arXiv subjects

MohammadTaghi Hajiaghayi

Publications and source records attributed to MohammadTaghi Hajiaghayi.

At least 109 records · Page 6Linked to original sources

arXiv2016

Price of Competition and Dueling Games

We study competition in a general framework introduced by Immorlica et al. and answer their main open question. Immorlica et al. considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user's satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an ordering over all possible webpages based on the submitted query. The user tends to choose the ordering which displays her requested webpage in a higher rank. The goal of both players is to maximize the probability that her ordering beats that of her opponent and gets the user's attention. Immorlica et al. show this game directs both players to provide suboptimal search results. However, they leave the following as their main open question: "does competition between algorithms improve or degrade expected performance?" In this paper, we resolve this question for the ranking duel and a more general class of dueling games. More precisely, we study the quality of orderings in a competition between two players. This game is a zero-sum game, and thus any Nash equilibrium of the game can be described by minimax strategies. Let the value of the user for an ordering be a function of the position of her requested item in the corresponding ordering, and the social welfare for an ordering be the expected value of the corresponding ordering for the user. We propose the price of competition which is the ratio of the social welfare for the worst minimax strategy to the social welfare obtained by a social planner. We use this criterion for analyzing the quality of orderings in the ranking duel. We prove the quality of minimax results is surprisingly close to that of the optimum solution.

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arXiv2016

Near-Optimal Disjoint-Path Facility Location Through Set Cover by Pairs

In this paper we consider two special cases of the "cover-by-pairs" optimization problem that arise when we need to place facilities so that each customer is served by two facilities that reach it by disjoint shortest paths. These problems arise in a network traffic monitoring scheme proposed by Breslau et al. and have potential applications to content distribution. The "set-disjoint" variant applies to networks that use the OSPF routing protocol, and the "path-disjoint" variant applies when MPLS routing is enabled, making better solutions possible at the cost of greater operational expense. Although we can prove that no polynomial-time algorithm can guarantee good solutions for either version, we are able to provide heuristics that do very well in practice on instances with real-world network structure. Fast implementations of the heuristics, made possible by exploiting mathematical observations about the relationship between the network instances and the corresponding instances of the cover-by-pairs problem, allow us to perform an extensive experimental evaluation of the heuristics and what the solutions they produce tell us about the effectiveness of the proposed monitoring scheme. For the set-disjoint variant, we validate our claim of near-optimality via a new lower-bounding integer programming formulation. Although computing this lower bound requires solving the NP-hard Hitting Set problem and can underestimate the optimal value by a linear factor in the worst case, it can be computed quickly by CPLEX, and it equals the optimal solution value for all the instances in our extensive testbed.

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arXiv2016

Designing FPT algorithms for cut problems using randomized contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. More precisely, we show the following: - We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2^{O(k^2\log |Σ|)}n^4\log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ| is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O(\sqrt{\log n}) to optimality, which improves over the trivial O(1) upper bound. - We prove that the Steiner Cut problem can be solved in 2^{O(k^2\log k)}n^4\log n deterministic time and \tilde{O}(2^{O(k^2\log k)}n^2) randomized time where k is the size of the cutset. This result improves the double exponential running time of the recent work of Kawarabayashi and Thorup (FOCS'11). - We show how to combine considering `cut' and `uncut' constraints at the same time. More precisely, we define a robust problem Node Multiway Cut-Uncut that can serve as an abstraction of introducing uncut constraints, and show that it admits an algorithm running in 2^{O(k^2\log k)}n^4\log n deterministic time where k is the size of the cutset. To the best of our knowledge, the only known way of tackling uncut constraints was via the approach of Marx, O'Sullivan and Razgon (STACS'10), which yields algorithms with double exponential running time. An interesting aspect of our technique is that, unlike important separators, it can handle real weights.

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arXiv2016

A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $ω: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} ω(e)\cdot ϕ(e)$ is minimized, where $ϕ(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. For each $k\geq 1$, we show the following: The \sharing problem can be solved in $n^{O(k)}$ time. A matching lower bound for our algorithm: the \sharing problem does not have an $f(k)\cdot n^{o(k)}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for \sharing relies on a structural result regarding an optimal solution followed by using the idea of a "token game" similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the \sharinggeneral problem if $\min\{k_1, k_2\}\geq 2$. Therefore \sharing is the most general problem one can attempt to solve with our techniques.

