Simple Approximation Algorithm Research Articles

Fair Projections as a Means toward Balanced Recommendations

The goal of recommender systems is to provide to users suggestions that match their interests, with the eventual goal of increasing their satisfaction, as measured by the number of transactions (clicks, purchases, and so forth). Often, this leads to providing recommendations that are of a particular type. For some contexts (e.g., browsing videos for information) this may be undesirable, as it may enforce the creation of filter bubbles. This is because of the existence of underlying bias in the input data of prior user actions. Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. In this article, we consider both the densest subgraph and the \(k\) -clustering problem, two primitives that are being used by some recommender systems. We are given a coloring on the nodes, respectively the points, and aim to compute a fair solution \(S\) , consisting of a subgraph or a clustering, such that none of the colors is disparately impacted by the solution. Unfortunately, introducing fair solutions typically makes these problems substantially more difficult. Unlike the unconstrained densest subgraph problem, which is solvable in polynomial time, the fair densest subgraph problem is NP-hard even to approximate, which means that with the standard computational model it is probably impossible to solve (or even approximate it sufficiently well) in polynomial time. For \(k\) -clustering, the fairness constraints make the problem very similar to capacitated clustering, which is a notoriously hard problem to even approximate. Despite such negative premises, we are able to provide positive results in important use cases. In particular, we are able to prove that a suitable spectral embedding allows recovery of an almost optimal, fair, dense subgraph hidden in the input data, whenever one is present, a result that is further supported by experimental evidence. We also show a polynomial-time, \(2\) -approximation algorithm to the problem of fair densest subgraph, assuming that there exist only two colors and both colors occur equally often in the graph. This result turns out to be optimal assuming the small set expansion hypothesis. For fair \(k\) -clustering, we show that we can recover high quality fair clusterings effectively and efficiently. For the special case of \(k\) -median and \(k\) -center, we offer additional, fast and simple approximation algorithms as well as new hardness results. The above theoretical findings drive the design of heuristics, which we experimentally evaluate on a scenario based on real data, in which our aim is to strike a good balance between diversity and highly correlated items from Amazon co-purchasing graphs and Facebook contacts. We additionally evaluated our algorithmic solutions for the fair \(k\) -median problem through experiments on various real-world datasets.

A local search approximation algorithm for the multiway cut problem

In the multiway cut problem we are given a weighted undirected graph G=(V,E) and a set T⊆V of k terminals. The goal is to find a minimum weight set of edges E′⊆E with the property that by removing E′ from G all the terminals become disconnected. In this paper we present a simple local search approximation algorithm for the multiway cut problem with approximation ratio 2−2k. We present an experimental evaluation of the performance of our local search algorithm and show that it greatly outperforms the isolation heuristic of Dalhaus et al. and it has similar performance as the much more complex algorithms of Calinescu et al., Sharma and Vondrak, and Buchbinder et al. which have the currently best known approximation ratios for this problem.

A simple polynomial-time approximation algorithm for the total variation distance between two product distributions

We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.

DNN-Approximations of Control Laws for Nonlinear Systems with Polytopic Constraints

This work focuses on deep artificial feed-forward neural networks as parametric function approximators in optimal control of discrete-time nonlinear but affine-in-control systems subject to polytopic constraints on the continuous states and control inputs. The neural networks are either considered as approximators of optimal control laws or as approximators of the corresponding cost functions. In both cases, an approach is developed for determining the approximated optimal control inputs without violating the constraints. A simple approximate policy iteration algorithm exploiting both approaches is finally presented and illustrated for a numerical example.

Upper and lower bounds on approximating weighted mixed domination

A mixed dominating set of a graph G=(V,E) is a mixed set D of vertices and edges, such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is NP-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight wv and all edges are assigned the same nonnegative weight we, and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include:1.for we≥wv, a 2-approximation algorithm;2.for we≥2wv, inapproximability within ratio 1.3606 unless P=NP and within ratio 2 under UGC;3.for 2wv>we≥wv, inapproximability within ratio 1.1803 unless P=NP and within ratio 1.5 under UGC;4.for we<wv, inapproximability within ratio (1−ϵ)ln⁡|V| unless P=NP for any ϵ>0.

Vertex-edge domination in unit disk graphs

Let G = ( V , E ) be a simple undirected graph. A set D ⊆ V is called a vertex-edge dominating set of G if for each edge e = u v ∈ E , either u or v is in D or one vertex from their neighbor is in D . Simply, a vertex v ∈ V , vertex-edge dominates every edge u v , as well as every edge adjacent to these edges. The objective of the vertex-edge domination problem is to find a minimum size vertex-edge dominating set of G . Herein, we study the vertex-edge domination problem in unit disk graphs and prove that the decision version of the problem belongs to the NP-complete class. We also show that the problem admits a simple polynomial-time 4-factor approximation algorithm and a polynomial-time approximation scheme (PTAS) in unit disk graphs.

