Viswanath Nagarajan

Non-monotone submodular maximization under matroid and knapsack constraints

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+ε)-approximation for the submodular maximization problem under k matroid constraints, and a (1/5-ε)-approximation algorithm for this problem subject to k knapsack constraints (ε>0 is any constant). We improve the approximation guarantee of our algorithm to 1/k+1+{1/k-1}+ε for k≥2 partition matroid constraints. This idea also gives a ({1/k+ε)-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.

Approximation algorithms for distance constrained vehicle routing problems

We study the distance constrained vehicle routing problem (DVRP) (Laporte et al., Networks 14 (1984), 47–61, Li et al., Oper Res 40 (1992), 790–799): given a set of vertices in a metric space, a specified depot, and a distance bound D, find a minimum cardinality set of tours originating at the depot that covers all vertices, such that each tour has length at most D. This problem is NP-complete, even when the underlying metric is induced by a weighted star. Our main result is a 2-approximation algorithm for DVRP on tree metrics; we also show that no approximation factor better than 1.5 is possible unless P = NP. For the problem on general metrics, we present a

Approximation algorithms for optimal decision trees and adaptive TSP problems

(O(\log {1 \over \varepsilon }),1 + \varepsilon )

Additive guarantees for degree bounded directed network design

**image** -bicriteria approximation algorithm: i.e., for any e > 0, it obtains a solution violating the length bound by a 1 + e factor while using at most

Approximating the k -multicut problem

O(\log {1 \over \varepsilon })

Additive Guarantees for Degree-Bounded Directed Network Design

**image** times the optimal number of vehicles.

On k -column sparse packing programs

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 the patient having any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? We settle the approximability of this problem by giving a tight O(logm)-approximation algorithm. We also consider a more substantial generalization, the Adaptive TSP problem, which can be used to model switching costs between tests in the optimal decision tree problem. Given an underlying metric space, a random subset S of cities is drawn from a known distribution, but S is initially unknown to us--we get information about whether any city is in S only when we visit the city in question. What is a good adaptive way of visiting all the cities in the random subset S while minimizing the expected distance traveled? For this adaptive TSP problem, we give the first poly-logarithmic approximation, and show that this algorithm is best possible unless we can improve the approximation guarantees for the well-known group Steiner tree problem.

Better Scalable Algorithms for Broadcast Scheduling

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 non-negative edge-costs, a connectivity requirement specified by an intersecting supermodular function f, and upper bounds av, bvv∈ 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 ≤ ε ≤ 1/2, computes an f-connected subgraph with in-degrees at most ⌈ av/1-ε ⌉ + 4, out-degrees at most ⌈ bv/1-ε ⌉ + 4, and cost at most 1/ε 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. Setting ε=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<ε <1, we show an instance where any arborescence with out-degrees at most bv/1-ε + O(1) has cost at least 1-o(1)/ε 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+ε)Δ*+O(log1+ε 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, which was used by Lau et al. and Singh and Lau 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 cut-constraints 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 Eii=1k, and non-negative integers bii=1k, 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 most r sets; this considerably improves the result of Bilo et al. A special case of this problem is degree-bounded minimum spanning tree, and our result gives a substantially easier proof of the recent +1 approximation of Singh and Lau.

Fairness and optimality in congestion games

We study the <i>k</i>-multicut problem: Given an edge-weighted undirected graph, a set of <i>l</i> pairs of vertices, and a target <i>k</i> ≤ <i>l</i>, find the minimum cost set of edges whose removal disconnects <i>at least k</i> pairs. This generalizes the well known multicut problem, where <i>k = l.</i> We show that the <i>k</i>-multicut problem on trees can be approximated within a factor of 8/3 + ε, for any fixed ε > 0, and within <i>O</i>(log² <i>n</i> log log <i>n</i>) on general graphs, where <i>n</i> is the number of vertices in the graph.For any fixed ε > 0, we also obtain a polynomial time algorithm for <i>k</i>-multicut on trees which returns a solution of cost at most (2 + ε) · <i>OPT</i>, that separates at least (1 - ε) · <i>k</i> pairs, where <i>OPT</i> is the cost of the optimal solution separating <i>k</i> pairs.Our techniques also give a simple 2-approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].

Poly-logarithmic Approximation Algorithms for Directed Vehicle Routing Problems

We present polynomial-time approximation algorithms for some degree-bounded directed network design problems. Our main result is for intersecting supermodular connectivity requirements with degree bounds: given a directed graph