Hardness Proofs Research Articles

Influence Maximization on Undirected Graphs

We study the influence maximization problem in undirected networks, specifically focusing on the independent cascade and linear threshold models. We prove APX-hardness (NP-hardness of approximation within factor (1-τ) for some constant τ > 0) for both models, which improves the previous NP-hardness lower bound for the linear threshold model. No previous hardness result was known for the independent cascade model. As part of the hardness proof, we show some natural properties of these cascades on undirected graphs. For example, we show that the expected number of infections of a seed set S is upper bounded by the size of the edge cut of S in the linear threshold model and a special case of the independent cascade model, the weighted independent cascade model. Motivated by our upper bounds, we present a suite of highly scalable local greedy heuristics for the influence maximization problem on both the linear threshold model and the weighted independent cascade model on undirected graphs that, in practice, find seed sets that on average obtain 97.52% of the performance of the much slower greedy algorithm for the linear threshold model and 97.39% of the performance of the greedy algorithm for the weighted independent cascade model. Our heuristics also outperform other popular local heuristics, such as the degree discount heuristic by Chen et al. [7].

On the Complexity of Some Variations of Sorting by Transpositions

One of the main challenges in Computational Biology is to find the evolutionary distance between two organisms. In the field of comparative genomics, one way to estimate such distance is to find a minimum cost sequence of rearrangements (large scale mutations) needed to transform one genome into another, which is called the rearrangement distance. In the past decades, these problems were studied considering many types of rearrangements (such as reversals, transpositions, transreversals, and revrevs) and considering the same weight for all rearrangements, or different weights depending on the types of rearrangements. The complexity of the problems involving reversals, transpositions, and both rearrangements is known, even though the hardness proof for the problem combining reversals and transpositions was recently given. In this paper, we enhance the knowledge for these problems by proving that models involving transpositions alongside reversals, transreversals, and revrevs are NP-hard, considering weights w1 for reversals and w2 for the other rearrangements such that w2/w1 ≤ 1.5. In addition, we address a cost function related to the number of fragmentations caused by a rearrangement, proving that the problem of finding a minimum cost sorting sequence, considering the fragmentation cost function with some restrictions, is NP-hard for transpositions and the combination of reversals and transpositions.

Adaptive Secure Nearest Neighbor Query Processing Over Encrypted Data

Nearest neighbor query processing is a fundamental problem that arises in many fields such as spatial databases and machine learning. This article aims to address the Secure Nearest Neighbor (SNN) problem in cloud computing. Prior SNN schemes are both insecure and inefficient. In this article, we formally prove and experimentally demonstrate that the SNN scheme ASPE is actually insecure against even ciphertext only attacks. Although prior work proved that it is impossible to construct an SNN scheme even in much relaxed standard security models, we point out the flaws of the hardness proof. We propose an SNN scheme and prove that it is secure against adaptive chosen keyword attacks. Our scheme is efficient as its query processing complexity is logarithmic. To evaluate the efficiency of our SNN scheme, we implemented our scheme in C++ and compared its performance with a plain text scheme, binary scheme, and a PIR scheme on a large set of over 10 million real-world data points. Experimental results show that our scheme is fast (0.124 millisecond per query when data set size is 10 million) and scalable in terms of the number of data points.

Revenue maximization in Stackelberg Pricing Games: beyond the combinatorial setting

