Non-approximability Results Research Articles

Normalized information distance and the oscillation hierarchy

We study the complexity of computing the normalized information distance. We introduce a hierarchy of limit-computable functions by considering the number of oscillations. This is a function version of the difference hierarchy for sets. We show that the normalized information distance is not in any level of this hierarchy, strengthening previous nonapproximability results. As an ingredient to the proof, we demonstrate a conditional undecidability result about the independence of pairs of random strings.

Additive non-approximability of chromatic number in proper minor-closed classes

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally, for every k≥1 and 1≤β≤4/3, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4−3β)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.Furthermore, we give somewhat weaker non-approximability bound for K4k+1-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

Competitive and Collaborative Influence in Social Networks

We consider revenue maximization in viral marketing of competitive and collaborative products through social networks, focusing on the word-of-mouth effect on personal decisions in adopting products, technologies, or Internet applications. In our model, each advertiser submits its value per consumer, and its total budget. The publisher pays a selected set of users on social networks as seed nodes. It demands a payment (equal to its value) from an advertiser for each influenced node in social networks. The publisher’s revenue equals to the total payment from the influenced users minus its cost of seed nodes. In this paper, we study the efficient allocation problem of the publisher to maximize its revenue. Our work is motivated by recent extensive studies on influence models for social networks. It has been noted that the promoted products could either be competitive or complementary such as Kindle versus Nook e-reader or Kindle cover with Kindle, respectively. Our models evaluate the revenue/cost effect of marketing those types of products through a social network, focusing on the issues of revenue maximization and complementary and competitive effects of products on each other. We take the algorithmic complexity approach for revenue maximization under those models and prove NP-hardness, non-approximability results for general structures, and polynomial time algorithms and applications for special classes of networks.

Scheduling maintenance jobs in networks

We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines.We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and in the non-preemptive case we give strong non-approximability results. Furthermore, we give tight bounds on the power of preemption, that is, the maximum ratio of the values of non-preemptive and preemptive optimal solutions.Interestingly, the preemptive and the non-preemptive problem can be solved efficiently on paths, whereas we show that mixing both leads to a weakly NP-hard problem that allows for a simple 2-approximation.

On the complexity of Wafer-to-Wafer Integration

In this paper we consider the Wafer-to-Wafer Integration problem. A wafer can be seen as a p-dimensional binary vector. The input of this problem is described by m multisets (called “lots”), where each multiset contains n wafers. The output of the problem is a set of n disjoint stacks, where a stack is a set of m wafers (one wafer from each lot). To each stack we associate a p-dimensional binary vector corresponding to the bit-wise AND operation of the wafers of the stack. The objective is to maximize the total number of “1” in the n stacks. We provide m1−ϵ and p1−ϵ non-approximability results even for n=2, f(n) non-approximability for any polynomial-time computable function f, as well as a pr-approximation algorithm for any constant r. Finally, we show that the problem is FPT when parameterized by p, and we use this FPT algorithm to improve the running time of the pr-approximation algorithm.

No-Wait Flowshop Scheduling Is as Hard as Asymmetric Traveling Salesman Problem

In this paper, we study the classical no-wait flowshop scheduling problem with makespan objective (F|no-wait|Cmax in the standard three-field notation). This problem is well known to be a special case of the asymmetric traveling salesman problem (ATSP) and as such has an approximation algorithm with logarithmic performance guarantee. In this work, we show a reverse connection, we show that any polynomial time α-approximation algorithm for the no-wait flowshop scheduling problem with makespan objective implies the existence of a polynomial time α(1 + ɛ)-approximation algorithm for the ATSP for any ɛ > 0. This, in turn, implies that all nonapproximability results for the ATSP (current or future) will carry over to its special case. In particular, it follows that the no-wait flowshop problem is APX-hard, which is the first nonapproximability result for this problem.

The Server Provisioning Problem for Continuous Distributed Interactive Applications

In this paper, we study the server provisioning problem for continuous Distributed Interactive Applications (DIAs) whose application states not only change because of the operations performed by participants, but also evolve along with the passing of time. We focus on finding the locations of servers for hosting continuous DIAs, with the goals of optimizing the interactivity performance while fulfilling the consistency and fairness requirements. We show that the server provisioning problem is challenging by presenting its NP-hardness and non-approximability results under several conditions. We propose two efficient server placement algorithms and analyze their approximation ratios. The approximation ratio of the proposed M-Better algorithm is quite close to a lower bound for any polynomial-time algorithm. We also conduct experimental evaluations to compare the proposed algorithms with several baseline server placements.

