal competitive ratio 3 among all myopic algorithms. • A far-sighted algorithm with proper waiting strategy can achieve 8/3-competitive ratio.

Inspired by a recent line of work in online algorithms with predictions, we study the constrained adversary model that utilizes predictions from a different perspective. Prior works mostly focused on designing simultaneously robust and consistent algorithms, without making assumptions on the quality of the predictions. In contrary, our model assumes the adversarial instance is consistent with the predictions and aim to design algorithms that have best worst-case performance against all such instances. We revisit classical online selection problems under the constrained adversary model. For the single item selection problem, we design an optimal algorithm in the adversarial arrival model and an improved algorithm in the random arrival model (a.k.a., the secretary problem). For the online edge-weighted bipartite matching problem, we extend the classical Water-filling and Ranking algorithms and achieve improved competitive ratios.

In the classic Directed Steiner Tree problem (DST), we are given an edge-weighted directed graph G = (V,E) with n nodes, a specified root node r ∈ V, and k terminals X ⊆ V-{r}. The goal is to find the cheapest F ⊆ E such that r can reach any terminal using only edges in F. Designing approximation algorithms for DST is quite challenging, to date the best approximation guarantee of a polynomial-time algorithm for DST is O(k^ε) for any constant ε > 0 [Charikar et al., 1999]. For network design problems like DST, one often relies on natural cut-based linear programming (LP) relaxations to design approximation algorithms. In general, the integrality gap of such an LP for DST is known to have a polynomial integrality gap lower bound [Zosin and Khuller, 2002; Li and Laekhanukit, 2021]. So particular interest has been invested in special cases or in strengthenings of this LP. In this work, we show the integrality gap is only O(log k) for instances of DST where no Steiner node has both an edge from another Steiner node and an edge to another Steiner node, i.e. the longest path using only Steiner nodes has length at most 1. This generalizes the well-studied case of quasi-bipartite DST where no edge has both endpoints being Steiner nodes. Our result is also optimal in the sense that the integrality gap can be as bad as poly(n) even if the longest path with only Steiner nodes has length 2.

Karp, Vazirani, and Vazirani (STOC 1990) initiated the study of online bipartite matching, which has held a central role in online algorithms ever since. Of particular importance are the Ranking algorithm for integral matching and the Water-filling algorithm for fractional matching. Most algorithms in the literature can be viewed as adaptations of these two in the corresponding models. Recently, Huang et al. (SODA 2019, JACM 2020) introduced a more general model called fully online matching, which considers general graphs and allows all vertices to arrive online. They also generalized Ranking and Water-filling to fully online matching and gave some tight analysis: Ranking is Ω ≈ 0.567-competitive on bipartite graphs where the Ω-constant satisfies ΩeΩ=1, and Water-filling is 2-√2 ≈ 0.585-competitive on general graphs. We propose fully online matching algorithms strictly better than Ranking and Water-filling. For integral matching on bipartite graphs, we build on the online primal dual analysis of Ranking and Water-filling to design a 0.569-competitive hybrid algorithm called Balanced Ranking. To our knowledge, it is the first integral algorithm in the online matching literature that successfully integrates ideas from Water-filling. For fractional matching on general graphs, we give a 0.592-competitive algorithm called Eager Water-filling, which may match a vertex on its arrival. By contrast, the original Water-filling algorithm always matches vertices at their deadlines. Our result for fractional matching further shows a separation between fully online matching and the general vertex arrival model by Wang and Wong (ICALP 2015), due to an upper bound of 0.5914 in the latter model by Buchbinder, Segev, and Tkach (ESA 2017).

Abstract. Three decades ago, Karp, Vazirani, and Vazirani [ Proceedings of the 22 nd Annual ACM Symposium on Theory of Computing, 1990, pp. 352–358] defined the online matching problem and gave an optimal [Formula: see text]-competitive algorithm. Fifteen years later, Mehta et al. [ J. ACM, 54 (2007), pp. 22:1–22:19] introduced the first generalization called AdWords driven by online advertising and obtained the optimal [Formula: see text] competitive ratio in the special case of small bids. It has been open ever since whether there is an algorithm for general bids better than the 0.5-competitive greedy algorithm. This paper presents a 0.5016-competitive algorithm for AdWords, answering this open question on the positive end. The algorithm builds on several ingredients, including a combination of the online primal dual framework and the configuration linear program of matching problems recently explored by Huang and Zhang [ Proceedings of the 52 nd ACM Symposium on Theory of Computing, 2020], a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach et al. [ Proceedings of the 61 st Annual IEEE Symposium on Foundations of Computer Science, 2020], which we call the panoramic online correlated selection.

There has been a long history for studying randomized greedy matching algorithms since the work by Dyer and Frieze(RSA 1991). We follow this trend and consider the problem formulated in the oblivious setting, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al.(RSA 1995) which is proved to be (0.5+epsilon)-approximate. In particular, we study a weaker version of the algorithm named Random Decision Order (RDO) that in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor if exists. We prove the RDO algorithm is 0.639-approximate and 0.531-approximate for bipartite graphs and general graphs respectively. As a corollary, we substantially improve the approximation ratio of MRG. Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501 approximation ratio. This result solves the open question by Chan et al.(SICOMP 2018) about the existence of an algorithm that beats greedy in this setting. As a corollary, it also solves the open questions by Gamlath et al.(SODA 2019) in the stochastic setting.

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, then the algorithm must irrevocably either match it to an unmatched neighbor or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, and so on. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step toward solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, then we show a tight competitive ratio ≈0.5671 of Ranking. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1- 1/e-competitive in our model, even for bipartite graphs.

Huang et al. (STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al. (STOC 1990) is strictly better than 0.5-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be (1 – 1/e)-competitive. This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the problem, we show that a natural instantiation of the water-filling algorithm is 2 – ≈ 0.585-competitive, together with a matching hardness result. Interestingly, our hardness result applies to arbitrary algorithms in the edge-arrival models of the online matching problem, improving the state-of-art upper bound. For integral algorithms, we show a tight competitive ratio of ≈ 0.567 for the ranking algorithm on bipartite graphs, matching a hardness result by Huang et al. (STOC 2018).

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and we prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time increases, and each online vertex matches the neighbor with the highest offer at its arrival.

We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of $(5 - \sqrt{5})/2 \approx 1.382$, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios. We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1−1/e-competitive in our model even for bipartite graphs.