In a Stackelberg Pricing Game a distinguished player, the leader, chooses prices for a set of items, and the other players, the followers, each seek to buy a minimum cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Most previously studied cases of such games can be captured by a combinatorial model where we have a base set of items, some with fixed prices, some priceable, and constraints on the subsets that are feasible for each follower. In this combinatorial setting, Briest et al. and Balcan et al. independently showed that the maximum revenue can be approximated to a factor of H_ksim log k, where k is the number of priceable items. Our results are twofold. First, we strongly generalize the model by letting the follower minimize any continuous function plus a linear term over any compact subset of {mathbb {R}}^n_{ge 0}; the coefficients (or prices) in the linear term are chosen by the leader and determine her revenue. In particular, this includes the fundamental case of linear programs. We give a tight lower bound on the revenue of the leader, generalizing the results of Briest et al. and Balcan et al. Besides, we prove that it is strongly NP-hard to decide whether the optimum revenue exceeds the lower bound by an arbitrarily small factor. Second, we study the parameterized complexity of computing the optimal revenue with respect to the number k of priceable items. In the combinatorial setting, given an efficient algorithm for optimal follower solutions, the maximum revenue can be found by enumerating the 2^k subsets of priceable items and computing optimal prices via a result of Briest et al., giving time O(2^k|I|^c) where |I| is the input size. Our main result here is a W[1]-hardness proof for the case where the followers minimize a linear program, ruling out running time f(k)|I|^c unless mathsf {FPT} =mathsf {W[1]} and ruling out time |I|^{o(k)} under the Exponential-Time Hypothesis.

Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are mathsf {NP}-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440–465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of “arranged marriages” first considered by Knuth (Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Université de Montréal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an mathsf {NP}-hardness proof and fixed-parameter algorithm, or mathsf {NP}-hardness proof and mathsf {W}[1]-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440–465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.

Graph homomorphism reconfiguration and frozen H‐colorings

AbstractFor a fixed graph H, the reconfiguration problem for H‐colorings (ie, homomorphisms to H) asks: given a graph G and two H‐colorings and of G, does there exist a sequence of H‐colorings such that , , and for every and ? If the graph G is loop‐free, then this is the equivalent to asking whether it possible to transform into by changing the color of one vertex at a time such that all intermediate mappings are H‐colorings. In the affirmative, we say that reconfigures to . Currently, the complexity of deciding whether an H‐coloring reconfigures to an H‐coloring is only known when H is a clique, a circular clique, a ‐free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE‐complete when H is an odd wheel. An important notion in the study of reconfiguration problems for H‐colorings is that of a frozen H‐coloring; that is, an H‐coloring such that does not reconfigure to any H‐coloring such that . We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph G admits a frozen H‐coloring. The hardness proof involves a reduction from a constraint satisfaction problem which is shown to be nondeterministic polynomial time NP‐complete by establishing the nonexistence of a certain type of polymorphism.

The complexity of computing a robust flow

The Maximum Robust Flow problem asks for a flow on the paths of a network maximizing the guaranteed amount of flow surviving the removal of any k arcs. We point out a flaw in a previous publication that claimed NP-hardness for this problem when k=2. For the case that k is part of the input, we present a new hardness proof. We also discuss the complexity of the integral version of the problem.

The complexity of simulating local measurements on quantum systems

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity classPQMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian isPQMA[log]-complete. In this paper, we continue the study ofPQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - ThePQMA[log]-completeness result of [Ambainis, CCC 2014] requiresO(log⁡n)-local observables and Hamiltonians. We show that simulating even asingle qubitmeasurement on ground states of5-local Hamiltonians isPQMA[log]-complete, resolving an open question of Ambainis.- We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarlyPQMA[log]-complete. - We identify a flaw in [Ambainis, CCC 2014] regarding aPUQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtainPUQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): -PQMA[log]is thought of as ``slightly harder'' than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to showPQMA[log]⊆PP. This improves the containmentQMA⊆PP[Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves apromiseproblem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.

Scheduling with common due date assignment to minimize generalized weighted earliness–tardiness penalties

We investigate a single-machine common due date assignment scheduling problem with the objective of minimizing the generalized weighted earliness/tardiness penalties. The earliness/tardiness penalty includes not only a variable cost which depends upon the job earliness/tardiness but also a fixed cost for each early/tardy job. We provide an $$O(n^3)$$ time algorithm for the case where all jobs have equal processing times. Under the agreeable ratio condition, we solve the problem by formulating a series of half-product problems, which permits us to devise a fully polynomial-time approximation scheme with $$O(n^3/\epsilon )$$ time. $$\mathcal {NP}$$ -hardness proof is proved for a very special case and fast FPTASes with $$O(n^2/\epsilon )$$ running time are identified for two special cases.