Graph orientation and flows over time

Flows over time are used to model many real‐world logistic and routing problems. The networks underlying such problems—streets, tracks, etc.—are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus, the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send one‐third of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is half, which is again tight. We present more results of similar flavor and also show nonapproximability results for finding the best orientation for single and multicommodity maximum flows over time. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(3), 196–209 2015

On the Computational Intractability of Exact and Approximate Dictionary Learning

The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete dictionary from training data sets of certain signal classes for use in sparse representations has turned out to be of particular importance regarding practical signal processing applications. Most popular dictionary learning algorithms involve NP-hard sparse recovery problems in each iteration, which may give some indication about the complexity of dictionary learning but does not constitute an actual proof of computational intractability. In this technical note, we show that learning a dictionary with which a given set of training signals can be represented as sparsely as possible is indeed NP-hard. Moreover, we also establish hardness of approximating the solution to within large factors of the optimal sparsity level. Furthermore, we give NP-hardness and non-approximability results for a recent dictionary learning variation called the sensor permutation problem. Along the way, we also obtain a new non-approximability result for the classical sparse recovery problem from compressed sensing.

Complexity of conflict-free colorings of graphs

We consider conflict-free colorings of graph neighborhoods: Each vertex of the graph must be assigned a color so that for each vertex v there is at least one color appearing exactly once in the neighborhood of v. The goal is to minimize the number of used colors. We consider both the case of closed neighborhoods, when the neighborhood of a node includes the node itself, and the case of open neighborhoods when a node does not belong to its neighborhood. In this paper, we study complexity aspects of the problem. We show that the problem of conflict-free coloring of closed neighborhoods is NP-complete. Moreover, we give non-approximability results for the conflict-free coloring of open neighborhoods. From a positive point of view, both problems become tractable if parameterized by the vertex cover number or the neighborhood diversity number of the graph. We present simple algorithms which improve on existing results.

The Complexity of Debate Checking

We study probabilistic debate checking, where a silent resource-bounded verifier reads a dialogue about the membership of a given string in the language under consideration between a prover and a refuter. We consider debates of partial and zero information, where the prover is prevented from seeing some or all of the messages of the refuter, as well as those of complete information. This model combines and generalizes the concepts of one-way interactive proof systems, games of possibly incomplete information, and probabilistically checkable debate systems. We give full characterizations of the classes of languages with debates checkable by verifiers operating under simultaneous bounds of O(1) space and O(1) random bits. It turns out such verifiers are strictly more powerful than their deterministic counterparts, and allowing a logarithmic space bound does not add to this power. PSPACE and EXPTIME have zero- and partial-information debates, respectively, checkable by constant-space verifiers for any desired error bound when we omit the randomness bound. No amount of randomness can give a verifier under only a fixed time bound a significant performance advantage over its deterministic counterpart. However, randomness does seem to help verifiers with simultaneous bounds on space and time. In the case of logspace and polynomial-time verifiers, we show that logarithmic randomness is sufficient to check complete- and partial-information debates for all languages in PSPACE. Any such language can be reduced to the quantified max word problem for matrices, which allows us to present a nonapproximability result for the optimization version of this problem.

Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called balanced separator) that arises in many settings. For this problem, and variants such as the uniform sparsest cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and nonapproximability results. While no constant factor approximation algorithms are known, even APX-hardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from 1, even for $\Omega(n)$ levels of the hierarchy. This complements recent algorithmic results in Guruswami and Sinop [Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011, pp. 482--491] which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting “polynomial constraints” (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy.

Approximability of the two-stage stochastic knapsack problem with discretely distributed weights

In this paper the two-stage knapsack problem with random weights is studied under the aspect of approximability. We assume finite probability distributions for the weights and show that, unless P=NP, the so obtained problem cannot be approximated in polynomial time within a better ratio than K−1/2 (where K is the number of second-stage scenarios). We further study the special cases where in the second stage items can only be added or only be removed, but not both. Positive approximation results are given for three particular cases, namely linearly dependent first- and second-stage rewards, the polynomial scenario model and the case where the number of scenarios is assumed to be a constant. To the best of our knowledge, this is the first study of a two-stage knapsack problem under the aspect of approximability and the first time a non-approximability result has been proven for a stochastic knapsack problem of any kind.

