Christos Koufogiannakis, Neal E. Young
We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual solutions whose costs are within a factor of 1+eps of the optimal cost in time O((r+c)log(n)/eps^2 + n).
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We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping packing of a subset of hypercubes such that the profit of the packed hypercubes is maximized. For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+epsilon). For d=2, Jansen and Solis-Oba (IPCO'08) showed that the problem admits a polynomial-time approximation scheme (PTAS); Heydrich and Wiese (SODA'17) further improve…
Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given $O(1/ε)$ advice bits for each input item allows approximating the dual bin packing problem online to within a factor of $1+ε$. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let $s$ be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice $O\left(\frac{s + \log n}{ε^2}\right)$ approximates the dual b…
We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve t…
We give an $α(1+ε)$-approximation algorithm for solving covering LPs, assuming the presence of a $(1/α)$-approximation algorithm for a certain optimization problem. Our algorithm is based on a simple modification of the Plotkin-Shmoys-Tardos algorithm (MOR 1995). We then apply our algorithm to $α(1+ε)$-approximately solve the configuration LP for a large class of bin-packing problems, assuming the presence of a $(1/α)$-approximate algorithm for the corresponding knapsack problem (KS). Previous results give us a PTAS for the configuration LP using a PTAS for KS. Those results don't extend to th…
We present new approximation schemes for bin packing based on the following two approaches: (1) partitioning the given problem into mostly identical sub-problems of constant size and then construct a solution by combining the solutions of these constant size sub-problems obtained through PTAS or exact methods; (2) solving bin packing using irregular sized bins, a generalization of bin packing, that facilitates the design of simple and efficient recursive algorithms that solve a problem in terms of smaller sub-problems such that the unused space in bins used by an earlier solved sub-problem is …
A PTAS for Non-Adaptive Stochastic Top-$k$ Sum under General Combinatorial Constraints
该研究针对一般组合约束下的非自适应随机Top-k和问题，利用近似最大和预言机与打包LP等技术，为固定维打包族（含背包问题）构造了PTAS，并证明其为通用最优方案
In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We first observe there is a Polynomial Time Approximation Scheme (PTAS) for the problem of packing weighted squares into rectangular bins with large resources, then apply the PTAS to the problem of packing squares with profits into a rectangular bin and get a $\frac65+ε$ approximation algorithm.
Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does compa…
We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+ε)$-approximate solutions in time (and work) $O^*(N/ε^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs.
Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it may be desirable to obtain {\it sparse} dual solutions, i.e., those with support size (almost) independent of the number of primal constraints. In this paper, we give an algorit…
Packing and covering linear programs belong to the narrow class of linear programs that are efficiently solvable in parallel and distributed models of computation, yet are a powerful modeling tool for a wide range of fundamental problems in theoretical computer science, operations research, and many other areas. Following recent progress in obtaining faster distributed and parallel algorithms for packing and covering linear programs, we present a simple algorithm whose iteration count matches the best known $\tilde{O}(\frac{1}{ε^2})$ for this class of problems. The algorithm is similar to the …
In many problems, the inputs arrive over time, and must be dealt with irrevocably when they arrive. Such problems are online problems. A common method of solving online problems is to first solve the corresponding linear program, and then round the fractional solution online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum …
In the dynamic linear program (LP) problem, we are given an LP undergoing updates and we need to maintain an approximately optimal solution. Recently, significant attention (e.g., [Gupta et al. STOC'17; Arar et al. ICALP'18, Wajc STOC'20]) has been devoted to the study of special cases of dynamic packing and covering LPs, such as the dynamic fractional matching and set cover problems. But until now, there is no non-trivial dynamic algorithm for general packing and covering LPs. In this paper, we settle the complexity of dynamic packing and covering LPs, up to a polylogarithmic factor in update…
