Strong Inapproximability Results Research Articles

Inapproximability of Counting Hypergraph Colourings

Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25 ; He et al. 23 ]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K/2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.

Escaping Braess's paradox through approximate Caratheodory's theorem

In selfish routing, Braess's paradox demonstrates the counterintuitive fact that removing a part of a network may improve the players' cost at equilibrium. In this work, we use the approximate version of Caratheodory's theorem (Barman, SICOMP 2018) and show how to efficiently approximate the best subnetwork on selfish routing instances with relatively short paths and Lipschitz continuous cost functions. Restriction to networks with short paths is necessary to escape the strong inapproximability result of (Roughgarden, JCSS 2006). We present the first polynomial-time algorithm that approximates (in a certain relaxed sense) the best subnetwork problem on such instances. Specifically, for any constant ε>0, our algorithm computes an ε-Nash flow with maximum latency at most (1+ε)L⁎+ε, where L⁎ is the equilibrium latency of the best subnetwork. Moreover, our algorithm runs in quasipolynomial time on networks with polylogarithmically long paths. As a corollary, we obtain the first polynomial-time approximation scheme for the best subnetwork in the class of random Gn,p instances proven prone to Braess's paradox by (Roughgarden and Valiant, RSA 2010) and (Chung Graham et al., RSA 2012).

Candidate selections with proportional fairness constraints

Selecting a subset of candidates with various attributes under fairness constraints has been attracting considerable attention from the AI community, with applications ranging from school admissions to committee selections. The fairness constraints are usually captured by absolute upper bounds and/or lower bounds on the number of selected candidates in specific attributes. In many scenarios, however, the total number of selected candidates is not predetermined. It is, therefore, more natural to express these fairness constraints in terms of proportions of the final selection size. In this paper, we study the proportional candidate selection problem, where the goal is to select a subset of candidates with maximum cardinality while meeting certain proportional fairness constraints. We first analyze the computational complexity of the problem and show strong inapproximability results. Next, we investigate the algorithmic aspects of the problem in two directions. First, by treating the proportional fairness constraints as soft constraints, we devise two polynomial-time algorithms that could return (near) optimal solutions with bounded violations on each fairness constraint. Second, we design an exact algorithm with a fast running time in practice. Simulations based on both synthetic and publicly available data confirm the effectiveness and efficiency of our proposed algorithms.

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

Approximation and hardness of Shift-Bribery

In the Shift-Bribery problem we are given an election, a preferred candidate, and the costs of shifting this preferred candidate up the voters' preference orders. The goal is to find such a set of shifts that ensures that the preferred candidate wins the election. We give the first polynomial-time approximation scheme for the Shift-Bribery problem for the case of positional scoring rules, and for the Copeland rule we show strong inapproximability results.

Obtaining a Proportional Allocation by Deleting Items

We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is {mathsf {W}}[3]-hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation pi in advance as part of the input, and our aim is to delete a minimum number of items such that pi is proportional in the remainder; this variant turns out to be {{mathsf {N}}}{{mathsf {P}}}-hard for six agents, but polynomial-time solvable for two agents, and we show that it is mathsf {W[2]}-hard when parameterized by the number k of

Approximation and Hardness of Shift-Bribery

In the SHIFT-BRIBERY problem we are given an election, a preferred candidate, and the costs of shifting this preferred candidate up the voters’ preference orders. The goal is to find such a set of shifts that ensures that the preferred candidate wins the election. We give the first polynomial-time approximation scheme for the case of positional scoring rules, and for the Copeland rule we show strong inapproximability results.

Cumulative activation in social networks

Most studies on influence maximization focus on one-shot propagation, i.e. the influence is propagated from seed users only once following a probabilistic diffusion model and users' activation are determined via single cascade. In reality it is often the case that a user needs to be cumulatively impacted by receiving enough pieces of information propagated to her before she makes the final purchase decision. In this paper we model such cumulative activation as the following process: first multiple pieces of information are propagated independently in the social network following the classical independent cascade model, then the user will be activated (and adopt the product) if the cumulative pieces of information she received reaches her cumulative activation threshold. Two optimization problems are investigated under this framework: seed minimization with cumulative activation (SM-CA), which asks how to select a seed set with minimum size such that the number of cumulatively active nodes reaches a given requirement $\eta$; influence maximization with cumulative activation (IM-CA), which asks how to choose a seed set with fixed budget to maximize the number of cumulatively active nodes. For SM-CA problem, we design a greedy algorithm that yields a bicriteria $O(\ln n)$-approximation when $\eta=n$, where $n$ is the number of nodes in the network. For both SM-CA problem with $\eta<n$ and IM-CA problem, we prove strong inapproximability results. Despite the hardness results, we propose two efficient heuristic algorithms for SM-CA and IM-CA respectively based on the reverse reachable set approach. Experimental results on different real-world social networks show that our algorithms significantly outperform baseline algorithms.