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arXiv2015

Prophet Secretary

Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold OPT/2 for every step, one can achieve the tight competitive ratio of 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive ratio of 1/e. In this paper, we introduce Prophet Secretary, a natural combination of the prophet inequality and the secretary problems. An example motivation for our problem is as follows. Consider a seller that has an item to sell on the market to a set of arriving customers. The seller knows the types of customers that may be interested in the item and he has a price distribution for each type: the price offered by a customer of a type is anticipated to be drawn from the corresponding distribution. However, the customers arrive in a random order. Upon the arrival of a customer, the seller makes an irrevocable decision whether to sell the item at the offered price. We address the question of finding a strategy for selling the item at a high price. We show that by using a uniform threshold one cannot break the 0.5 barrier. However, we show that i) using n distinct non-adaptive thresholds one can obtain a competitive ratio that goes to (1-1/e) as n grows; and ii) no online algorithm can achieve a competitive ratio better than 0.75. Our results improve the (asymptotic) approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1-1/e) when the order of agents (customers) is chosen randomly.

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arXiv2015

Applications of Uniform Sampling: Densest Subgraph and Beyond

Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bhattacharya et al. by providing a $(1-ε)$-approximation algorithm using $\tilde{O}(n)$ space. Our algorithm is conceptually simple - it samples $\tilde{O}(n)$ edges uniformly at random, and finds the densest subgraph on the sampled graph. We also show how to perform this sampling with update time $\tilde{O}(1)$. In addition to this, we show that given oracle access to the edge set, we can implement our algorithm in time $\tilde{O}(n)$ on a graph in the standard RAM model. To the best of our knowledge this is the fastest $(0.5-ε)$-approximation algorithm for the densest subgraph problem in the RAM model given such oracle access. Further, we extend our results to a general class of graph optimization problems that we call heavy subgraph problems. This class contains many interesting problems such as densest subgraph, directed densest subgraph, densest bipartite subgraph, $d$-cut and $d$-heavy connected component. Our result, by characterizing heavy subgraph problems, partially addresses open problem 13 at the IITK Workshop on Algorithms for Data Streams in 2006 regarding the effects of subsampling in this context.

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arXiv2015

Revenue Maximization for Selling Multiple Correlated Items

We study the problem of selling $n$ items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e., desirabilities of the buyer for the items are not drawn independently. Ideally, the goal is to design a mechanism to maximize the revenue. However, it has been shown that a revenue optimal mechanism might be very complicated and as a result inapplicable to real-world auctions. Therefore, our focus is on designing a simple mechanism that achieves a constant fraction of the optimal revenue. Babaioff et al. propose a simple mechanism that achieves a constant fraction of the optimal revenue for independent setting with a single additive buyer. However, they leave the following problem as an open question: "Is there a simple, approximately optimal mechanism for a single additive buyer whose value for $n$ items is sampled from a common base-value distribution?" Babaioff et al. show a constant approximation factor of the optimal revenue can be achieved by either selling the items separately or as a whole bundle in the independent setting. We show a similar result for the correlated setting when the desirabilities of the buyer are drawn from a common base-value distribution. It is worth mentioning that the core decomposition lemma which is mainly the heart of the proofs for efficiency of the mechanisms does not hold for correlated settings. Therefore we propose a modified version of this lemma which is applicable to the correlated settings as well. Although we apply this technique to show the proposed mechanism can guarantee a constant fraction of the optimal revenue in a very weak correlation, this method alone can not directly show the efficiency of the mechanism in stronger correlations.

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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.

