Natural Optimization Problems Research Articles

Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation and inapproximability results.

Linear feature conflation: An optimization‐based matching model with connectivity constraints

AbstractGeospatial data conflation is the process of combining multiple datasets about a geographic phenomenon to produce a single, richer dataset. It has received increased research attention due to its many applications in map making, transportation, planning, and temporal geospatial analyses, among many others. One approach to conflation, attempted from the outset in the literature, is the use of optimization‐based conflation methods. Conflation is treated as a natural optimization problem of minimizing the total number of discrepancies while finding corresponding features from two datasets. Optimization‐based conflation has several advantages over traditional methods including conciseness, being able to find an optimal solution, and ease of implementation. However, current optimization‐based conflation methods are also limited. A main shortcoming with current optimized conflation models (and other traditional methods as well) is that they are often too weak and cannot utilize the spatial context in each dataset while matching corresponding features. In particular, current optimal conflation models match a feature to targets independently from other features and therefore treat each GIS dataset as a collection of unrelated elements, reminiscent of the spaghetti GIS data model. Important contextual information such as the connectivity between adjacent elements (such as roads) is neglected during the matching. Consequently, such models may produce topologically inconsistent results. In this article, we address this issue by introducing new optimization‐based conflation models with structural constraints to preserve the connectivity and contiguity relation among features. The model is implemented using integer linear programming and compared with traditional spaghetti‐style models on multiple test datasets. Experimental results show that the new element connectivity (ec‐bimatching) model reduces false matches and consistently outperforms traditional models.

The board packing problem

We introduce the board packing problem (BoPP). In this problem we are given a rectangular board with a given number of rows and columns. Each position of the board has an integer value representing a gain, or revenue, that is obtained if the position is covered. A set of rectangles is also given, each with a given size and cost. The objective is to purchase some rectangles to place on the board so as to maximize the profit, which is the sum of the gain values of the covered cells minus the total cost of purchased rectangles. This problem subsumes several natural optimization problems that arise in practice. A mixed-integer programming model for the BoPP problem is provided, along with a proof that BoPP is NP-hard by reduction from the satisfiability problem. An evolutionary algorithm is also developed that can solve large instances of BoPP. We introduce benchmark instances and make extensive computer examinations.

Harmonizing Full and Partial Matching in Geospatial Conflation: A Unified Optimization Model

Spatial data conflation is aimed at matching and merging objects in two datasets into a more comprehensive one. Starting from the “map assignment problem” in the 1980s, optimized conflation models treat feature matching as a natural optimization problem of minimizing certain metrics, such as the total discrepancy. One complication in optimized conflation is that heterogeneous datasets can represent geographic features differently. Features can correspond to target features in the other dataset either on a one-to-one basis (forming full matches) or on a many-to-one basis (forming partial matches). Traditional models consider either full matching or partial matches exclusively. This dichotomy has several issues. Firstly, full matching models are limited and cannot capture any partial match. Secondly, partial matching models treat full matches just as partial matches, and they are more prone to admit false matches. Thirdly, existing conflation models may introduce conflicting directional matches. This paper presents a new model that captures both full and partial matches simultaneously. This allows us to impose structural constraints differently on full/partial matches and enforce the consistency between directional matches. Experimental results show that the new model outperforms conventional optimized conflation models in terms of precision (89.2%), while achieving a similar recall (93.2%).

Algorithms for flows over time with scheduling costs

Flows over time have received substantial attention from both an optimization and (more recently) a game-theoretic perspective. In this model, each arc has an associated delay for traversing the arc, and a bound on the rate of flow entering the arc; flows are time-varying. We consider a setting which is very standard within the transportation economic literature, but has received little attention from an algorithmic perspective. The flow consists of users who are able to choose their route but also their departure time, and who desire to arrive at their destination at a particular time, incurring a scheduling cost if they arrive earlier or later. The total cost of a user is then a combination of the time they spend commuting, and the scheduling cost they incur. We present a combinatorial algorithm for the natural optimization problem, that of minimizing the average total cost of all users (i.e., maximizing the social welfare). Based on this, we also show how to set tolls so that this optimal flow is induced as an equilibrium of the underlying game.

