Previous Hardness Result Research Articles

Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

We study the existence of approximate pure Nash equilibria (α-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d. Previously, it was known that d-PNE always exist, whereas nonexistence was established only for small constants, namely, for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have [Formula: see text]-PNE, which provides the first superconstant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, by deploying this technique, we are able to show that deciding whether a weighted congestion game has an [Formula: see text]-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest, and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of α-PNE, in which a certain set of players plays a specific strategy profile, is NP-hard for any [Formula: see text], even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n. We show that n-PNE always exists, matched by an almost tight nonexistence bound of [Formula: see text], which we can again transform into an NP-completeness proof for the decision problem. Funding: Supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). D. Poças was also funded by FCT via LASIGE Research Unit, ref. UIDB/00408/2020 and ref. UIDP/00408/2020.

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].

Ordered Level Planarity and Its Relationship to Geodesic Planarity, Bi-Monotonicity, and Variations of Level Planarity

We introduce and study the problem Ordered Level Planarity, which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y -monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge e is realized as a polygonal path p composed of line segments with two adjacent directions from a given set S of directions that is symmetric with respect to the origin. Our results on Ordered Level Planarity imply NP -hardness for any S with ∣S∣ ≥ 4, even if the given graph is a matching. Manhattan Geodesic Planarity is the special case where S contains precisely the horizontal and vertical directions. Katz, Krug, Rutter, and Wolff claimed that Manhattan Geodesic Planarity can be solved in polynomial time for the special case of matchings [GD’09]. Our results imply that this is incorrect unless P = NP . Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Štefankovič. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity, and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli.

An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems

We investigate the computational complexity of the Densestk-Subgraph problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with the maximum number of edges. We extend previous work on Densestk-Subgraph by studying its parameterized complexity for parameters describing the sparseness of the input graph and for parameters related to the solution size k.On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, Densestk-Subgraph becomes fixed-parameter tractable with respect to either of the parameters maximum degree of G and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for Densestk-Subgraph with respect to the combined parameter “degeneracy of G and |V|−k”.On the negative side, we find that Densestk-Subgraph is W[1]-hard with respect to the combined parameter “solution size k and degeneracy of G”. We furthermore strengthen a previous hardness result for Densestk-Subgraph (Cai, 2008) by showing that for every fixed μ, 0<μ<1, the problem of deciding whether G contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V|−k.Our positive results are obtained by an algorithmic framework that can be applied to a wide range of Fixed-Cardinality Optimization problems.

Internet routing between autonomous systems: Fast algorithms for path trading

Routing traffic on the internet efficiently has become an important research topic over the past decade. In this article we consider a generalization of the shortest path problem, the path-trading problem, which has applications in inter-domain traffic routing. When traffic is forwarded between autonomous systems (ASes), such as competing internet providers, each AS selfishly routes the traffic inside its own network. Efficient solutions to the path trading problem can lead to higher global performance in such systems, while maintaining the objectives and costs of the individual ASes. First, we extend a previous hardness result for the path trading problem. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give a theoretical explanation for that fact. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than a previously known algorithm.

Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and yet simpler (albeit using the “It Ain't Over Till It's Over” theorem) UG-hardness result for the Feedback Vertex Set problem than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. An extended abstract of this paper appeared in the Proceedings of the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2012 (APPROX'12).

Complexity of Wavelength Assignment in Optical Network Optimization

We study the complexity of a set of design problems for optical networks. Under wavelength division multiplexing (WDM) technology, demands sharing a common fiber are transported on distinct wavelengths. Multiple fibers may be deployed on a physical link. Our basic goal is to design networks of minimum cost, minimum congestion and maximum throughput. This translates to three variants in the design objectives: 1) MIN-SUMFIBER: minimizing the total cost of fibers deployed to carry all demands; 2) MIN-MAXFIBER: minimizing the maximum number of fibers per link to carry all demands; and 3) MAX-THROUGHPUT: maximizing the carried demands using a given set of fibers. We also have two variants in the design constraints: 1) CHOOSEROUTE: Here we need to specify both a routing path and a wavelength for each demand; 2) FIXEDROUTE: Here we are given demand routes and we need to specify wavelengths only. The FIXEDROUTE variant allows us to study wavelength assignment in isolation. Combining these variants, we have six design problems. Previously we have shown that general instances of the problems MIN-SUMFIBER-CHOOSEROUTE and MIN-MAXFIBER-FIXEDROUTE have no constant-approximation algorithms. In this paper, we prove that a similar statement holds for all four other problems. Our main result shows that MIN-SUMFIBER-FIXEDROUTE cannot be approximated within any constant factor unless NP-hard problems have efficient algorithms. This, together with the previous hardness result of MIN-MAXFIBER-FIXEDROUTE, shows that the problem of wavelength assignment is inherently hard by itself. We also study the complexity of problems that arise when multiple demands can be time-multiplexed onto a single wavelength (as in time-domain wavelength interleaved networking (TWIN) networks) and when wavelength converters can be placed along the path of a demand.

The complexity of properly learning simple concept classes

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes: • We show that unless NP = RP , there is no polynomial-time PAC learning algorithm for DNF formulas where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless NP = RP . Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NP -hard to learn the intersection of ℓ ⩾ 2 halfspaces by the intersection of k halfspaces for any constant k ⩾ 0 . Previous work held for the case when k = ℓ . • Assuming that NP ⊈ DTIME ( 2 n ϵ ) for a certain constant ϵ < 1 we show that it is not possible to learn size s decision trees by size s k decision trees for any k ⩾ 0 . Previous hardness results for learning decision trees held for k ⩽ 2 . • We present the first non-trivial upper bounds on properly learning DNF formulas. More specifically, we show how to learn size s DNF by DNF in time 2 O ˜ ( n log s ) . The hardness results for DNF formulas and intersections of halfspaces are obtained via specialized graph products for amplifying the hardness of approximating the chromatic number as well as applying recent work on the hardness of approximate hypergraph coloring. The hardness results for decision trees, as well as the new upper bounds, are obtained by developing a connection between automatizability in proof complexity and learnability, which may have other applications.

Microstructural changes in polyethylene-polypropylene blends as revealed by microhardness

Microhardness is used to examine the microstructural changes of a series of polyethylene (PE) /polypropylene (PP) blends in a wide composition range. This study complements previous hardness results obtained on high-density/low-density polyethylene systems. The use of isotactic polypropylene, as a blend component allows investigation of a material in which the hardness of the amorphous phase, contrary to PE, differs from zero. The influence of treatments such as crystallization of the PP-phase in the presence of molten PE, within the blend, or annealing the PE phase, while leaving the PP component unmodified, are discussed with reference to the additivity hardness values of the single componentsH PE andH PP. It is shown that the coexistence of the PP and PE phases inhibits the crystallization capability of one phase and modifies the annealing behaviour of the other phase leading, as a result, to depressedH PP andH PE values. The observed deviations ofH blend, throughout the composition range, from the additivity law of single components are quantitatively justified in the light of crystallinity changes of the PP phase and in terms of the population of modified lamellae of the PE component.