Bootstrap Inference in Partially Identified Models Defined by Moment Inequalities: Coverage of the Identified Set

This paper introduces a novel bootstrap procedure to perform inference in a wide class of partially identified econometric models. We consider econometric models defined by finitely many weak moment inequalities, ² We can also admit models defined by moment equalities by combining pairs of weak moment inequalities.
which encompass many applications of economic interest. The objective of our inferential procedure is to cover the identified set with a prespecified probability. ³ We deal with the objective of covering each element of the identified set with a prespecified probability in Bugni (2010a).
We compare our bootstrap procedure, a competing asymptotic approximation, and subsampling procedures in terms of the rate at which they achieve the desired coverage level, also known as the error in the coverage probability. Under certain conditions, we show that our bootstrap procedure and the asymptotic approximation have the same order of error in the coverage probability, which is smaller than that obtained by using subsampling. This implies that inference based on our bootstrap and asymptotic approximation should eventually be more precise than inference based on subsampling. A Monte Carlo study confirms this finding in a small sample simulation.     

Estimating hybrid frequency moments of data streams

We consider the problem of estimating hybrid frequency moments of two dimensional data streams. In this model, data is viewed to be organized in a matrix form (A _i,j )_1≤i,j,≤n . The entries A _i,j are updated coordinate-wise, in arbitrary order and possibly multiple times. The updates include both increments and decrements to the current value of A _i,j . The hybrid frequency moment F _p,q (A) is defined as \(\sum_{j=1}^{n}(\sum_{i=1}^{n}{A_{i,j}}^{p})^{q}\) and is a generalization of the frequency moment of one-dimensional data streams.We present the first \(\tilde{O}(1)\) space algorithm for the problem of estimating F _p,q for p∈[0,2] and q∈[0,1] to within an approximation factor of 1±ε. The \(\tilde{O}\) notation hides poly-logarithmic factors in the size of the stream m, the matrix size n and polynomial factors of ε ^?1. We also present the first \(\tilde{O}(n^{1-1/q})\) space algorithm for estimating F _p,q for p∈[0,2] and q∈(1,2].     

An improved time-space lower bound for tautologies

We show that for all reals c and d such that c ² d<4 there exists a positive real e such that tautologies of length n cannot be decided by both a nondeterministic algorithm that runs in time n ^c , and a nondeterministic algorithm that runs in time n ^d and space n ^e . In particular, for every \(d<\sqrt[3]{4}\) there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n ^d and space n ^e .     

Smart elements in combinatorial group testing problems

In combinatorial group testing problems the questioner needs to find a special element \(x \in [n]\) by testing subsets of [n]. Tapolcai et al. (in: Proceedings of IEEE INFOCOM, Toronto, Canada, pp 1860–1868, 2014; IEEE Trans Commun 64(6):2527–2538, 2016) introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the special one. Using classical results of extremal set theory we prove that if \(\mathcal {F}_n \subseteq 2^{[n]}\) solves the non-adaptive version of this problem and has minimal cardinality, then

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{|\mathcal {F}_n|}{\log _2 n} = \log _{(3/2)}2. \end{aligned}$$

This improves results in Tapolcai et al. (2014, 2016). We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the special one. Finally the adaptive versions of the different models are investigated.

     

Approximating <Emphasis Type="Italic">k</Emphasis>-generalized connectivity via collapsing HSTs

An instance of the k -generalized connectivity problem consists of an undirected graph G=(V,E), whose edges are associated with non-negative costs, and a collection \({\mathcal{D}}=\{(S_{1},T_{1}),\ldots,(S_{d},T_{d})\}\) of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph ??G connects a demand (S _i ,T _i ) when it contains a path with one endpoint in S _i and the other in T _i . Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in \({\mathcal{D}}\).Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiple-choice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log?² nlog?² d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k -subgraph.In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n ^2/3?polylog(n,k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.     