Learning from survey propagation: a neural network for MAX-E-3-SAT

Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, it suffices to get reasonably good solutions for all (or even most) instances in practice. This paper presents a new algorithm for computing approximate solutions in Θ(N) for the maximum exact 3-satisfiability (MAX-E-3-SAT) problem by using supervised learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random conjunctive normal form instances of the MAX-E-3-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art maximum satisfiability solvers, it can solve substantially larger and more complicated problems than it ever saw during training.

Reliable broadcast with trusted nodes: Energy reduction, resilience, and speed

Broadcast is an important primitive in large-scale distributed systems. One application is to enable reliable and efficient communication in large-scale networks such as sensor networks and Internet-of-Things. There is a rich study on achieving reliable broadcast under various kind of failures. In this paper, we use the notion of “trust” to improve the performance of reliable broadcast. We focus on Certified Propagation Algorithm (CPA), one of the simple algorithms that does not rely on a cryptographic infrastructure and has a proven guarantee on resilience (number of node failures tolerated).Specifically, the paper has two main contributions: (i) A new algorithm Trust-CPA – which integrates CPA with trusted nodes – has been proposed and shown to increase the resilience from the original CPA, and (ii) Natural optimization problems related to Trust-CPA have been proposed and studied as well.First, we study the location to place trusted nodes to reduce power consumption while maintaining the same level of resilience. Second, we study two optimizations problems on finding out which set of nodes to “trustify” to increase resilience and reduce speed. For the speed-related optimization problem, we present a greedy heuristic algorithm, and its efficacy has been examined using simulation. We show that our algorithm performs relatively well in geometric random graphs, an appropriate model for large-scale wireless sensor networks.

Weighted minimum backward Fréchet distance

The minimum backward Fréchet distance (MBFD) problem is a natural optimization problem for the weak Fréchet distance, a variant of the well-known Fréchet distance. In this problem, a threshold ε and two polygonal curves, T1 and T2, are given. The objective is to find a pair of walks on T1 and T2, which minimizes the union of the portions of backward movements (backtracking) while maintaining, at any time, a distance between the moving entities of at most ε. In this paper, we generalize this model to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. More specifically, each edge of T1 and T2 has an associated non-negative weight. The cost of backtracking on an edge is the Euclidean length of backward movement on that edge multiplied by the corresponding weight. The objective is to find a pair of walks that minimizes the sum of the costs on the edges of the curves, while guaranteeing that the weak traversal of the curves maintains a weak Fréchet distance of at most ε. We propose two exact algorithms, a simple algorithm with O(n4) time and space complexities and an improved algorithm whose time and space complexities are O(n2log3/2⁡n), where n is the maximum number of the edges of T1 and T2. A solution to weighted MBFD also implies a solution to the more general optimization problem in which both backward and forward movements have associated costs.

Constrained dynamic multi-objective evolutionary optimization for operational indices of beneficiation process

Operational indices optimization of beneficiation process is a dynamic optimization problem in nature. It is difficult to solve because the related dynamic models of operational indices cannot be achieved easily. Focusing on the operational indices optimization under uncertain environments in production process, this paper first formulates a constrained dynamic multi-objective optimization problem based on the collected data, which considers the changing factors in production and the constraints of operational and production indices, and takes the production indices as optimization objectives and the operational indices as decision variables. To solve the established constrained dynamic multi-objective problem, a prediction with modification mechanism based dynamic multi-objective evolutionary optimization algorithm is proposed. The algorithm first divides the population into several sub-populations and then predicts each sub-population center of new environment independently. New population is generated by Gaussian and uniform distribution based on the estimated centers to improve the convergence speed. At the same time, to ensure the population diversity, a modification strategy is adopted to detect which reference point has no individual associated and produces some individuals around it. The proposed algorithm is applied to solve the dynamic operational indices optimization problem and compared with a constrained and a modified unconstrained dynamic multi-objective optimization algorithm. The statistical results demonstrate the efficiency and effectiveness of the proposed algorithm to solve the real-world dynamic operational indices optimization problem.