Upper Dominating Set: Tight algorithms for pathwidth and sub-exponential approximation

In the Upper Dominating Set problem, the goal is to find a minimal dominating set of maximum size. We study the complexity of parameterized algorithms for Upper Dominating Set, as well as its sub-exponential approximation. First, we prove that, under the ETH, Upper Dominating Set cannot be solved in time O(no(k)) (improving on O(no(k))), and in the same time we show under the same complexity assumption that for any constant ratio r and any ε>0, there is no r-approximation algorithm running in time O(nk1−ε). Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time O⁎(6pw) (improving the current best O⁎(7pw)), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an r-approximation in time nO(n/r), for any desired approximation ratio r<n. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio r>1 and ε>0, no algorithm can output an r-approximation in time n(n/r)1−ε. Hence, we completely characterize the approximability of the problem in sub-exponential time.

Burning Graphs Through Farthest-First Traversal

Graph burning is a process to determine the spreading of information in a graph. If a sequence of vertices burns all the vertices of a graph by following the graph burning process, then such a sequence is known as a burning sequence. The graph burning problem consists in finding a minimum length burning sequence for a given graph. The solution to this NP-hard combinatorial optimization problem helps quantify a graph&#x2019;s vulnerability to contagion. This paper introduces a simple farthest-first traversal-based approximation algorithm for this problem over arbitrary graphs. We refer to this proposal as the Burning Farthest-First (BFF) algorithm. BFF runs in <inline-formula> <tex-math notation="LaTeX">$O(n^{3})$ </tex-math></inline-formula> steps and has a tight approximation factor of <inline-formula> <tex-math notation="LaTeX">$3-2/b(G)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$b(G)$ </tex-math></inline-formula> is the size of an optimal solution. The main attribute of BFF is that it has a better approximation factor than the state-of-the-art approximation algorithms for arbitrary graphs, which report an approximation factor of 3. Despite being simple, BFF proved practical when tested over some benchmark datasets.

A scaling approach to size effect modeling of Jc CDF for 20MnMoNi55 reactor steel in transition temperature region

Ever since the 1980s there is a sustained interest in the size effect, as one of the most pronounced consequences of fracture mechanics. In the present study, the investigation of the size effect is focused on estimation of the Weibull cumulative distribution function (CDF) of the critical value of the J-integral (Jc) in the transition temperature region under constraint of a small statistical sample size. Specifically, the Jc experimental data correspond to the C(T) specimen testing of the reactor pressure-vessel steel 20MnMoNi55 at only two geometrically-similar sizes. Thus, a simple approximate scaling algorithm has been developed to tackle the effect of the C(T) specimen size on the Jc CDF under these circumstances. Due to the specific form of the two-parameter Weibull CDF, F(Jc|β,η), the scaling procedure is applied according to a two-step scheme. First, the Jc-scaling is performed to ensure the approximate overlap of the points that correspond to the CDF value F(Jc = η) = 1 − 1/e ≈ 0.632 for different C(T) specimen widths (W), which assumes η·Wκ = const. Second, the F-scaling is performed to ensure the equality of the slopes of the CDF in the scaled (F·Wξ vs. Jc·Wκ) space. The objective of the sketched approach is to obtain a size-dependent Jc CDF that encapsulates a reasonably conservative data extrapolation.

A greedy algorithm for the social golfer and the Oberwolfach problem

Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the number of rounds that can be guaranteed in a Swiss-system tournament. Matches of these tournaments are usually determined in a myopic round-based way dependent on the results of previous rounds. Together with the hard constraint that no two players meet more than once during the tournament, at some point it might become infeasible to schedule a next round. For tournaments with n players and match sizes of k≥2 players, we prove that we can always guarantee ⌊nk(k−1)⌋ rounds. We show that this bound is tight. This provides a simple polynomial time constant factor approximation algorithm for the social golfer problem.We extend the results to the Oberwolfach problem. We show that a simple greedy approach guarantees at least ⌊n+46⌋ rounds for the Oberwolfach problem. This yields a polynomial time 13+ε-approximation algorithm for any fixed ε>0 for the Oberwolfach problem. Assuming that El-Zahar’s conjecture is true, we improve the bound on the number of rounds to be essentially tight.