Resource-sharing systems and hypergraph colorings

A resource-sharing system is modeled by a hypergraph H in which a vertex represents a process and an edge represents a resource consisting of all vertices (processes) that have access to it. A schedule of H=(V,E) is a mapping f:?→2 ^V , where f(i) is an independent set of H which consists of processes that operate at round i. The rate of f is defined as \({\rm rate}(f)=\limsup_{n\to\infty}\sum_{i=1}^{n}|f(i)|/(n|V|)\), which is the average fraction of operating processes at each round. The purpose of this paper is to study optimal rates for various classes of schedules under different fairness conditions. In particular, we give relations between these optimal rates and fractional/circular chromatic numbers. For the special case of the hypergraph is a connected graph, a new derivation for the previous result by Yeh and Zhu is also given.     

Optimal on-line algorithms for one batch machine with grouped processing times

In this paper, we study on-line scheduling problems on a batch machine with the assumption that all jobs have their processing times in [p, (1+φ)p], where p>0 and \(\phi=(\sqrt{5}-1)/2\). Jobs arrive over time. First, we deal with the on-line problem on a bounded batch machine with the objective to minimize makespan. A class of algorithms with competitive ratio \((\sqrt{5}+1)/2\) are given. Then we consider the scheduling on an unbounded batch machine to minimize the time by which all jobs have been delivered, and provide a class of on-line algorithms with competitive ratio \((\sqrt{5}+1)/2\). The two class of algorithms are optimal for the problems studied here.     

Maximum flows in generalized processing networks

Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios \(\alpha _e \in [0,1]\) are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes \({\mathcal {NP}}\)-hard to solve and approximate in this case.     

Semi-online scheduling on two uniform machines with the known largest size

This paper investigates semi-online scheduling on two uniform machines with the known largest size. Denote by s _j the speed of each machine, j=1,2. Assume 0<s ₁≤s ₂, and let s=s ₂/s ₁ be the speed ratio. First, for the speed ratio \(s\in [1,\sqrt{2}]\), we present an optimal semi-online algorithm \(\mathcal{LSMP}\) with the competitive ratio \(\mathrm{max}\{\frac {2(s+1)}{2s+1},s\}\). Second, we present a semi-online algorithm \(\mathcal{HSMP}\). And for \(s\in(\sqrt{2},1+\sqrt{3})\), the competitive ratio of \(\mathcal{HSMP}\) is strictly smaller than that of the online algorithm \(\mathcal{LS}\). Finally, for the speed ratio s≥s ^*≈3.715, we show that the known largest size cannot help us to design a semi-online algorithm with the competitive ratio strictly smaller than that of \(\mathcal{LS}\). Moreover, we show a lower bound for \(s\in(\sqrt{2},s^{*})\).     

A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable

A graph G is \((d_1, d_2)\)-colorable if its vertices can be partitioned into subsets \(V_1\) and \(V_2\) such that in \(G[V_1]\) every vertex has degree at most \(d_1\) and in \(G[V_2]\) every vertex has degree at most \(d_2\). Let \(\mathcal {G}_5\) denote the family of planar graphs with minimum cycle length at least 5. It is known that every graph in \(\mathcal {G}_5\) is \((d_1, d_2)\)-colorable, where \((d_1, d_2)\in \{(2,6), (3,5),(4,4)\}\). We still do not know even if there is a finite positive d such that every graph in \(\mathcal {G}_5\) is (1, d)-colorable. In this paper, we prove that every graph in \(\mathcal {G}_5\) without adjacent 5-cycles is (1, 7)-colorable. This is a partial positive answer to a problem proposed by Choi and Raspaud that is every graph in \(\mathcal {G}_5\;(1, 7)\)-colorable?.     

Approximation and hardness results for label cut and related problems

We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first non-trivial approximation and hardness results for the Label Cut problem. Firstly, we present an \(O(\sqrt{m})\)-approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NP-hard to approximate Label Cut within \(2^{\log ^{1-1/\log\log^{c}n}n}\) for any constant c<1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions).     

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }?V and a function f : S→? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ∑ _i=1 ^d f(s _i ) in-trees denoted by \(T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}\) for every i=1,…,d such that \(T_{i,1},\ldots,T_{i,f(s_{i})}\) are rooted at s _i , each T _i,j spans vertices from which s _i is reachable and the union of all arc sets of T _i,j for i=1,…,d and j=1,…,f(s _i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑ _i=1 ^d f(s _i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.     

