Consider a random graph model where each possible edge e is present independently with some probability p e . Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most t i times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following:• We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.1 of unsuccessful dates a person might be willing to participate in, "timeouts" on vertices are also provided. More precisely, valid policies are allowed, for each vertex i, to only probe at most t i edges incident to i. Similar considerations arise in kidney exchanges, details of which appear in [7]. Chen et al. asked the question: how can we devise probing policies to maximize the expected cardinality (or weight) of the matching? They showed that the greedy algorithm that probes edges in decreasing order of p ij (as long as their endpoints had not timed out) was a 4-approximation to the cardinality version of the stochastic matching problem. This greedy algorithm (and other simple greedy schemes) can be seen to be arbitrarily bad in the presence of weights, and they left open the question of obtaining good algorithms to maximize the expected weight of the matching produced. In addition to being a natural generalization, weights can be used as a proxy for revenue generated in matchmaking services. (The unweighted case can be thought of as maximizing the social welfare.) In this paper, we resolve the main open question from Chen et al. [7]:Theorem 1 There is a 4-approximation algorithm for the weighted stochastic matching problem. For bipartite graphs, there is a 3-approximation algorithm.Our main idea is to use the knowledge of edge probabilities to solve a linear program where each edge e has a variable 0 ≤ y e ≤ 1 corresponding to the probability that a strategy probes e (over all possible realizations of the graph). This is similar to the approach for stochastic packing problems considered by Dean et al. [9,8]. We then give two d...

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard.In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a " 1 k+2+ 1 k + " -approximation for the submodular maximization problem under k matroid constraints, and a`1 5 − ´-approximation algorithm for this problem subject to k knapsack constraints ( > 0 is any constant). We improve the approximation guarantee of our algorithm to 1 k+1+ 1 k−1 + for k ≥ 2 partition matroid constraints. This idea also gives a " 1 k+ "-approximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which improves over the previously best known guarantee of 1 k+1 .

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [ST04], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor.Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set S ⊆ V of size |S| ≤ ρn with minimum edge-expansion. We give an O( log n log (1/ρ)) bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.

We study a general stochastic probing problem defined on a universe V , where each element e ∈ V is "active" independently with probability p e . Elements have weights {w e : e ∈ V } and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p e values-to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation:• the set Q of probed elements satisfy an "outer" packing constraint,• the set S of chosen elements satisfy an "inner" packing constraint.The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching [12,3] and Bayesian mechanism design [9], and can also handle more general constraints. As an application, we obtain the first polynomial-time Ω(1/k)-approximate "Sequential Posted Price Mechanism" under k-matroid intersection feasibility constraints, improving on prior work [9,25,19].

We present polynomial-time approximation algorithms for some degree-bounded directed network design problems. Our main result is for intersecting supermodular connectivity with degree bounds: given a directed graph G = (V, E) with nonnegative edge-costs, a connectivity requirement specified by an intersecting supermodular function f , and upper bounds {av, bv}v∈V on in-degrees and out-degrees of vertices, find a minimum-cost f -connected subgraph of G that satisfies the degree bounds. We give a bicriteria approximation algorithm that for any 0 ≤ ≤ 1 2 , computes an f -connected subgraph with in-degrees at most av 1− +4, out-degrees at most bv 1− + 4, and cost at most 1 times the optimum. This includes, as a special case, the minimum-cost degree-bounded arborescence problem. We also obtain similar results for the (more general) class of crossing supermodular requirements. Our result extends and improves the (3av + 4, 3bv + 4, 3)-approximation of Lau et al. [13]. Setting = 0, our result gives the first purely additive guarantee for the unweighted versions of these problems. Our algorithm is based on rounding an LP relaxation for the problem.We also prove that the above cost-degree trade-off (even for the degree-bounded arborescence problem) is optimal relative to the natural LP relaxation. For every 0 < < 1, we show an instance where any arborescence with out-degrees at most bv 1− +O(1) has cost at least 1−o(1) times the optimal LP value.For the special case of finding a minimum degree arborescence (without costs), we give a stronger +2 additive approximation. This improves on a result of Lau et al. [13] that gives a 2Δ * + 2 guarantee, and Klein et al. [11] that gives a (1 + )Δ * + O(log 1+ n) bound, where Δ * is the degree of the optimal arborescence. As a corollary of our result, we (almost) settle a conjecture of Bang-Jensen et al. [1] on low-degree arborescences. Our algorithms use the iterative rounding technique of Jain [9], which was used by Lau et al. [13] and Singh and Lau [19] in the context of degree-bounded network design. It is however non-trivial to extend these techniques to the directed setting without incurring a multiplicative violation in the degree bounds. This is due to the fact that known polyhedral characterization of arborescences has the cutconstraints which, along with degree-constraints, are unsuitable for arguing the existence of integral variables in a basic feasible solution. We overcome this difficulty by enhancing the iterative rounding steps and by means of stronger counting arguments. Our counting technique is quite general, and it also simplifies the proofs of many previous results.We also apply the technique to undirected graphs. We consider the minimum crossing spanning tree problem: given an undirected edge-weighted graph G, edge-subsets {Ei} k i=1 , and non-negative integers {bi} k i=1 , find a minimum-cost spanning tree (if it exists) in G that contains at most bi edges from each set Ei. We obtain a +(r − 1) additive approximation for this problem, when each edge lies in at...

We consider the problem of constructing optimal decision trees: given a collection of tests which can disambiguate between a set of m possible diseases, each test having a cost, and the a-priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This problem has been studied in several works, with O(log m)-approximations known in the special cases when either costs or probabilities are uniform. In this paper, we settle the approximability of the general problem by giving a tight O(log m)-approximation algorithm.We also consider a substantial generalization, the adaptive traveling salesman problem. Given an underlying metric space, a random subset S of vertices is drawn from a known distribution, but S is initially unknown-we get information about whether any vertex is in S only when it is visited. What is a good adaptive strategy to visit all vertices in the random subset S while minimizing the expected distance traveled? This problem has applications in routing message ferries in ad-hoc networks, and also models switching costs between tests in the optimal decision tree problem. We give a poly-logarithmic approximation algorithm for adaptive TSP, which is nearly best possible due to a connection to the well-known group Steiner tree problem. Finally, we consider the related adaptive traveling repairman problem, where the goal is to compute an adaptive tour minimizing the expected sum of arrival times of vertices in the random subset S; we obtain a poly-logarithmic approximation algorithm for this problem as well.

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