Approximating the Chvátal--Gomory Closure of Capacity-Bounded Min-Closed Systems
本文针对容量受限最小闭系统，证明了常数次平方和松弛可为其第一CG闭包上的线性目标最大化提供PTAS，扩展了混合符号约束下的近似保证。
We study the online convex covering problem and online convex packing problem. The (offline) convex covering problem is modeled by the following convex program: $\min_{x \in R_+^n} f(x) \ \text{s.t}\ A x \ge 1$, where $f : R_+^n \mapsto R_+$ is a monotone and convex cost function, and $A$ is an $m \times n$ matrix with non-negative entries. Each row of the constraint matrix $A$ corresponds to a covering constraint. In the online problem, each row of $A$ comes online and the algorithm must maintain a feasible assignment $x$ and may only increase $x$ over time. The (offline) convex packing probl…
We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest(EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal we need to augment our solution subgraph to connect the new terminal to the…
An important objective in scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. The currently best known polynomial time algorithm for the problem is a (2+eps)-approximation by Rohwedder and Wiese [STOC 2021] which builds on the prior break-through result by…
We consider a variant of bin packing called multiple-choice vector bin packing. In this problem we are given a set of items, where each item can be selected in one of several $D$-dimensional incarnations. We are also given $T$ bin types, each with its own cost and $D$-dimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in networks with guaranteed quality of service (QoS), but due to its general formulation it has many other applications as well. We present an approximation algorithm that is guaranteed to produce a sol…
Dynamic Matroids: Base Packing and Covering
本文提出动态Matroid中基集打包和覆盖的高效算法，以(1±ε)精度近似基集相关参数，适用于动态更新场景。
We consider online fractional covering problems with a convex objective, where the covering constraints arrive over time. Formally, we want to solve $\min\,\{f(x) \mid Ax\ge \mathbf{1},\, x\ge 0\},$ where the objective function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex, and the constraint matrix $A_{m\times n}$ is non-negative. The rows of $A$ arrive online over time, and we wish to maintain a feasible solution $x$ at all times while only increasing coordinates of $x$. We also consider "dual" packing problems of the form $\max\,\{c^\intercal y - g(μ) \mid A^\intercal y \le μ,\, y\ge 0\}…
An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed…
Covering and packing mixed-integer linear programs with a fixed number of constraints: Approximation and convex hull
本文提出了一种针对固定数量约束的覆盖混合整数线性规划问题的近似算法，并改进了背包覆盖问题的近似算法。
We present a parallel approximation algorithm for a class of mixed packing and covering semidefinite programs which generalize on the class of positive semidefinite programs as considered by Jain and Yao [2011]. As a corollary we get a faster approximation algorithm for positive semidefinite programs with better dependence of the parallel running time on the approximation factor, as compared to that of Jain and Yao [2011]. Our algorithm and analysis is on similar lines as that of Young [2001] who considered analogous linear programs.
We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $κ$ distinct lengths, and another interval $B$, can we pack the intervals in $\mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $\mathcal{I}$) by moving $τ$ intervals and keeping the other $σ= n - τ$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $κ$, $τ$, and $σ$ as single parameter, but are FPT with combined parameters $κ$ and $τ$. We also obtain improved polynomial-time algorithms for packing and covering, inc…
Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement a…
We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the ILP Feasibility problem admits no polynomial kernelization when parameterized by both the number of variables and the number of constraints, unless NP \subseteq coNP/poly. This extends to the restricted cases of bounded variable degree and bounded number of variables per constraint, and to covering and packing ILPs. Second, we give a polynomial kernelization …
We show algorithms for computing representative families for matroid intersections and use them in fixed-parameter algorithms for set packing, set covering, and facility location problems with multiple matroid constraints. We complement our tractability results by hardness results.
Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple target locations. Herein a configuration describes a possible placement on one of the target locations, and the IP is used to chose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and therefore be solved efficiently. As an application, we consider sche…
This paper gives poly-logarithmic-round, distributed D-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio D is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with D=2). Via duality, the paper also gives poly-logarithmic-round, distributed D-approximation algorithms for Fractional Packing linear programs (where D is the maximum number of constraints in which any variable occurs), and for Max Wei…
A Simple Distributed Algorithm for Sparse Fractional Covering and Packing Problems
本文提出一种简单的CONGEST模型分布式算法，对RS-FCP和CS-FPP实现$(1+ε)$-近似，较已知最佳算法更简单且显著改善了对$ε$的依赖。
Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $ε>0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-ε$ while violating only the covering constraints by a multiplicative factor of $1-ε$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above mai…
Approximation Algorithms for Norm-Budgeted Packing Problems
研究范数预算打包问题，通过涉及单调、对称范数的范数预算约束建模资源，将多种经典打包问题纳入框架，开发框架获得常数因子近似保证及相关问题的PTAS，并为子模版本开发常数因子近似算法。
Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all "reasonable" algorithms have a competitive ratio of 1/2. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restric…
Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems, no single method has emerged that matches the effectiveness and simplicity afforded by the covering case. We offer a simple iterative packing technique that retains features of Jain's seminal approach, including the property that the magnitude of the fractional value of the element rounded during each iteration has a direct impact on the approximation guaran…
An $n^{O(\log\log n)}$ time approximation scheme for capacitated VRP in the Euclidean plane
提出欧几里得平面上带容量车辆路径问题的拟多项式时间近似方案，运行时间为 $n^{f(\epsilon)\cdot\log\log n}$。先将其归约为无容量的 $m$ -路径问题，再给出该问题的拟多项式时间近似方案。
Given a tournament T and a positive integer k, the C_3-Pakcing-T problem asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Pakcing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse t…
Routing and scheduling problems are fundamental problems in combinatorial optimization, and also have many applications. Most variations of these problems are NP-Hard, so we need to use heuristics to solve these problems on large instances, which are fast and yet come close to the optimal value. In this thesis, we study the design and analysis of approximation algorithms for such problems. We focus on two important class of problems. The first is the Unsplittable Flow Problem and some of its variants and the second is the Resource Allocation for Job Scheduling Problem and some of its variants.…
The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size $o(\log \log n)$ has a competitive ratio of at most 0.5. In other words, advice of size $o(\log \log n)$ is useless for improving the competitive ratio of 0.5, attainable by an online algorit…
Finite Pinwheel Covering
该研究引入k次访问覆盖，证明其为强NP完全，还给出特殊情形的算法，且发现其密度阈值无界，是覆盖场景首个强NP难结果。
Approximation algorithms for the square min-sum bin packing problem
针对平方最小和装箱问题，研究人员证明经典二维装箱排序算法性能差，提出近似比约2.3178的O(n log n)算法，并给出其多项式时间近似方案。
In the recent paper \cite{BDT10} we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a "good and fast" packing or covering. In that paper we mainly dealt with only one possible online BP/CD model, and proposed a new method that we call the Evolution of Algorithms. In case of such methods a neighborhood structure is defined among algorithms, and using a metaheuristic (for example simulated annealing) in some sense the best algorithm is chosen to solve the problem. Now we turn …
FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less
本文提出算法元定理，证明可由单态二阶逻辑定义且具有限整数索引的覆盖/装填问题，以mod_H、ed_H、tw_H参数化的FPT近似方案存在性等价，并给出多个经典图问题的对应近似方案。
NP-Hardness and a PTAS for the Pinwheel Problem
研究针轮问题的NP难性及其近似算法，证明其NP难并提出PTAS，改进了近似因子。
We give a lower bound on the iteration complexity of a natural class of Lagrangean-relaxation algorithms for approximately solving packing/covering linear programs. We show that, given an input with $m$ random 0/1-constraints on $n$ variables, with high probability, any such algorithm requires $Ω(ρ\log(m)/ε^2)$ iterations to compute a $(1+ε)$-approximate solution, where $ρ$ is the width of the input. The bound is tight for a range of the parameters $(m,n,ρ,ε)$. The algorithms in the class include Dantzig-Wolfe decomposition, Benders' decomposition, Lagrangean relaxation as developed by Held an…
Routing-Controlled Spanners
本文提出路由控制的生成子图及更一般的打包-覆盖生成子图概念，在距离约束外引入路由、打包与覆盖约束，并给出首个近似算法以统一解决有向网络设计中的多种连通性问题。