Approximation algorithms for sorting permutations by extreme block-interchanges

Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two substrings that are called as blocks in literature, in a given permutation. When the blocks are restricted to be adjacent then one obtains a well studied operation: transposition. We call either a prefix or a suffix as an extreme. Restricting one of the swapped blocks to be an extreme in block-interchange operation yields a prefix or a suffix block-interchange respectively, the two types of extreme block-interchanges. For prefix block-interchange operation over permutations we design: (i) an optimum algorithm to sort reverse permutation, Rn, in n/2 moves, (ii) a simple 2-approximation algorithm, and (iii) for permutations with O(1) cycles, a 4/3 approximation algorithm. Due to symmetry, these results apply to suffix block-interchange operation also.

Hybrid precoding algorithm for massive MIMO system with imperfect channel knowledge

Millimeter wave MIMO system offers orders of magnitude high data rate due to the excess bandwidth available then conventional MIMO system. But, at the cost of excessive free space losses that highly influence the transmission at very high millimeter wave frequencies. The resulting small Wavelength at the millimeter wave frequency can overcome the path losses using beam forming gain offered by the massive antennas which is known as pre-coding. In millimeter wave massive MIMO system, hybrid pre-coding is the technique that combines analog pre-coding and digital pre-coding to reduce the requirement of high number of RF chains and thus energy consumption. Hybrid beam forming uses number of RF chains equal to the number of simulations users being served rather than using separate RF chains in the case of digital pre-coding. In designing analog and digital pre-coders, large number of calculations that uses complex mathematical process such as block diagonalization, SVD for channel matrix inversion required for the computation of pre-coder weight vector computation. In this paper, we propose low complexity hybrid pre-coding algorithm using simple linear approximation algorithms for matrix inversion operations. We first used the simple inversion algorithm in the computation of analog beam forming matrix without any SVD or direct inversion of matrices. Then, the same is being used in digital pre-coding matrix. In the proposed algorithm, there is no need for the complicated operations as SVD as in traditional pre coding algorithms.

Two competitive agents to minimize the weighted total late work and the total completion time

This paper studies deterministic constraint optimization problem with two competitive agents in which the following objective functions on a single machine: the total weighted late work and the total completion time. We show that the constraint optimization problem is the binary NP-hard by Knapsack problem reduction. Furthermore, we present a pseudo-polynomial time algorithm by early due date maximum not-late sequence, and an approximation Pareto curve by dynamic programming algorithm and two eliminated states, which time complexity of the two approximation algorithms are O(nA2nBQ∑(pjA+pjB)) and O(n4θ2logUBAlogUBB), where pj,θare processing time of job Jj, a given positive constant, and UBx an upper bound of the objective function of agent x,x∈{A,B}. Finally, we present a simple approximation algorithm by the earliest due date (EDD) rule, which jobs of agent B are assigned an dummy due date.

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

New results on multi-level aggregation

In the Multi-Level Aggregation Problem (MLAP ), requests for service arrive at the nodes of an edge-weighted rooted tree T. Each service is represented by a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs a waiting cost between its arrival and service time. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees.The currently best online algorithms for the MLAP achieve competitive ratios polynomial in the tree depth, while the best lower bound is only 3.618. In this paper, we report some progress towards closing this gap, by improving this lower bound and providing several tight bounds for restricted variants of MLAP: (1) We first study a Single-Phase variant of MLAP where all requests are released at the beginning and expire at some unknown time θ, for which we provide an online algorithm with optimal competitive ratio of 4. (2) We prove a lower bound of 4 on the competitive ratio for MLAP, even when the tree is a path. We complement this with a matching upper bound for the deadline variant of MLAP on paths.Additionally, we provide two results for the offline case: (3) We prove that the Single-Phase variant can be solved optimally in polynomial time, and (4) we give a simple 2-approximation algorithm for offline MLAP with deadlines.

An Approximation Algorithm to Maximize User Capacity for an Auto-Scaling VoD System

In a video-on-demand (VoD) service, blockbuster videos have stable and predictable popularity, but the traffic can vary significantly within short timescale. To efficiently serve the user pool in a geographic region, we consider a regional auto-scaling cloud-based data center consisting of multiple servers. For efficient storage, we partition the videos into fixed-size blocks. To respond to dynamic user traffic in a timely and cost-effective manner, we may activate or deactivate each server according to the traffic while keeping at least one replica for each block in the active servers. We maximize the user capacity of the active servers (and hence minimizing the number of active servers at any time) by jointly optimizing block allocation in the servers, server selection at each traffic level, and request dispatching to a server. We believe that this is the first work to study such problem for an auto-scaling cloud-based VoD data center. We first formulate the problem and show its NP-hardness. We then propose AVARDO ( A uto-scaling V ideo A llocation and R equest D istribution O ptimization), a simple but efficient approximation algorithm with proven optimality. AVARDO operates the servers like a stack, with a server being pushed into or popped from the existing active server set according to some optimized traffic thresholds. We prove that AVARDO approaches the theoretical optimum as the block size reduces. Trace-driven experimental results based on large-scale real-world video data further validate that AVARDO is closely optimal. It achieves significantly higher user capacity as compared with other state-of-the-art and traditional schemes, and reduces the optimality gap by multiple times.