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arXiv2015

Randomized Revenue Monotone Mechanisms for Online Advertising

Online advertising is the main source of revenue for many Internet firms. A central component of online advertising is the underlying mechanism that selects and prices the winning ads for a given ad slot. In this paper we study designing a mechanism for the Combinatorial Auction with Identical Items (CAII) in which we are interested in selling $k$ identical items to a group of bidders each demanding a certain number of items between $1$ and $k$. CAII generalizes important online advertising scenarios such as image-text and video-pod auctions [GK14]. In image-text auction we want to fill an advertising slot on a publisher's web page with either $k$ text-ads or a single image-ad and in video-pod auction we want to fill an advertising break of $k$ seconds with video-ads of possibly different durations. Our goal is to design truthful mechanisms that satisfy Revenue Monotonicity (RM). RM is a natural constraint which states that the revenue of a mechanism should not decrease if the number of participants increases or if a participant increases her bid. [GK14] showed that no deterministic RM mechanism can attain PoRM of less than $\ln(k)$ for CAII, i.e., no deterministic mechanism can attain more than $\frac{1}{\ln(k)}$ fraction of the maximum social welfare. [GK14] also design a mechanism with PoRM of $O(\ln^2(k))$ for CAII. In this paper, we seek to overcome the impossibility result of [GK14] for deterministic mechanisms by using the power of randomization. We show that by using randomization, one can attain a constant PoRM. In particular, we design a randomized RM mechanism with PoRM of $3$ for CAII.

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arXiv2015

Kernelization via Sampling with Applications to Dynamic Graph Streams

In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: -- Matching: First, there exists an $\tilde{O}(k^2)$ space algorithm that returns an exact maximum matching on the assumption the cardinality is at most $k$. The best previous algorithm used $\tilde{O}(kn)$ space where $n$ is the number of vertices in the graph and we prove our result is optimal up to logarithmic factors. Our algorithm has $\tilde{O}(1)$ update time. Second, there exists an $\tilde{O}(n^2/α^3)$ space algorithm that returns an $α$-approximation for matchings of arbitrary size. (Assadi et al. (2015) showed that this was optimal and independently and concurrently established the same upper bound.) We generalize both results for weighted matching. Third, there exists an $\tilde{O}(n^{4/5})$ space algorithm that returns a constant approximation in graphs with bounded arboricity. -- Vertex Cover and Hitting Set: There exists an $\tilde{O}(k^d)$ space algorithm that solves the minimum hitting set problem where $d$ is the cardinality of the input sets and $k$ is an upper bound on the size of the minimum hitting set. We prove this is optimal up to logarithmic factors. Our algorithm has $\tilde{O}(1)$ update time. The case $d=2$ corresponds to minimum vertex cover. Finally, we consider a larger family of parameterized problems (including $b$-matching, disjoint paths, vertex coloring among others) for which our subgraph sampling primitive yields fast, small-space dynamic graph stream algorithms. We then show lower bounds for natural problems outside this family.

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arXiv2014

Directed Subset Feedback Vertex Set is Fixed-Parameter Tractable

Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in the graph. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. In the subset versions of this problems, we are given an additional subset $S$ of vertices (resp., edges) and we want to hit all cycles passing through a vertex of $S$ (resp. an edge of $S$). Recently, the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al. (SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time $2^{O(k^3)}n^{O(1)}$. By our result, we complete the picture for feedback vertex set problems and their subset versions in undirected and directed graphs. Besides proving the fixed-parameter tractability of Directed Subset Feedback Vertex Set, we reformulate the random sampling of important separators technique in an abstract way that can be used for a general family of transversal problems. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from $2^{2^{O(k)}}$ to $2^{O(k^2)}$ in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).

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arXiv2014

Parameterized Streaming Algorithms for Vertex Cover

As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems where we are given a parameter $k$ and the objective is to decide whether there is a solution bounded by $k$. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of the Vertex Cover problem. We consider the following three models for a graph stream on $n$ nodes: 1. The insertion-only model where the edges can only be added. 2. The dynamic model where edges can be both inserted and deleted. 3. The \emph{promised} dynamic model where we are guaranteed that at each timestamp there is a solution of size at most $k$. In each of these three models we are able to design parameterized streaming algorithms for the Vertex Cover problem. We are also able to show matching lower bound for the space complexity of our algorithms. (Due to the arXiv limit of 1920 characters for abstract field, please see the abstract in the paper for detailed description of our results)