Orthogonal Terrain Guarding is NP-complete

A terrain is an x-monotone polygonal curve, that is, every vertical line crosses the curve at most once. In the Terrain Guarding problem, a special case of the famous art gallery problem, one has to place at most $k$ guards on the vertices of a $n$-vertex terrain, in order to fully see it. In 2010, King and Krohn showed that Terrain Guarding is NP-hard [SODA '10, SIAM J. Comput. '11] thereby solving a long-standing open question. They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or orthogonal. Recently, Ashok et al. [SoCG'17] presented an FPT algorithm running in time $k^{O(k)}n^{O(1)}$ for Dominating Set in the visibility graphs of rectilinear terrains without 180-degree vertices. They ask if Orthogonal Terrain Guarding is in P or NP-hard. In the same paper, they give a subexponential-time algorithm running in $n^{O(\sqrt n)}$ (actually even $n^{O(\sqrt k)}$) for the general Terrain Guarding and notice that the hardness proof of King and Krohn only disproves a running time $2^{o(n^{1/4})}$ under the ETH. Hence, there is a significant gap between their $2^{O(n^{1/2} \log n)}$-algorithm and the no $2^{o(n^{1/4})}$ ETH-hardness implied by King and Krohn's result.In this paper, we adapt the gadgets of King and Krohn to rectilinear terrains in order to prove that even Orthogonal Terrain Guarding is NP-complete. Then, we show how to obtain an improved ETH lower bound of $2^{\Omega(n^{1/3})}$ by refining the quadratic reduction from Planar 3-SAT into a cubic reduction from 3-SAT. [In the conference version of this paper, we mistakenly claim a tighter lower bound.] This works for both Orthogonal Terrain Guarding and Terrain Guarding.

The Clever Shopper Problem

We investigate a variant of the so-called Internet Shopping problem introduced by Blazewicz et al. (Appl. Math. Comput. Sci. 20, 385–390, 2010), where a customer wants to buy a list of products at the lowest possible total cost from shops which offer discounts when purchases exceed a certain threshold. Although the problem is NP-hard, we provide exact algorithms for several cases, e.g. when each shop sells only two items, and an FPT algorithm for the number of items, or for the number of shops when all prices are equal. We complement each result with hardness proofs in order to draw a tight boundary between tractable and intractable cases. Finally, we give an approximation algorithm and hardness results for the problem of maximising the sum of discounts

Cooperative games with overlapping coalitions: Charting the tractability frontier

The framework of cooperative games with overlapping coalitions (OCF games), which was proposed by Chalkiadakis et al. [1], generalizes classic cooperative games to settings where agents may belong to more than one coalition. OCF games can be used to model scenarios where agents distribute resources, such as time or energy, among several tasks, and then divide the payoffs generated by these tasks in a fair and/or stable manner. As the framework of OCF games is very expressive, identifying settings that admit efficient algorithms for computing ‘good’ outcomes of OCF games is a challenging task. In this work, we put forward two approaches that lead to tractability results for OCF games. First, we propose a discretized model of overlapping coalition formation, where each agent i has a weight Wi∈N and may allocate an integer amount of weight to any task. Within this framework, we focus on the computation of outcomes that are socially optimal and/or stable. We discover that the algorithmic complexity of this task crucially depends on the amount of resources that each agent possesses, the maximum coalition size, and the pattern of communication among the agents. We identify several constraints that lead to tractable subclasses of discrete OCF games, and supplement our tractability results by hardness proofs, which clarify the role of our constraints. Second, we introduce and analyze a natural class of (continuous) OCF games—the Linear Bottleneck Games. We show that such games always admit a stable outcome, even assuming a large space of feasible deviations, and provide an efficient algorithm for computing such outcomes.