Sensitivity Oracles for Matroid Packing, Matroid Covering, and Matching Problems with Applications
该研究提出统一代数框架，得到支持任意f次更新的拟阵打包等问题的灵敏度预言机，解决开放问题并证明下界，应用于流、割、匹配等优化问题。
Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx < b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting the covering constraints (Ax > a) and multiplicity constraints (x < d), and satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums beta_i = sum_j B_ij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost covering i…
Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction
本文通过统一的直径缩减方法，改进了打包和覆盖问题的收敛率，解决了线性耦合方法在两者间表现不一致的问题。
Packing and covering linear programs (PC-LPs) form an important class of linear programs (LPs) across computer science, operations research, and optimization. In 1993, Luby and Nisan constructed an iterative algorithm for approximately solving PC-LPs in nearly linear time, where the time complexity scales nearly linearly in $N$, the number of nonzero entries of the matrix, and polynomially in $\varepsilon$, the (multiplicative) approximation error. Unfortunately, all existing nearly linear-time algorithms for solving PC-LPs require time at least proportional to $\varepsilon^{-2}$. In this pape…
In this paper, we give a faster width-dependent algorithm for mixed packing-covering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a $1+\eps$ approximate solution in time $O(Nw/ \eps)$, where $N$ is number of nonzero entries in the constraint matrix and $w$ is the maximum number of nonzeros in any constraint. This run-time is better than Nesterov's smoothing algorithm which requires $O(N\sqrt{n}w/ \eps)$ where $n$ is the dimension of the problem. Our work utilizes the framework of area convexity in…
We consider the setting of online computation with advice, and study the bin packing problem and a number of scheduling problems. We show that it is possible, for any of these problems, to arbitrarily approach a competitive ratio of $1$ with only a constant number of bits of advice per request. For the bin packing problem, we give an online algorithm with advice that is $(1+\varepsilon)$-competitive and uses $O\left(\frac{1}{\varepsilon}\log \frac{1}{\varepsilon} \right)$ bits of advice per request. For scheduling on $m$ identical machines, with the objective function of any of makespan, machi…
We study several stochastic combinatorial problems, including the expected utility maximization problem, the stochastic knapsack problem and the stochastic bin packing problem. A common technical challenge in these problems is to optimize some function of the sum of a set of random variables. The difficulty is mainly due to the fact that the probability distribution of the sum is the convolution of a set of distributions, which is not an easy objective function to work with. To tackle this difficulty, we introduce the Poisson approximation technique. The technique is based on the Poisson appro…
Fair resource allocation is a fundamental optimization problem with applications in operations research, networking, and economic and game theory. Research in these areas has led to the general acceptance of a class of $α$-fair utility functions parameterized by $α\in [0, \infty]$. We consider $α$-fair packing -- the problem of maximizing $α$-fair utilities under positive linear constraints -- and provide a simple first-order method for solving it with relative-error guarantees. The method has a significantly lower convergence time than the state of the art, and to analyze it, we leverage the …
We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on…
We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph $G$ and a family $\mathcal{D}$ of objects, each being a connected subgraph of $G$ with a prescribed weight, and the task is to find a maximum-weight subfamily of $\mathcal{D}$ consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given an edge-weighted graph $G$, two sets $\mathcal{D},\mathcal{C}$ of vertices of $G$, where vertices of $\mathcal{D}$ have prescribed weights, and a nonnegative radius $r$. The task is to find a minimum-weight subs…
Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems. BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This work deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and…
Improved Hardness of Approximation for Geometric Bin Packing
本文针对几何装箱问题，证明除非NP=P，否则该问题无法在d^{1-ε}因子内近似，并引入集合族的几何装箱维度概念，证明其关键性质以揭示几何装箱与向量装箱的根本差异。
Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$. The \textsc{$(p,r,\mathcal{F})$-Packing} problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $…
In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximabilit…
The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs
本研究提出LOCAL模型下的低直径分解算法，进一步发展出近似求解一般装箱与覆盖整数线性规划的分布式算法，改进了已有结果，且对多种基础组合图优化问题近似最优。
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