Parinya Chalermsook

Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset S c ⊆ [n] of items of interest, together with a budget B c , and we assume that there is an unlimited supply of each item. Once the prices are fixed for all items, each customer c buys a subset of items in S c , according to its buying rule. The goal is to set the item prices so as to maximize the total profit.

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.

Simple distributed algorithms for approximating minimum steiner trees

Given a network G=(V,E), edge weights w(.), and a set of terminals S⊆V, the minimum-weight Steiner tree problem is to find a tree in G that spans S with minimum weight. Most provable heuristics treat the network G is a metric; This assumption, in a distributed setting, cannot be easily achieved without a subtle overhead. We give a simple distributed algorithm based on a minimum spanning tree heuristic that returns a solution whose cost is within a factor of two of the optimal. The algorithm runs in time O(|V|log|V|) on a synchronous network. We also show that another heuristic based on iteratively finding shortest paths gives a Θ(log |V|)-approximation using a novel charging scheme based on low-congestion routing on trees. Both algorithms work for unit-cost and general cost cases. The algorithms also have applications in finding multicast trees in wireless ad hoc networks.

Improved hardness of approximation for stackelberg shortest-path pricing

We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2-o(1). We also argue that the nicely structured type of instance resulting from our reduction captures most of the challenges we face in dealing with the problem in general and, in particular, we show that the gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large.

Nearly Tight Approximability Results for Minimum Biclique Cover and Partition

In this paper, we consider the minimum biclique cover and minimum biclique partition problems on bipartite graphs. In the minimum biclique cover problem, we are given an input bipartite graph G = (V,E), and our goal is to compute the minimum number of complete bipartite subgraphs that cover all edges of G. This problem, besides its correspondence to a well-studied notion of bipartite dimension in graph theory, has applications in many other research areas such as artificial intelligence, computer security, automata theory, and biology. Since it is NP-hard, past research has focused on approximation algorithms, fixed parameter tractability, and special graph classes that admit polynomial time exact algorithms. For the minimum biclique partition problem, we are interested in a biclique cover that covers each edge exactly once.

Coloring and maximum independent set of rectangles

In this paper, we consider two geometric optimization problems: Rectangle Coloring problem (RCOL) and Maximum Independent Set of Rectangles (MISR). In RCOL, we are given a collection of n rectangles in the plane where overlapping rectangles need to be colored differently, and the goal is to find a coloring using minimum number of colors. Let q be the maximum clique size of the instance, i.e. the maximum number of rectangles containing the same point. We are interested in bounding the ratio σ(q) between the total number of colors used and the clique size. This problem was first raised by graph theory community in 1960 when the ratio of σ(q) = O(q) was proved. Over decades, except for special cases, only the constant in front of q has been improved. In this paper, we present a new bound for σ(q) that significantly improves the known bounds for a broad class of instances. The bound σ(q) has a strong connection with the integrality gap of natural LP relaxation for MISR, in which the input is a collection of rectangles where each rectangle is additionally associated with non-negative weight, and our objective is to find a maximum-weight independent set of rectangles. MISR has been studied extensively and has applications in various areas of computer science. Our new bounds for RCOL imply new approximation algorithms for a broad class of MISR, including (i) O(log log n) approximation algorithm for unweighted MISR, matching the result by Chalermsook and Chuzhoy, and (ii) O(log log n)-approximation algorithm for the MISR instances arising in the Unsplittable Flow Problem on paths. Our technique builds on and generalizes past works.

Pre-reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term f(G * H), e.g., to show that f(G * H) = f(G)f(H). In this paper, we study graph products in a non-standard form f(R[G * H]) where R is a “reduction”, a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. The first problem is minimum consistent deterministic finite automaton (DFA). We show a tight n1-ϵ approximation hardness, improving the n1/14-ϵ hardness of [Pitt and Warmuth, STOC 1989 and JACM 1993], where n is the sample size. (In fact, we also give improved hardnesses for the case of acyclic DFA and NFA.) Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NP ≠ RP (the weakest possible assumption). This affirmatively answers an open problem raised 25 years ago in the paper of Pitt and Warmuth and the survey of Pitt [All 1989]. Prior to our results, this hardness only follows from the stronger hardness of improperly learning DFAs, which requires stronger assumptions, i.e., either a cryptographic or an average case complexity assumption [Kearns and Valiant STOC 1989 and J. ACM 1994; Daniely et al. STOC 2014]. The second problem is edge-disjoint paths (EDP) on directed acyclic graphs (DAGs). This problem admits an O(√n)-approximation algorithm [Chekuri, Khanna, and Shepherd, Theory of Computing 2006] and a matching Ω(√n) integrality gap, but so far only an n1/26-ϵ hardness factor is known [Chuzhoy et al., STOC 2007]. (n denotes the number of vertices.) Our techniques give a tight n1/2-ϵ hardness for EDP on DAGs, thus resolving its approximability status. As by-products of our techniques: (i) We give a tight hardness of packing vertex-disjoint k-cycles for large k, complimenting [Guruswami and Lee, ECCC 2014] and matching [Krivelevich et al., SODA 2005 and ACM Transactions on Algorithms 2007]. (ii) We give an alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst. Sci. 2008]. Our new concept reduces the task of proving hardnesses to merely analyzing graph product inequalities, which are often as simple as textbook exercises. This concept was inspired by, and can be viewed as a generalization of, the graph product subadditivity technique we previously introduced in SODA 2013. This more general concept might be useful in proving other hardness results as well.

Submodular unsplittable flow on trees

AbstractWe study the Unsplittable Flow problem (

New Integrality Gap Results for the Firefighters Problem on Trees

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