Complexity of the robust weighted independent set problems on interval graphs

This paper deals with the max-min and min-max regret versions of the maximum weighted independent set problem on interval graphswith uncertain vertex weights. Both problems have been recently investigated by Nobibon and Leus (2014), who showed that they are NP-hard for two scenarios and strongly NP-hard if the number of scenarios is a part of the input. In this paper, new complexity and approximation results on the problems under consideration are provided, which extend the ones previously obtained. Namely, for the discrete scenario uncertainty representation it is proven that if the number of scenarios $K$ is a part of the input, then the max-min version of the problem is not at all approximable. On the other hand, its min-max regret version is approximable within $K$ and not approximable within $O(\log^{1-\epsilon}K)$ for any $\epsilon>0$ unless the problems in NP have quasi polynomial algorithms. Furthermore, for the interval uncertainty representation it is shown that the min-max regret version is NP-hard and approximable within 2

Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove the following results for any sufficiently large $d$: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{d}{2} - o(d)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{d}{3} - o(d)\right)$. Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish the following hardness results for approximating Maximum Independent Set on $k$-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{k}{4} - o(k)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$. In comparison, known approximation algorithms achieve $\left(\frac{k}{2} - o(k)\right)$-approximation in polynomial time [Neuwohner, STACS 2021; Thiery and Ward, SODA 2023] and $(\frac{k}{3} + o(k))$-approximation in quasi-polynomial time [Cygan et al., SODA 2013]

Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP\not\subseteq DTIME(N^{O(loglog N)}), There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for some constant c > 0. There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for some constant c > 0. For any fixed k >= 4, there is no polynomial time algorithm that finds a partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log n)^{-c} beyond the random assignment threshold for Max-E3-SAT and Max-Ek-Set-Splitting is an exponential improvement over the previous bounds proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al.[MOS13].Comment: 23 Page

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If, moreover, each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H$, the minimum cost homomorphism problem for$H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, decide whether there exists a homomorphism of$G$to$H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes

Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the ``Max-Plus martingale,'' we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quasi-periodic Solutions of the Spatial Lunar Three-body Problem

By application of KAM theorem to Lidov-Ziglin's global study of the quadrupolar approximation of the spatial lunar three-body problem, we establish the existence of several families of quasi-periodic orbits in the spatial lunar three-body problem.Comment: 24 pages, 3 figure

On the Entropy Region of Discrete and Continuous Random Variables and Network Information Theory

We show that a large class of network information theory problems can be cast as convex optimization over the convex space of entropy vectors. A vector in 2^(n) - 1 dimensional space is called entropic if each of its entries can be regarded as the joint entropy of a particular subset of n random variables (note that any set of size n has 2^(n) - 1 nonempty subsets.) While an explicit characterization of the space of entropy vectors is well-known for n = 2, 3 random variables, it is unknown for n > 3 (which is why most network information theory problems are open.) We will construct inner bounds to the space of entropic vectors using tools such as quasi-uniform distributions, lattices, and Cayley's hyperdeterminant