In the Network Flow Interdiction problem an adversary attacks a network in order to minimize the maximum s-t-flow. Very little is known about the approximatibility of this problem despite decades of interest in it. We present the first approximation hardness, showing that Network Flow Interdiction and several of its variants cannot be much easier to approximate than Densest k-Subgraph. In particular, any $n^{o(1)}$-approximation algorithm for Network Flow Interdiction would imply an $n^{o(1)}$-approximation algorithm for Densest k-Subgraph. We complement this hardness results with the first approximation algorithm for Network Flow Interdiction, which has approximation ratio 2(n-1). We also show that Network Flow Interdiction is essentially the same as the Budgeted Minimum s-t-Cut problem, and transferring our results gives the first approximation hardness and algorithm for that problem, as well.
Studying on networked systems, in which a communication between nodes is functional if their distance under a given metric is lower than a pre-defined threshold, has received significant attention recently. In this work, we propose a metric to measure network resilience on guaranteeing the pre-defined performance constraint. This metric is investigated under an optimization problem, namely \textbf{Length-bounded Paths Interdiction in Continuous Domain} (cLPI), which aims to identify a minimum set of nodes whose changes cause routing paths between nodes become undesirable for the network servic…
We study a bilevel optimization problem which is a zero-sum Stackelberg game. In this problem, there are two players, a leader and a follower, who pick items from a common set. Both the leader and the follower have their own (multi-dimensional) budgets, respectively. Each item is associated with a profit, which is the same to the leader and the follower, and will consume the leader's (follower's) budget if it is selected by the leader (follower). The leader and the follower will select items in a sequential way: First, the leader selects items within the leader's budget. Then the follower sele…
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.
Indian Statistical Institute Kolkata（印度统计学院加尔各答）；Indian Institute of Technology Delhi（印度理工学院德里分校）
The Euclidean $k$-median problem is defined in the following manner: given a set $\mathcal{X}$ of $n$ points in $\mathbb{R}^{d}$, and an integer $k$, find a set $C \subset \mathbb{R}^{d}$ of $k$ points (called centers) such that the cost function $Φ(C,\mathcal{X}) \equiv \sum_{x \in \mathcal{X}} \min_{c \in C} \|x-c\|_{2}$ is minimized. The Euclidean $k$-means problem is defined similarly by replacing the distance with squared distance in the cost function. Various hardness of approximation results are known for the Euclidean $k$-means problem. However, no hardness of approximation results wer…
We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the $d$-to-$1$ conjecture, and hardness results for $2$-CSP's that can be obtained via standard techniques, such as Parallel Repetition combined with standard $2$-prover protocols for the 3SAT problem. We hope that this work will motivate fu…
We consider the following class of submodular k-multiway partitioning problems: (Sub-$k$-MP) $\min \sum_{i=1}^k f(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k = V \mbox{ and } S_i \neq \emptyset \mbox{ for all }i\in [k]$. Here $f$ is a non-negative submodular function, and $\uplus$ denotes the union of disjoint sets. Hence the goal is to partition $V$ into $k$ non-empty sets $S_1,S_2,\ldots,S_k$ such that $\sum_{i=1}^k f(S_i)$ is minimized. These problems were introduced by Zhao et al. partly motivated by applications to network reliability analysis, VLSI design, hypergraph cut, and other par…
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
我们提出了一系列几乎解决的不可逼近性结果的三个基本问题。在我们的系列中的第一个是最大独立集问题的次指数时间不可近似性，这是在参数化复杂性领域研究的一个问题。第二个问题是有界度二部图的最大诱导匹配问题的逼近困难。在我们的系列中的最后一个是严格的硬度近似的k-超图定价问题，一个基本的问题，从算法博弈论的领域。特别地，假设指数时间假设，我们的两个主要结果是：-对于任何大于某个常数的r，任何求解最大独立集问题的r-近似算法必须至少运行2^{n^{1-ε}/r^{1+ε}}时间。这几乎与2^{n/r}的上限相匹配（Cygan等人，2008年）。它还改进了参数化复杂性域中的一些困难结果（例如，Escoffier等人，2012和Chitnis等人，2013）-对于任何大于某个常数的k，k-超图定价问题不存在多项式时间min（k^{1-ε}，n^{1/2-ε}）-近似算法，其中n是输入图中的顶点数。这几乎匹配min（O（k），\tilde O（\sqrt{n}））的上界（Balcan and Blum，2007 and an algorithm in this paper）。我们注意到一个有趣的事实，与多项式时间算法的n^{1/2-ε}困难性相反，k-超图定价问题允许在准多项式时间中对任何δ>0进行n^δ逼近。这使得这个问题在一个罕见的近似类，其中近似阈值可以显着提高，允许算法运行在准多项式时间。
Weitzman (1979) introduced the Pandora Box problem as a model for sequential search with inspection costs, and gave an elegant index-based policy that attains provably optimal expected payoff. In various scenarios, the searching agent may select an option without making a costly inspection. The variant of the Pandora box problem with non-obligatory inspection has attracted interest from both economics and algorithms researchers. Various simple algorithms have proved suboptimal, with the best known 0.8-approximation algorithm due to Guha et al. (2008). No hardness result for the problem was kno…
We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost $k$-cycle free graphs, for any constant $k\geq 4$. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many $4$- or $5$-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with $m^{1+o(1)}$ preprocessing time and …
Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph a…
In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. On…
We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in term…