The Constant Inapproximability of the Parameterized Dominating Set Problem

A set $D$ of vertices of a graph $G$ is a dominating set if every vertex of $G$ is contained in $D$ or adjacent to some vertex of $D$. The number of vertices in a smallest dominating set of $G$ is denoted by $\gamma(G)$. We prove that, under the assumption ${FPT}\ne {W}[1]$ from parameterized complexity, for any constant $c\in \mathbb{N}^+$ and computable function $f: \mathbb{N}\to \mathbb{N}$ there is no algorithm which on every input graph $G$ finds a dominating set of size at most $c\cdot \gamma(G)$ in $f(\gamma(G))\cdot |G|^{O(1)}$ time. In other words, any constant approximation of the parameterized dominating set problem is ${W}[1]$-hard. Furthermore, assuming the exponential time hypothesis (ETH) [R. Impagliazzo and R. Paturi, J. Comput. System Sci., 62 (2001), pp. 367--375], we can even rule out the existence of a $f(\gamma(G))\cdot |G|^{{({log}\;\gamma(G))}^{{\varepsilon}/{12}}}$-time algorithm which on every input graph $G$ outputs a dominating set of size at most $\sqrt[3+\varepsilon]{{log}\; (\gamma(G))} \cdot \gamma(G)$ for every $0<\varepsilon<1$. Our hardness reduction is built on the second author's recent ${W}[1]$-hardness proof of the biclique problem [B. Lin, The parameterized complexity of $k$-Biclique, in Proceedings of the 26th Annual ACM--SIAM Symposium on Discrete Algorithms, SODA 2015 (San Diego, CA), ACM, New York, SIAM, Philadelphia, 2015, pp. 605--615]. This yields, among other things, a proof without the probabilistically checkable proof (PCP) machinery that the classic dominating set problem has no polynomial time constant approximation under ETH.

Clique-width III

M AX -C UT , E DGE D OMINATING S ET , G RAPH C OLORING , and H AMILTONIAN C YCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO 2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO 1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g ( k ) n f ( k ) on graphs of clique-width k . Fomin et al. (2010) showed that the running times cannot be improved to g ( k ) n O (1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f ( k ) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for E DGE D OMINATING S ET and M AX -C UT to n O ( k ) , and proved that these problems cannot be solved in time g ( k ) n o ( k ) unless ETH fails. Thus, prior to this work, E DGE D OMINATING S ET and M AX -C UT were known to have tight n Θ ( k ) algorithmic upper and lower bounds. In this article, we provide lower bounds for H AMILTONIAN C YCLE and G RAPH C OLORING . For H AMILTONIAN C YCLE , our lower bound g ( k ) n o ( k ) matches asymptotically the recent upper bound n O ( k ) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n Θ( k ) bounds for E DGE D OMINATING S ET , M AX -C UT , and H AMILTONIAN C YCLE , the G RAPH C OLORING problem has an upper bound of n O (2 k ) and a lower bound of merely n o (√ [4] k ) (implicit from the W[1]-hardness proof). In this article, we close the gap for G RAPH C OLORING by proving a lower bound of n 2 o ( k ) . This shows that G RAPH C OLORING behaves qualitatively different from the other three problems. To the best of our knowledge, G RAPH C OLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n .

Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

Wu et al. (2014) showed that under the small set expansion hypothesis (SSEH) there is no polynomial time approximation algorithm with any constant approximation factor for several graph width parameters, including tree-width, path-width, and cut-width (Wu et al. 2014). In this paper, we extend this line of research by exploring other graph width parameters: We obtain similar approximation hardness results under the SSEH for rank-width and maximum induced matching-width, while at the same time we show the approximation hardness of carving-width, clique-width, NLC-width, and boolean-width. We also give a simpler proof of the approximation hardness of tree-width, path-width, and cut-widththan that of Wu et al.