Improved approximability and non-approximability results for graph diameter decreasing problems

In this paper, we study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G = ( V , E ) , and two positive integers D and B , the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem is to find a minimum-cardinality set F of edges to be added to G in such a way that the diameter of G + F is less than or equal to D , while the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G + F is minimized. Both problems are well known to be NP-hard, as well as approximable within O ( log n log D ) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O ( log n ) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of MCBD, which was known to be not approximable within c log n , for some constant c > 0 , unless P = NP . Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution whose diameter is optimal up to a multiplicative factor approaching 5 3 . On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter. Some of our results extend to the edge-weighted version of the problems.

Bicriteria problems to minimize maximum tardiness and due date assignment cost in various scheduling environments

We study bicriteria problems of minimizing maximum tardiness and total due date assignment cost in various scheduling environments. We assume that each job can be assigned a different due date without any restriction, and that each due date assignment cost is a non-decreasing function of the quoted due date. We settle the complexity of most of the problems studied by either proving that they are NP -hard or finding a polynomial time solution for them. We also include approximation and non-approximability results for several parallel-machine problems.

On the complexity of minimizing interference in ad-hoc and sensor networks

One of the most critical factors for lifetime and operability of ad-hoc and sensor networks is the limited amount of available energy. To this respect, minimizing the interference in the network (i.e., the overlapping of signals at network nodes) has certainly a positive effect, because it induces a reduction of the number of conflicting transmissions, and then results in an overall saving of energy consumption. Along this direction, in this paper we study the computational hardness of several interference minimization problems which arise while supporting some classic network communication patterns such as broadcasting (one-to-all), gossiping (all-to-all), and symmetric gossiping (symmetric all-to-all). In particular, concerning the non-approximability results, we prove that for any of the above communication patterns, the prominent problem of minimizing the maximum interference experienced by any node in the network is hard to approximate within better than a logarithmic factor, unless NP admits slightly superpolynomial time algorithms. On a positive side, we show that any approximation algorithm for the problem of minimizing the total transmission power assigned to the nodes in order to guarantee any of the above communication patterns, can be transformed, by maintaining the same performance ratio, into an approximation algorithm for the problem of minimizing the total interference experienced by all the nodes in the network.

General scheduling non-approximability results in presence of hierarchical communications

We investigate on the issue of minimizing the makespan (resp. the sum of the completion times) for the multiprocessor scheduling problem in presence of hierarchical communications. We consider a model with two levels of communication: interprocessor and intercluster. The processors are grouped in fully connected clusters. We propose general non-approximability results in the case where all the tasks of the precedence graph have unit execution times, and where the multiprocessor is composed of an unrestricted number of machines with l ⩾ 4 identical processors each.

Optimal due date assignment in multi-machine scheduling environments

We study two due date assignment problems in various multi-machine scheduling environments. We assume that each job can be assigned an arbitrary non-negative due date, but longer due dates have higher cost. The first problem is to minimize a cost function, which includes earliness, tardiness and due date assignment costs. In the second problem, we minimize an objective function which includes the number of tardy jobs and due date assignment costs. We settle the complexity of many of these problems by either showing that they are $\mathcal{NP}$ -hard or by providing a polynomial time solution for them. We also include approximation and non-approximability results for several parallel-machine problems.

Approximation of min–max and min–max regret versions of some combinatorial optimization problems

This paper investigates, for the first time in the literature, the approximation of min–max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min–max versions of these problems, using relationships between multi-objective and min–max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min–max regret shortest path. We also establish a fully polynomial-time approximation scheme for min–max regret spanning tree and prove that min–max regret knapsack is not at all approximable. For a non-constant number of scenarios, in which case min–max and min–max regret versions of polynomial-time solvable problems usually become strongly NP-hard, non-approximability results are provided for min–max (regret) versions of shortest path and spanning tree.

On approximation by ⊕-OBDDs

We show that for every ∨-OBDD of quasipolynomial size there is a ⊕-OBDD of quasipolynomial size computing the same function up to a small fraction of the inputs. We also prove that the final step in the approximation cannot be improved and discuss possibilities of proving non-approximability results and connections to lower bounds on matrix rigidity.