The synthesis problem of decentralized energy systems is strongly NP-hard

We analyze the computational complexity of the synthesis problem of decentralized energy systems. This synthesis problem consists of combining various types of energy conversion units and determining their sizing as well as operations in order to meet time-varying energy demands while maximizing an objective function, e.g., the net present value. In this paper, we prove that the synthesis problem of decentralized energy systems is strongly NP-hard. Furthermore, we prove a strong inapproximability result. This paper provides the first complexity findings in the long scientific history of the synthesis problem of decentralized energy systems.

Strong Inapproximability Results on Balanced Rainbow-Colorable Hypergraphs

Consider a K-uniform hypergraph H = (V, E). A coloring c: V ? {1, 2, ?, k} with k colors is rainbow if every hyperedge e contains at least one vertex from each color, and is called perfectly balanc...

Stable matchings of teachers to schools

Several countries successfully use centralized matching schemes for school or higher education assignment, or for entry-level labour markets. In this paper we explore the computational aspects of a possible similar scheme for assigning teachers to schools. Our model is motivated by a particular characteristic of the education system in many countries where each teacher specializes in two subjects. We seek stable matchings, which ensure that no teacher and school have the incentive to deviate from their assignments. Indeed we propose two stability definitions depending on the precise format of schools' preferences. If the schools' ranking of applicants is independent of their subjects of specialism, we show that the problem of deciding whether a stable matching exists is NP-complete, even if there are only three subjects, unless there are master lists of applicants or of schools. By contrast, if the schools may order applicants differently in each of their specialization subjects, the problem of deciding whether a stable matching exists is NP-complete even in the presence of subject-specific master lists plus a master list of schools. Finally, we prove a strong inapproximability result for the problem of finding a matching with the minimum number of blocking pairs with respect to both stability definitions.

On the computational complexity of the virtual network embedding problem

Given a graph representing a substrate (or physical) network with node and edge capacities and a set of virtual networks with node capacity demands and node-to-node traffic demands, the Virtual Network Embedding problem (VNE) calls for an embedding of (a subset of) the virtual networks onto the substrate network which maximizes the total profit while respecting the physical node and edge capacities. In this work, we investigate the computational complexity of VNE. In particular, we present a polynomial-time reduction from the maximum stable set problem which implies strong NP-hardness for VNE even for very special subclasses of graphs and yields a strong inapproximability result for general graphs. We also consider the special cases obtained when fixing one of the dimensions of the problem to one. We show that VNE is still strongly NP-hard when a single virtual network request is present or when each virtual network request consists of a single virtual node and that it is weakly NP-hard for the case with a single physical node.

Routing Regardless of Network Stability

How effective are interdomain routing protocols, such as the border gateway protocol, at routing packets? Theoretical analyses have attempted to answer this question by ignoring the packets and instead focusing upon protocol stability. To study stability, it suffices to model only the control plane (which determines the routing graph)--an approach taken in the stable paths problem. To analyse packet routing requires modelling the interactions between the control plane and the forwarding plane (which determines where packets are forwarded), and our first contribution is to introduce such a model. We then examine the effectiveness of packet routing in this model for the broad class next-hop preferences with filtering. Here each node $$v$$ v has a filtering list $$\mathcal {D}(v)$$ D ( v ) consisting of nodes it does not want its packets to route through. Acceptable paths (those that avoid nodes in the filtering list) are ranked according to the next-hop, that is, the neighbour of $$v$$ v that the path begins with. On the negative side, we present a strong inapproximability result. For filtering lists of cardinality at most one, given a network in which an equilibrium is guaranteed to exist, it is NP-hard to approximate the maximum number of packets that can be routed to within a factor of $$n^{1-\epsilon }$$ n 1 - ∈ , for any constant $$\epsilon >0$$ ∈ > 0 . On the positive side, we give algorithms to show that, in two fundamental cases, there exist activation sequences under which every packet will route. The first case is when each node's filtering list contains only itself, that is, $$\mathcal {D}(v)=\{v\}$$ D ( v ) = { v } ; this is the fundamental case in which a node does not want its packets to cycle. Moreover, every packet will be routed before the control plane reaches an equilibrium. The second case is when all the filtering lists are empty, that is, $$\mathcal {D}(v)=\emptyset $$ D ( v ) = ? . Thus, every packet will route even when the nodes do not care if their packets cycle! Furthermore, under these activation sequences, every packet will route even when the control plane has no equilibrium at all. Our positive results require the periodic application of route verification. To our knowledge, these are the first results to guarantee the possibility that all packets get routed without stability. These positive results are tight--for the general case of filtering lists of cardinality one, it is not possible to ensure that every packet will eventually route.