Reliable Mobile Ad-Hoc Network Routing Using Firefly Algorithm

Routing in Mobile Ad-hoc NETwork (MANET) is a contemporary graph problem that is solved using various shortest path search techniques. The routing algorithms employed in modern routers use deterministic algorithms that extract an exact nondominated set of solutions from the search space. The search efficiency of these algorithms is found to have an exponential time complexity in the worst case. Moreover this problem is a multi-objective optimization problem in nature for MANET and it is required to consider changing topology layout. This study attempts to employ a formulation incorporating objectives viz., delay, hopdistance, load, cost and reliability that has significant impact on network performance. Simulation with different random topologies has been carried out to illustrate the implementation of an exhaustive search algorithm and it is observed that the algorithm could handle small-scale networks limited to 15 nodes. A random search meta-heuristic that adopts the nature of firefly swarm has been proposed for larger networks to yield an approximated non-dominated path set. Firefly Algorithm is found to perform better than the exact algorithm in terms of scalability and computational time.

A game-theoretical and cryptographical approach to crypto-cloud computing and its economical and financial aspects

Recently, crypto-cloud computing has become an interesting research area with many technical, security, commercial and financial aspects, goals and consequences. The original intention of the cloud is to link computing stations (services) for collaboration. Sharing and coordination of computational resources is important because of the activities between service providers and service requesters. Considering the cooperative functionality of crypto-cloud computing, the use of game theory in that area has became very beneficial. In the sequel, we mathematically associate that area with game theory, i.e., the bargaining and compromising of interests of various “players”, by using a game-theoretical approach which arises from networks, servers, operating systems, storage devices, etc. Further, we propose a novel efficient encryption system by using XTR (effective and compact subgroup trace representation) which has the property of semantic security. Those interactions has been constructed in the direction of how cryptographic tools can be used to address a natural optimization problem in the fields of game theory and financial economics. It is believed that game theory and its optimization is going to provide a suitable framework for the design of a crypto-cloud computing system that will be perceived as a strong technique and satisfy the needs of many participants and users of the cloud. The paper ends with a conclusion and an outlook to future studies.

From competition to complementarity

Influence maximization is a well-studied problem that asks for a small set of influential users from a social network, such that by targeting them as early adopters, the expected total adoption through influence cascades over the network is maximized. However, almost all prior work focuses on cascades of a single propagating entity or purely-competitive entities. In this work, we propose the Comparative Independent Cascade (Com-IC) model that covers the full spectrum of entity interactions from competition to complementarity. In Com-IC, users' adoption decisions depend not only on edge-level information propagation, but also on a node-level automaton whose behavior is governed by a set of model parameters, enabling our model to capture not only competition, but also complementarity, to any possible degree. We study two natural optimization problems, Self Influence Maximization and Complementary Influence Maximization , in a novel setting with complementary entities. Both problems are NP-hard, and we devise efficient and effective approximation algorithms via non-trivial techniques based on reverse-reachable sets and a novel "sandwich approximation" strategy. The applicability of both techniques extends beyond our model and problems. Our experiments show that the proposed algorithms consistently outperform intuitive baselines on four real-world social networks, often by a significant margin. In addition, we learn model parameters from real user action logs.

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

The Complexity of the Simultaneous Cluster Problem

We study clustering over multiple graphs each encoding a distinct set of similarity relationships (edges) over the same set of objects (nodes) where the aim is to identify clusters that are supported across the collection of graphs. This problem of simultaneous clustering is readily motivated by the recent deluge of datasets in several domains (including the biological sciences, social sciences, and marketing), where the same objects are repeatedly measured in different conditions, populations or time points. Whilst there has been a vast amount of heuristic work on practical simultaneous clustering problems, little is known on the theoretical side – we present theoretical results that help explain why such heuristics typically come without quantitative guarantees. We give algorithmic and complexity results for simultaneous clustering using two standard measures on clustering quality: density and connectivity. Specifically, we focus on the basic problem of finding a single cluster (rather than an entire clustering) that is simultaneously of high quality in every graph. When the quality of a cluster is its minimum density over all graphs, we show the problem is not approximable within a factor of 2 1−e , unless NP ⊆ DTIME(n). Furthermore, this problem appears very difficult even when there are just two graphs; the resulting problem is approximately as hard as the problem of finding a dense subgraph on at most k vertices. When cluster quality is a fixed connectivity requirement between terminals within the cluster, there are two natural optimization problems: a maximization version (find a good quality cluster with as many terminals as possible) and a minimization version (find a good quality cluster that is as small as possible). We show that the maximization problem is tractable in polynomial time for any fixed connectivity requirement k. On the other hand the minimization problem is hard to approximate within a factor of 2 1−e , unless NP ⊆ DTIME(n). The number of graphs in our reduction depends on n. If instead the number of graphs is fixed, we show there is an e > 0 for which the minimization problem is not approximable within g1/2−e for any fixed number g of graphs unless NP = ZPP . These hardness results for the minimization problem hold even in the simple cases where the connectivity requirement is one and there are either just two terminal nodes or every node is a terminal node. We remark that our results extend to case where more robust variants of the quality measure are used.