Bounding the scaling window of random constraint satisfaction problems

The model \(k\)-CSP is a random CSP model with moderately growing arity \(k\) of constraints. By incorporating certain linear structure, \(k\)-CSP is revised to a random linear CSP, named \(k\)-hyper-\({\mathbb F}\)-linear CSP. It had been shown theoretically that the two models exhibit exact satisfiability phase transitions when the constraint density \(r\) is varied accordingly. In this paper, we use finite-size scaling analysis to characterize the threshold behaviors of the two models with finite problem size \(n\). A series of experimental studies are carried out to illustrate the scaling window of the model \(k\)-CSP.     

Online lazy bureaucrat scheduling with a machine deadline

The lazy bureaucrat scheduling problem was first introduced by Arkin et al. (Inf Comput 184:129–146, 2003). Since then, a number of variants have been addressed. However, very little is known on the online version. In this note we focus on the scenario of online scheduling, in which the jobs arrive over time. The bureaucrat (machine) has a working time interval. Namely, he has a deadline by which all scheduled jobs must be completed. A decision is only based on released jobs without any information on the future. We consider two objective functions of [min-makespan] and [min-time-spent]. Both admit best possible online algorithms with competitive ratio of \(\frac{\sqrt{5}+1}{2}\approx 1.618\).     

Neighbor product distinguishing total colorings

A total-[k]-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}\) such that any two adjacent elements in \(V (G) \cup E(G)\) receive different colors. Let f(v) denote the product of the color of a vertex v and the colors of all edges incident to v. A total-[k]-neighbor product distinguishing-coloring of G is a total-[k]-coloring of G such that \(f(u)\ne f(v)\), where \(uv\in E(G)\). By \(\chi ^{\prime \prime }_{\prod }(G)\), we denote the smallest value k in such a coloring of G. We conjecture that \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs and subcubic graphs. Furthermore, we show that if G is a \(K_4\)-minor free graph with \(\Delta (G)\ge 4\), then \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).     

An improved approximation algorithm for the shortest link scheduling in wireless networks under SINR and hypergraph models

Link scheduling is a fundamental problem in wireless ad hoc and sensor networks. In this paper, we focus on the shortest link scheduling (SLS) under Signal-to-Interference-plus-Noise-Ratio and hypergraph models, and propose an approximation algorithm \(SLS_{pc}\) (A link scheduling algorithm with oblivious power assignment for the shortest link scheduling) with oblivious power assignment for better performance than GOW* proposed by Blough et al. [IEEE/ACM Trans Netw 18(6):1701–1712, 2010]. For the average scheduling length of \(SLS_{pc}\) is 1 / m of GOW*, where \(m=\lfloor \varDelta _{max}\cdot p \rfloor \) is the expected number of the links in the set V returned by the algorithm HyperMaxLS (Maximal links schedule under hypergraph model) and \(0<p<1\) is the constant. In the worst, ideal and average cases, the ratios of time complexity of our algorithm \(SLS_{pc}\) to that of GOW* are \(O(\varDelta _{max}/\overline{k})\), \(O(1/(\overline{k}\cdot \varDelta _{max}))\) and \(O(\varDelta _{max}/(\overline{k}\cdot m))\), respectively. Where \(\overline{k}\) (\(1<\overline{k}<\varDelta _{max}\)) is a constant called the SNR diversity of an instance G.     

Signed mixed Roman domination numbers in graphs

Let \(G = (V;E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A signed mixed Roman dominating function (SMRDF) of \(G\) is a function \(f: V\cup E\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{y\in N_m[x]}f(y)\ge 1\) for each \(x\in V\cup E\), where \(N_m[x]\) is the set, called mixed closed neighborhood of \(x\), consists of \(x\) and the elements of \(V\cup E\) adjacent or incident to \(x\) (ii) every element \(x\in V\cup E\) for which \(f(x) = -1\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y) = 2\). The weight of a SMRDF \(f\) is \(\omega (f)=\sum _{x\in V\cup E}f(x)\). The signed mixed Roman domination number \(\gamma _{sR}^*(G)\) of \(G\) is the minimum weight of a SMRDF of \(G\). In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.