On the complexity of constructing minimum changeover cost arborescences

The reload cost concept refers to the cost that occurs at a vertex along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. This cost depends only on the colors of the traversed edges. Reload costs arise in various applications such as transportation networks, energy distribution networks, and telecommunications. Previous work on reload costs focuses on two problems of finding a spanning tree with minimum cost with respect to two different cost measures. In both problems the cost is associated with a set of paths from a given vertex r to all the leaves of the constructed tree. The first cost measure is the sum of the reload costs of all paths from r to the leaves. The second cost measure is the changeover cost, in which the cost of traversing a vertex by using two specific incident edges is paid only once regardless of the number of paths traversing it. The first problem is inapproximable within any polynomial time computable function of the input size [1], and the second problem is inapproximable within n1−ϵ for any ϵ>0[2]. In this paper we show that the first hardness result holds also for the second problem. Given this strong inapproximability result, we study the complexity and approximability properties of numerous special cases of this second problem. We mainly focus on bounded costs, and consider both directed and undirected graphs, bounded and unbounded number of colors, and both bounded and unbounded degree graphs. We also present polynomial time exact algorithms and an approximation algorithm for some special case. To the best of our knowledge, these are the first algorithms with a provable performance guarantee for the problem. Moreover, our approximation algorithm shows a tight bound on the approximability of the problem for a specific family of instances.

Approximating the balanced minimum evolution problem

We prove a strong inapproximability result for the Balanced Minimum Evolution Problem. Our proof also implies that the problem remains NP-hard even when restricted to metric instances. Furthermore, we give a MST-based 2-approximation algorithm for the problem for such instances.

On the approximability of clique and related maximization problems

We consider approximations of the form n 1− o(1) for the Maximum Clique problem, where n is the number of vertices in the input graph and where the “ o(1)” term goes to zero as n increases. We show that sufficiently strong negative results for such problems, which we call strong inapproximability results, have interesting consequences for exact computation. A simple sampling method underlies most of our results.

Differential approximation for optimal satisfiability and related problems

We study the differential approximability of several optimization satisfiability problems. We prove that, unless co − RP = NP , MIN SAT is not differential 1/ m 1− ε -approximable for any ε>0, where m is the number of clauses. We also prove that any differential approximation algorithm for MAX minimal vertex cover can be transformed into a differential approximation algorithm for MIN kSAT achieving the same differential performance ratio. This leads us to study the differential approximability of MAX minimal vertex cover and MIN independent dominating set. Both of them are equivalent for the differential approximation. For these problems we prove a strong inapproximability result, informally, unless P = NP , any approximation algorithm has worst-case approximation ratio equal to 0.

Hardness of approximation of the discrete time-cost tradeoff problem

We consider the discrete version of the well-known time-cost tradeoff problem for project networks, which has been extensively studied in the project management literature. We prove a strong in-approximability result with respect to polynomial time bicriteria approximation algorithms for this problem.

COMPLEXITY MEASURES FOR ASSEMBLY SEQUENCES

Our work focuses on various complexity measures for two-handed assembly sequences. For many products, there exist an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Although there has been a great deal of algorithmic success for finding feasible assembly sequences, there has been very little success towards optimizing the costs of sequences. We attempt to explain this lack of progress, by proving the inherent difficulty in finding optimal, or even near-optimal, assembly sequences. To begin, we define, "virtual assembly sequencing," a graph-theoretic problem that is a generalization of assembly sequencing, focusing on the combinatorial aspect of the family of feasible assembly sequences while temporarily separating out the specific geometric assumptions inherent to assembly sequencing. We formally prove the hardness of finding even near-optimal sequences for most cost measures ion our generalized framework. As a special case, we prove equally strong inapproximability results for the problem of scheduling with AND/OR precedence constraints. Finally, we re-introduce the geometry, and continue by realizing several of these hardness results in rather simple geometric settings. We are able to show strong inapproximability results, for example using an assembly consisting solely of unit disks in the plane.