A parallel branch-and-bound algorithm to compute a tighter tardiness bound for preemptive global EDF

In this paper we present a parallel exact algorithm to compute an upper bound to tardiness of preemptive global EDF (G-EDF) schedulers, named harmonic bound, which has been proved to be up to 30% tighter than previously proposed bounds. Tightness is a crucial property of tardiness bounds: a too loose bound may cause a feasible soft real-time application to be mistakenly deemed unfeasible. Unfortunately, no polynomial-time algorithm is known to date to compute the harmonic bound. Although there is no proof of hardness of any sort either, the complex formula of the bound apparently provides no hints to devise algorithms with sub-exponential worst-case cost. In this paper we address this issue by proposing a parallel, exact, branch-and-bound algorithm to compute the harmonic bound, called harm-BB, which proves to be extremely fast in a large number of experiments. More specifically, we compare its execution times with those of existing polynomial-time algorithms for other known tardiness bounds on 630,000 random task sets. harm-BB outperforms, or is comparable to, the competitor algorithms in all scenarios but the ones with the highest number of processors (7 and 8) and tasks ( $$\sim $$ 50). In the latter scenarios harm-BB is indeed slower than the other algorithms; yet, it was still feasible, as it takes only about 2.8 s to compute the bound on a commodity dual-core CPU. Even better, we show that harm-BB has a high parallel efficiency, thus its execution time may be largely cut down on highly-parallel platforms.

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X_1,dots ,X_k of length at most n, the task is to determine the length of the longest common subsequence of X_1,dots ,X_k that is also strictly increasing. Especially for the case of k=2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as Longest Common Subsequence. We further strengthen this lower bound (1) to rule out {mathcal {O}}left( (nL)^{1-varepsilon }right) time algorithms for LCIS, where L denotes the solution size, (2) to rule out {mathcal {O}}left( n^{k-varepsilon }right) time algorithms for k-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

Deadlock resolution in wait-for graphs by vertex/arc deletion

A deadlock occurs in a distributed computation if a group of processes wait indefinitely for resources from each other. In this paper we study actions to be taken after deadlock detection, especially the action of searching for a small deadlock-resolution set. More precisely, given a “snapshot” graph G representing a deadlocked state of a distributed computation governed by a certain deadlock model $${\mathbb {M}}$$ , we investigate the complexity of vertex/arc deletion problems that aim at finding minimum vertex/arc subsets whose removal turns G into a deadlock-free graph (according to model $${\mathbb {M}}$$ ). Our contributions include polynomial-time algorithms and hardness proofs, for general graphs and for special graph classes. Among other results, we show that the arc deletion problem in the OR model can be solved in polynomial time, and the vertex deletion problem in the OR model remains NP-complete even for graphs with maximum degree $${\varDelta }(G) = 4$$ , but is solvable in $$O (m \sqrt{n})$$ time for graphs with $${\varDelta }(G)\le 3$$ .

Biophysical mechanisms complementing "classical" cell biology.

This overview addresses phenomena in cell- and molecular biology which are puzzling by their fast and highly coordinated way of organization. Generally, it appears that informative processes probably involved are more on the biophysical than on the classical biochemical side. The coordination problem is explained within the first part of the review by the topic of endogenous electrical phenomena. These are found e.g. in fast tissue organization and reorganization processes like development, wound healing and regeneration. Here, coupling into classical biochemical signaling and reactions can be shown by modern microscopy, electronics and bioinformatics. Further, one can follow the triggered reactions seamlessly via molecular biology till into genetics. Direct observation of intracellular electric processes is very difficult because of e.g. shielding through the cell membrane and damping by other structures. Therefore, we have to rely on photonic and photon - phonon coupling phenomena like molecular vibrations, which are addressed within the second part. Molecules normally possess different charge moieties and thus small electromagnetic (EMF) patterns arise during molecular vibration. These patterns can now be measured best within the optical part of the spectrum - much less in the lower terahertz till kHz and lower Hz part (third part of this review). Finally, EMFs facilitate quantum informative processes in coherent domains of molecular, charge and electron spin motion. This helps to coordinate such manifold and intertwined processes going on within cells, tissues and organs (part 4). Because the phenomena described in part 3 and 4 of the review still await really hard proofs we need concerted efforts and a combination of biophysics, molecular biology and informatics to unravel the described mysteries in "physics of life".

Additivity, subadditivity and linearity: Automatic continuity and quantifier weakening

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.