Bundit Laekhanukit

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the bipartite induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: For any r larger than some constant, any r-approximation algorithm for the independent set problem must run in at least 2n1-ε/r1+ε time. This nearly matches the upper bound of 2n/r [23]. It also improves some hardness results in the domain of parameterized complexity (e.g., [26], [19]). For any k larger than some constant, there is no polynomial time min{k1-ε, n1/2-ε} time min -approximation algorithm for the k-hypergraph pricing problem , where n is the number of vertices in an input graph. This almost matches the upper bound of min{O(k), Õ(√n) } min (by Balcan and Blum [3] and an algorithm in this paper). We note an interesting fact that, in contrast to n1/2-ε hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits nδ approximation for any δ > 0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time. The proofs of our hardness results rely on unexpectedly tight connections between the three problems. First, we establish a connection between the first and second problems by proving a new graph-theoretic property related to an induced matching number of dispersers. Then, we show that the n1/2-ε hardness of the last problem follows from nearly tight subexponential time inapproximability of the first problem, illustrating a rare application of the second type of inapproximability result to the first one. Finally, to prove the subexponential-time inapproximability of the first problem, we construct a new PCP with several properties; it is sparse and has nearly-linear size, large degree, and small free-bit complexity. Our PCP requires no ground-breaking ideas but rather a very careful assembly of the existing ingredients in the PCP literature.

Faster Algorithms for Semi-Matching Problems

We consider the problem of finding <i>semi-matching</i> in bipartite graphs, which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted cases. For the weighted case, we give an <i>O</i>(<i>nm</i>log <i>n</i>)-time algorithm, where <i>n</i> is the number of vertices and <i>m</i> is the number of edges, by exploiting the geometric structure of the problem. This improves the classical <i>O</i>(<i>n</i>³)-time algorithms by Horn [1973] and Bruno et al. [1974b]. For the unweighted case, the bound can be improved even further. We give a simple divide-and-conquer algorithm that runs in <i>O</i>(√<i>nm</i>log <i>n</i>) time, improving two previous <i>O</i>(<i>nm</i>)-time algorithms by Abraham [2003] and Harvey et al. [2003, 2006]. We also extend this algorithm to solve the <i>Balanced Edge Cover</i> problem in <i>O</i>(√<i>nm</i>log <i>n</i>) time, improving the previous <i>O</i>(<i>nm</i>)-time algorithm by Harada et al. [2008].

Approximating Rooted Steiner Networks

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques, we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(kε) hardness bound for the rooted k-connectivity problem in undirected graphs. As a consequence, we obtain an Ω(kε) hardness bound for the undirected subset k-connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k-connectivity problem.

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We present an

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-Approximation Algorithm for the

-approximation algorithm for the problem of finding a

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-Vertex Connected Spanning Subgraph Problem

-vertex connected spanning subgraph of minimum cost, where

Improved hardness of approximation for stackelberg shortest-path pricing

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An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem

is the number of vertices in an input graph, and