Parameterized complexity of k-anonymity: hardness and tractability

The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization that has been recently proposed is the $k$-anonymity. This approach requires that the rows of a table are partitioned in clusters of size at least $k$ and that all the rows in a cluster become the same tuple, after the suppression of some entries. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be APX-hard even when the records values are over a binary alphabet and $k=3$, and when the records have length at most 8 and $k=4$ . In this paper we study how the complexity of the problem is influenced by different parameters. In this paper we follow this direction of research, first showing that the problem is W[1]-hard when parameterized by the size of the solution (and the value $k$). Then we exhibit a fixed parameter algorithm, when the problem is parameterized by the size of the alphabet and the number of columns. Finally, we investigate the computational (and approximation) complexity of the $k$-anonymity problem, when restricting the instance to records having length bounded by 3 and $k=3$. We show that such a restriction is APX-hard.

Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter T) on some box of finite size L. In this article, we replace the regularized corrector (which is the solution of a problem posed on ) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error. In order to improve the convergence, one may also consider N independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing T, R, L and N in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

The complexity of Boolean formula minimization

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ 2 P -complete and indeed appears as an open problem in Garey and Johnson (1979) [5]. The depth-2 variant was only shown to be Σ 2 P -complete in 1998 (Umans (1998) [13], Umans (2001) [15]) and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth- k version is Σ 2 P -complete under Turing reductions for all k ⩾ 3 . We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ 2 P -complete under Turing reductions.

Anonymizing binary and small tables is hard to approximate

The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k=3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achieve on two restrictions of the problem, namely (i) when the records values are over a binary alphabet and k=3, and (ii) when the records have length at most 8 and k=4, showing that these restrictions of the problem are APX-hard.

Some Relations between Approximation Problems and PCPs over the Real Numbers

In [10] it was recently shown that $\mbox{\rm NP}_{\Bbb R} \subseteq \mbox{\rm PCP}_{\Bbb R}(\,{\it poly}, O(1)),$ that is the existence of transparent long proofs for $\mbox{\rm NP}_{\Bbb R}$ was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number $\mbox{\rm PCP}_{\Bbb R}$ theorem $\mbox{\rm NP}_{\Bbb R} = \mbox{\rm PCP}_{\Bbb R}(O(\log{n}), O(1))$ would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits. Our main results combine the PCP framework over ${\Bbb R}$ with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS problem.

Integrality Ratio for Group Steiner Trees and Directed Steiner Trees

The natural relaxation for the group Steiner tree problem, as well as for its generalization, the directed Steiner tree problem, is a flow-based linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the group Steiner tree problem, we show that the integrality ratio is $\Omega(\log^2 k)$, where $k$ denotes the number of groups; this holds even for input graphs that are hierarchically well-separated trees, introduced by Bartal [in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, 1996, pp. 184-193], in which case this lower bound is tight. This also applies for the directed Steiner tree problem. In terms of the number $n$ of vertices, our results for the directed Steiner problem imply an $\Omega(\frac{\log^2 n}{(\log \log n)^2})$ integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is known to be superlogarithmic but subpolynomial. Our results and techniques have been used by Halperin and Krauthgamer [in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585-594] to show comparable inapproximability results, assuming that NP has no quasi-polynomial Las Vegas algorithms. We also show algorithmically that the integrality ratio for the group Steiner tree problem is much better for certain families of instances, which helps pinpoint the types of instances (parametrized by optimal solutions to their flow-based relaxations) that appear to be most difficult to approximate.