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arXiv2014

Network Cournot Competition

Cournot competition is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility---a function of the production cost as well as market price of the product---by deciding on the amount of production. In today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i.e., each market might be shared among a subset of these firms. In this situation, a bipartite graph models the access restriction where firms are on one side, markets are on the other side, and edges demonstrate whether a firm has access to a market or not. We call this game \emph{Network Cournot Competition} (NCC). In this paper, we propose algorithms for finding pure Nash equilibria of NCC games in different situations. First, we carefully design a potential function for NCC, when the price functions for markets are linear functions of the production in that market. However, for nonlinear price functions, this approach is not feasible. We model the problem as a nonlinear complementarity problem in this case, and design a polynomial-time algorithm that finds an equilibrium of the game for strongly convex cost functions and strongly monotone revenue functions. We also explore the class of price functions that ensures strong monotonicity of the revenue function, and show it consists of a broad class of functions. Moreover, we discuss the uniqueness of equilibria in both of these cases which means our algorithms find the unique equilibria of the games. Last but not least, when the cost of production in one market is independent from the cost of production in other markets for all firms, the problem can be separated into several independent classical \emph{Cournot Oligopoly} problems. We give the first combinatorial algorithm for this widely studied problem.

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arXiv2014

How to Influence People with Partial Incentives

We study the power of fractional allocations of resources to maximize influence in a network. This work extends in a natural way the well-studied model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a (small) seed set of nodes in a social network to influence directly, this influence cascades when other nodes reach certain thresholds of neighbor influence, and the goal is to maximize the final number of influenced nodes. Despite extensive study from both practical and theoretical viewpoints, this model limits the designer to a binary choice for each node, with no way to apply intermediate levels of influence. This model captures some settings precisely, e.g. exposure to an idea or pathogen, but it fails to capture very relevant concerns in others, for example, a manufacturer promoting a new product by distributing five "20% off" coupons instead of giving away one free product. While fractional versions of problems tend to be easier to solve than integral versions, for influence maximization, we show that the two versions have essentially the same computational complexity. On the other hand, the two versions can have vastly different solutions: the added flexibility of fractional allocation can lead to significantly improved influence. Our main theoretical contribution is to show how to adapt the major positive results from the integral case to the fractional case. Specifically, Mossel and Roch (2006) used the submodularity of influence to obtain their integral results; we introduce a new notion of continuous submodularity, and use this to obtain matching fractional results. We conclude that we can achieve the same greedy $(1-1/e-ε)$-approximation for the fractional case as the integral case. In practice, we find that the fractional model performs substantially better than the integral model, according to simulations on real-world social network data.

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arXiv2013

Scheduling a Cascade with Opposing Influences

Adoption or rejection of ideas, products, and technologies in a society is often governed by simultaneous propagation of positive and negative influences. Consider a planner trying to introduce an idea in different parts of a society at different times. How should the planner design a schedule considering this fact that positive reaction to the idea in early areas has a positive impact on probability of success in later areas, whereas a flopped reaction has exactly the opposite impact? We generalize a well-known economic model which has been recently used by Chierichetti, Kleinberg, and Panconesi (ACM EC'12). In this model the reaction of each area is determined by its initial preference and the reaction of early areas. We generalize previous works by studying the problem when people in different areas have various behaviors. We first prove, independent of the planner's schedule, influences help (resp., hurt) the planner to propagate her idea if it is an appealing (resp., unappealing) idea. We also study the problem of designing the optimal non-adaptive spreading strategy. In the non-adaptive spreading strategy, the schedule is fixed at the beginning and is never changed. Whereas, in adaptive spreading strategy the planner decides about the next move based on the current state of the cascade. We demonstrate that it is hard to propose a non-adaptive spreading strategy in general. Nevertheless, we propose an algorithm to find the best non-adaptive spreading strategy when probabilities of different behaviors of people in various areas drawn i.i.d from an unknown distribution. Then, we consider the influence propagation phenomenon when the underlying influence network can be any arbitrary graph. We show it is $\#P$-complete to compute the expected number of adopters for a given spreading strategy.

