Conference on Integer Programming and Combinatorial Optimization最新文献

Information Complexity of Mixed-Integer Convex Optimization 混合整数凸优化的信息复杂度

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2023-08-22 DOI: 10.1007/978-3-031-32726-1_1

A. Basu, Hongyi Jiang, Phillip A. Kerger, M. Molinaro

引用次数: 0

The Simultaneous Semi-random Model for TSP TSP的同时半随机模型

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2023-08-11 DOI: 10.1007/978-3-031-06901-7_4

Eric Balkanski, Yuri Faenza, Mathieu Kubik

引用次数: 1

Constant-Competitiveness for Random Assignment Matroid Secretary Without Knowing the Matroid 不知道矩阵的随机分配矩阵秘书的持续竞争力

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2023-05-09 DOI: 10.48550/arXiv.2305.05353

Richard Santiago, I. Sergeev, R. Zenklusen

{"title":"Constant-Competitiveness for Random Assignment Matroid Secretary Without Knowing the Matroid","authors":"Richard Santiago, I. Sergeev, R. Zenklusen","doi":"10.48550/arXiv.2305.05353","DOIUrl":"https://doi.org/10.48550/arXiv.2305.05353","url":null,"abstract":"The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an $O(1)$-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. $O(1)$-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto [SIAM Journal on Computing 2013] and Oveis Gharan&Vondrak [Algorithmica 2013] raised as an open question whether RA-MSP admits an $O(1)$-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an $O(1)$-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.","PeriodicalId":421894,"journal":{"name":"Conference on Integer Programming and Combinatorial Optimization","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114179594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Optimizing Low Dimensional Functions over the Integers 整数上的低维函数优化

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2023-03-04 DOI: 10.48550/arXiv.2303.02474

D. Dadush, A. L'eonard, Lars Rohwedder, José Verschae

{"title":"Optimizing Low Dimensional Functions over the Integers","authors":"D. Dadush, A. L'eonard, Lars Rohwedder, José Verschae","doi":"10.48550/arXiv.2303.02474","DOIUrl":"https://doi.org/10.48550/arXiv.2303.02474","url":null,"abstract":"We consider box-constrained integer programs with objective $g(Wx) + c^T x$, where $g$ is a\"complicated\"function with an $m$ dimensional domain. Here we assume we have $n gg m$ variables and that $W in mathbb Z^{m times n}$ is an integer matrix with coefficients of absolute value at most $Delta$. We design an algorithm for this problem using only the mild assumption that the objective can be optimized efficiently when all but $m$ variables are fixed, yielding a running time of $n^m(m Delta)^{O(m^2)}$. Moreover, we can avoid the term $n^m$ in several special cases, in particular when $c = 0$. Our approach can be applied in a variety of settings, generalizing several recent results. An important application are convex objectives of low domain dimension, where we imply a recent result by Hunkenschr\"oder et al. [SIOPT'22] for the 0-1-hypercube and sharp or separable convex $g$, assuming $W$ is given explicitly. By avoiding the direct use of proximity results, which only holds when $g$ is separable or sharp, we match their running time and generalize it for arbitrary convex functions. In the case where the objective is only accessible by an oracle and $W$ is unknown, we further show that their proximity framework can be implemented in $n (m Delta)^{O(m^2)}$-time instead of $n (m Delta)^{O(m^3)}$. Lastly, we extend the result by Eisenbrand and Weismantel [SODA'17, TALG'20] for integer programs with few constraints to a mixed-integer linear program setting where integer variables appear in only a small number of different constraints.","PeriodicalId":421894,"journal":{"name":"Conference on Integer Programming and Combinatorial Optimization","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123400431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

A linear time algorithm for linearizing quadratic and higher-order shortest path problems 线性化二次和高阶最短路径问题的线性时间算法

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2023-03-01 DOI: 10.48550/arXiv.2303.00569

E. Çela, Bettina Klinz, Stefan Lendl, G. Woeginger, Lasse Wulf

引用次数: 2

Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products 隐显双线性积的RLT切的有效分离

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2022-11-24 DOI: 10.1007/978-3-031-32726-1_2

Ksenia Bestuzheva, A. Gleixner, Tobias Achterberg

引用次数: 2

A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSP TSP半积分环切实例的4/3逼近算法

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2022-11-09 DOI: 10.48550/arXiv.2211.04639

Billy Jin, N. Klein, David P. Williamson

{"title":"A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSP","authors":"Billy Jin, N. Klein, David P. Williamson","doi":"10.48550/arXiv.2211.04639","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04639","url":null,"abstract":"A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP is at most 4/3. Despite significant efforts, the conjecture remains open. We consider the half-integral case, in which the LP has solution values in ${0, 1/2, 1}$. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis Gharan, in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993. This result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap is at most $1.5- 10^{-36}$ in the non-half-integral case. For the half-integral case, the current best-known ratio is 1.4983, a result by Gupta et al. With the improvements on the 3/2 bound remaining very incremental even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct. The previous works on the half-integral case perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to\"cycle cuts\"and the others to\"degree cuts\". We show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that the known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.","PeriodicalId":421894,"journal":{"name":"Conference on Integer Programming and Combinatorial Optimization","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127113767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Decomposition of Probability Marginals for Security Games in Abstract Networks 抽象网络安全博弈的概率边际分解

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2022-11-09 DOI: 10.1007/978-3-031-32726-1_22

J. Matuschke

引用次数: 0

Towards an Optimal Contention Resolution Scheme for Matchings 一种最优匹配争用解决方案

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03599

Pranav Nuti, Jan Vondr'ak

引用次数: 2

From approximate to exact integer programming 从近似到精确整数规划

Conference on Integer Programming and Combinatorial Optimization Pub Date : 2022-11-07 DOI: 10.48550/arXiv.2211.03859

D. Dadush, F. Eisenbrand, T. Rothvoss

{"title":"From approximate to exact integer programming","authors":"D. Dadush, F. Eisenbrand, T. Rothvoss","doi":"10.48550/arXiv.2211.03859","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03859","url":null,"abstract":"Approximate integer programming is the following: For a convex body $K subseteq mathbb{R}^n$, either determine whether $K cap mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity $c$. Approximate integer programming can be solved in time $2^{O(n)}$ while the fastest known methods for exact integer programming run in time $2^{O(n)} cdot n^n$. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $x^* in (K cap mathbb{Z}^n)$ can be found in time $2^{O(n)}$, provided that the remainders of each component $x_i^* mod{ell}$ for some arbitrarily fixed $ell geq 5(n+1)$ of $x^*$ are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a $2^{O(n)}n^n$ algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carath'eodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $Ax = b, 0 leq x leq u, , x in mathbb{Z}^n$ . Such a problem can be reduced to the solution of $prod_i O(log u_i +1)$ approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range $0 leq x_i leq p(n)$ can be solved in time $(log n)^{O(n)}$. For these problems, the best running time so far was $n^n cdot 2^{O(n)}$.","PeriodicalId":421894,"journal":{"name":"Conference on Integer Programming and Combinatorial Optimization","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126866212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2