The price of optimum in Stackelberg games on arbitrary single commodity networks and latency functions

Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [23, 43]. A Leader can decrease the coordination ratio by assigning flow αr on M, and then all Followers assign selfishly the (1 − α)r remaining flow. This is a Stackelberg Scheduling Instance (M, r, α), 0 ≤ α ≤ 1. It was shown [38] that it is weakly NP-hard to compute the optimal Leader’s strategy. For any such network M we efficiently compute the minimum portion βM of flow r> 0 needed by a Leader to induce M ’s optimum cost, as well as her optimal strategy. This shows that the optimal Leader’s strategy on instances (M, r, α ≥ βM) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden pre-sented a modification of Braess’s Paradox graph, such that no strategy controlling αr flow can induce ≤ 1α times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess’s graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [16]. Some preliminary results have also appeare

Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.Comment: 7 pages; 3 figure

Coloring random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR

Resolving Braess's Paradox in Random Networks

Braess’s paradox states that removing a part of a network may improve the players’ latency at equilibrium. In this work, we study the approximability of the best subnetwork problem for the class of random Gn,p instances proven prone to Braess’s paradox by Valiant and Roughgarden RSA ’10 (Random Struct Algorithms 37(4):495–515, 2010), Chung and Young WINE ’10 (LNCS 6484:194–208, 2010) and Chung et al. RSA ’12 (Random Struct Algorithms 41(4):451–468, 2012). Our main contribution is a polynomial-time approximation-preserving reduction of the best subnetwork problem for such instances to the corresponding problem in a simplified network where all neighbors of source s and destination t are directly connected by 0 latency edges. Building on this, we consider two cases, either when the total rate r is sufficiently low, or, when r is sufficiently high. In the first case of low r=O(n+), here n+ is the maximum degree of {s,t}, we obtain an approximation scheme that for any constant ε>0 and with high probability, computes a subnetwork and an ε-Nash flow with maximum latency at most (1+ε)L∗+ε, where L∗ is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree O(poly(lnn)) and the traffic rate is O(poly(lnlnn)), and in quasipolynomial time for average degrees up to o(n) and traffic rates of O(poly(lnn)). Finally, in the second case of high r=Ω(n+), we compute in strongly polynomial time a subnetwork and an ε-Nash flow with maximum latency at most (1+2ε+o(1))L∗

Minimizing energy below the glass thresholds

Focusing on the optimization version of the random K-satisfiability problem, the MAX-K-SAT problem, we study the performance of the finite energy version of the Survey Propagation (SP) algorithm. We show that a simple (linear time) backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states. A comparative numerical study on one of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio

Polynomial iterative algorithms for coloring and analyzing random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This lead us to propose a new algorithm able to color in polynomial time random graphs in the hard but colorable region, i.e when $c\in [c_d,c_q]$.Comment: 23 pages, 10 eps figure

The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic

We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time is quantitatively predicted, in agreement with simulations.Comment: 5 figure

Relaxation and Metastability in the RandomWalkSAT search procedure

An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte

Expression of IL-23/Th17-related cytokines in basal cell carcinoma and in the response to medical treatments

Several immune-related markers have been implicated in basal cell carcinoma (BCC) pathogenesis. The BCC inflammatory infiltrate is dominated by Th2 cytokines, suggesting a specific state of immunosuppression. In contrast, regressing BCC are characterized by a Th1 immune response with IFN-γ promoting a tumor suppressive activity. IL-23/Th17-related cytokines, as interleukin (IL)-17, IL-23 and IL-22, play a significant role in cutaneous inflammatory diseases, but their involvement in skin carcinogenesis is controversial and is poorly investigated in BCC. In this study we investigated the expression of IFN-γ, IL-17, IL-23 and IL-22 cytokines in BCC at the protein and mRNA level and their modulation during imiquimod (IMQ) treatment or photodynamic therapy (PDT). IFN-γ, IL-17, IL-23 and IL-22 levels were evaluated by immunohistochemistry and quantitative Real Time PCR in 41 histopatho-logically-proven BCCs (28 superficial and 13 nodular) from 39 patients. All BCC samples were analyzed at baseline and 19 of 41 also during medical treatment (9 with IMQ 5% cream and 10 with MAL-PDT). Association between cytokines expression and clinico-pathological variables was evaluated. Higher levels of IFN-γ, IL-17, IL-23 and IL-22 were found in BCCs, mainly in the peritumoral infiltrate, compared to normal skin, with the expression being correlated to the severity of the inflammatory infiltrate. IFN-γ production was higher in superficial BCCs compared to nodular BCCs, while IL-17 was increased in nodular BCCs. A significant correlation was found between IFN-γ and IL-17 expression with both cytokines expressed by CD4+ and CD8+ T-cells. An increase of all cytokines occurred during the inflammatory phase induced by IMQ and at the early time point of PDT treatment, with significant evidence for IFN-γ, IL-23, and IL-22. Our results confirm the role of IFN-γ and support the involvement of IL-23/Th17-related cytokines in BCC pathogenesis and in the inflammatory response during IMQ and MAL-PDT treatments