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arXiv2013

Fixed-Parameter and Approximation Algorithms: A New Look

A Fixed-Parameter Tractable (\FPT) $ρ$-approximation algorithm for a minimization (resp. maximization) parameterized problem $P$ is an FPT algorithm that, given an instance $(x, k)\in P$ computes a solution of cost at most $k \cdot ρ(k)$ (resp. $k/ρ(k)$) if a solution of cost at most (resp. at least) $k$ exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard {Clique} and W[2]-hard {Set Cover} problems, the natural question is whether we can get any \FPT-approximation. It is widely believed that both {Clique} and {Set-Cover} admit no FPT $ρ$-approximation algorithm, for any increasing function $ρ$. Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) \cite{eth-paturi} and the Projection Games Conjecture (PGC) \cite{r3}, we make the first progress towards proving this conjecture by showing that 1. Under the ETH and PGC, there exist constants $F_1, F_2 >0$ such that the {Set Cover} problem does not admit an FPT approximation algorithm with ratio $k^{F_1}$ in $2^{k^{F_2}}\cdot \text{poly}(N,M)$ time, where $N$ is the size of the universe and $M$ is the number of sets. 2. Unless $\NP\subseteq \SUBEXP$, for every $1> δ> 0$ there exists a constant $F(δ)>0$ such that {Clique} has no FPT cost approximation with ratio $k^{1-δ}$ in $2^{k^{F}}\cdot \text{poly}(n)$ time, where $n$ is the number of vertices in the graph. In the second part of the paper we consider various W[1]-hard problems such as {\dst}, {\dsf}, Directed Steiner Network and {\mec}. For all these problem we give polynomial time $f(\text{OPT})$-approximation algorithms for some small function $f$ (the largest approximation ratio we give is $\text{OPT}^2$).

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arXiv2013

Approximation Algorithms for Movement Repairmen

In the {\em Movement Repairmen (MR)} problem we are given a metric space $(V, d)$ along with a set $R$ of $k$ repairmen $r_1, r_2, ..., r_k$ with their start depots $s_1, s_2, ..., s_k \in V$ and speeds $v_1, v_2, ..., v_k \geq 0$ respectively and a set $C$ of $m$ clients $c_1, c_2, ..., c_m$ having start locations $s'_1, s'_2, ..., s'_m \in V$ and speeds $v'_1, v'_2, ..., v'_m \geq 0$ respectively. If $t$ is the earliest time a client $c_j$ is collocated with any repairman (say, $r_i$) at a node $u$, we say that the client is served by $r_i$ at $u$ and that its latency is $t$. The objective in the (\smr{}) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery \cite{amazon} and USPS gopost \cite{gopost}. We give the first $O(\log n)$-approximation algorithm for the \smr{} problem.

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arXiv2013

Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems

Moss and Rabani[12] study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(log n)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST). They use the algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm for the Budgeted node-weighted Steiner tree problem. Their solution may cost up to twice the budget, but collects a factor Omega(1/log n) of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual O(log h)-approximation algorithm for a more general problem, prize-collecting node-weighted Steiner forest, where we have (h) demands each requesting the connectivity of a pair of vertices. Our algorithm can be seen as a greedy algorithm which reduces the number of demands by choosing a structure with minimum cost-to-reduction ratio. This natural style of argument (also used by Klein and Ravi[10] and Guha et al.[8]) leads to a much simpler algorithm than that of Moss and Rabani[12] for PCST. Our second main contribution is for the Budgeted node-weighted Steiner tree problem, which is also an improvement to [12] and [8]. In the unrooted case, we improve upon an O(log^2(n))-approximation of [8], and present an O(log n)-approximation algorithm without any budget violation. For the rooted case, where a specified vertex has to appear in the solution tree, we improve the bicriteria result of [12] to a bicriteria approximation ratio of (1+eps, O(log n)/(eps^2)) for any positive (possibly subconstant) (eps). That is, for any permissible budget violation (1+eps), we present an algorithm achieving a tradeoff in the guarantee for prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in [12].

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