An experimental analysis on the similarity of argumentation semantics

In this paper we ask whether approximation for abstract argumentation is useful in practice, and in particular whether reasoning with grounded semantics – which has polynomial runtime – is already an approximation approach sufficient for several practical purposes. While it is clear from theoretical results that reasoning with grounded semantics is different from, for example, skeptical reasoning with preferred semantics, we investigate how significant this difference is in actual argumentation frameworks. As it turns out, in many graphs models, reasoning with grounded semantics actually approximates reasoning with other semantics almost perfectly. An algorithm for grounded reasoning is thus a conceptually simple approximation algorithm that not only does not need a learning phase – like recent approaches – but also approximates well – in practice – several decision problems associated to other semantics.

Stable fractional matchings

We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). In this cardinal setting, stable fractional matchings can have much larger social welfare than stable integral ones. Our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) stable fractional matching. We consider both exact and approximate stability notions, and provide simple approximation algorithms with weak welfare guarantees. Our main result is that, somewhat surprisingly, achieving better approximations is computationally hard. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings in the cardinal model. En route to these results, we provide a number of structural observations that could be of independent interest.

Approaching the One-Sided Exemplar Adjacency Number Problem.

The one-sided Exemplar Adjacency Number (EAN) is a known problem for computing the exemplar similarity between a generic linear genome G with gene duplications and an exemplar genome H (over the same set of n gene families). In this problem, we need to compute an exemplar genome G, which is a permutation obtained from G, such that the number of common adjacencies between G and H is maximized. Unfortunately, the problem is not only NP-hard but also NP-hard to approximate. In this paper, we approach the problem by relaxing the constraint such that a sub-permutation G+ obtained from G does not have to include all the gene families, but still needs to have a length at least k. Hence G+ is called a pseudo-exemplar genome. Then, a slightly more general problem (One-sided EAN+) is defined: compute a pseudo-exemplar genome G+ from G such that the number of common adjacencies between H and G+ is maximized. Certainly One-sided EAN+ contains One-sided EAN as a special case; moreover, it presents some flexibility in designing algorithms. First, we relax and formulate the One-sided EAN+ problem as the maximum independent set (MIS) on a colored interval graph and hence reduce the appearance of each gene to at most two times. We show that this new relaxation is still NP-complete, though a simple factor-2 approximation algorithm can be designed; moreover, we also prove that the problem cannot be approximated within 2-ε by a local search technique. We then show that this relaxed version is fixed-parameter tractable (FPT). Second, to ensure that each gene appears in G+ at most once, we use integer linear programming (ILP) to solve this problem. Finally, we implement our algorithm and compare it with the up-to-date software GREDU, with simulated signed and unsigned genomes. It turns out that our algorithm is more stable and can process genomes of length up to 12,000 for signed genomes (while GREDU can falter on such a large signed genome and it cannot handle unsigned genomes at all).

How to pack directed acyclic graphs into small blocks

This paper studies the following variant of clustering or laying out problems for directed acyclic graphs (DAG for short), called the minimum block transfer problem. The objective of this problem is to find a partition of a node set which satisfies the following two conditions: (i) Each element (called a block) of the partition has a size that is at most B and (ii) the maximum number of external arcs among directed paths from the roots to the leaves is minimized. Here, an external arc is defined as an arc connecting two distinct blocks.This paper mainly studies the case B=2. First, we show that the problem is NP-hard even if the height of DAGs is three and its maximum indegree and outdegree are two and three, respectively. Then, we design (i) linear-time optimal algorithms for DAGs of height at most two, (ii) a very simple 2-approximation algorithm, and moreover, (iii) a (2−2∕h)-approximation algorithm for the case that the height h of the input DAG is even and the other one for odd h, whose approximation ratio is 2−2∕(h+1). As for the inapproximability of the problem, for any ε>0 and unless P = NP, we show that the problem does not admit any polynomial time (3∕2−ε)-approximation ((4∕3−ε)-approximation, resp.) algorithm if the height of the input DAGs is restricted to at most five (at least six, resp.). Also, in the case B≥3, we show the NP-hardness and prove a (3∕2−ε)-inapproximability of this case.