Hard Satisfaction Rate (HSR)

• HSR is a rigorously defined metric that quantifies the fraction of system states fully satisfying hard constraints in optimization and decision systems.
• In Max CSPs, HSR=1 characterizes the NP-hard threshold where distinguishing full feasibility from strong infeasibility becomes intractable.
• In decentralized optimization and collective decision models, HSR empirically measures feasibility, linking performance with classic power indices and algorithmic trade-offs.

Hard Satisfaction Rate (HSR) is a rigorously defined metric and computational phenomenon measuring the ability of an optimization or collective decision system to strictly satisfy hard constraints, appearing across constraint satisfaction, distributed optimization, and collective decision models. HSR plays a central role as both an impossibility threshold in the complexity theory of constraint satisfaction problems ("hard gap at 1" in Max CSPs), as a principal feasibility rate in decentralized multi-agent algorithms, and as a quantitative satisfaction index in models of social choice, intimately related to power indices in simple games.

1. Formal Definitions and General Concepts

The concept of Hard Satisfaction Rate (HSR) quantifies the fraction or count of system states, solution attempts, or input configurations in which all hard constraints or actor preferences are satisfied. Its precise form is domain-dependent:

• Max CSP formulations: HSR corresponds to the "hard gap at 1," meaning that distinguishing fully satisfiable instances from those with at most an ε-fraction of satisfiable constraints is NP-hard for certain constraint languages—thereby equating HSR=1 with the inapproximability threshold (0712.1532).
• Multi-agent decentralized optimization: HSR $r$ is given by

$$r = \frac{1}{T}\sum_{t=1}^{T} \mathbf{1}\left[\forall\,u\in\{1,\dots,m\}: \mathcal{L}_u \leq g_u^{(t)} \leq \mathcal{U}_u\right]$$

where $g^{(t)}$ is the global plan in trial $t$ and $\mathcal{L}, \mathcal{U}$ are lower and upper hard-constraint vectors; $r$ measures the empirical frequency of full feasibility over algorithmic trials (Majumdar et al., 2023).

• Influence models and collective decision: HSR for actor $i$ in model $M$ is

$\HSR_M(i) = |\{x \in \{0,1\}^n : C_M(x) = x_i\}|$

counting profiles for which the collective decision aligns with actor $i$'s private choice (Molinero et al., 2016).

2. HSR in Constraint Satisfaction and Max CSPs

Within Max CSP, the Hard Satisfaction Rate is synonymous with the existence of a "hard gap at location 1." For a constraint language $\Gamma$ over domain $D$:

• An instance $I=(V, \mathcal{C})$ consists of variables $V$ and constraints $\mathcal{C}$, each of form $(R, (v_1,\dots,v_k))$ for $R \in \Gamma$.
• The value $\operatorname{val}(I, s)$ of assignment $s: V \rightarrow D$ is the count of satisfied constraints; $$\operatorname{OPT}(I) = \max_s \operatorname{val}(I, s)$$.
• $\maxCSP(\Gamma)$ has a hard gap at $\alpha$ if it is NP-hard to distinguish between $\operatorname{OPT}(I) \geq \alpha |\mathcal{C}|$ and $$\operatorname{OPT}(I) \leq \varepsilon |\mathcal{C}|$$ for some constant $\varepsilon < \alpha$.
• For $\alpha = 1$, HSR=1 corresponds to the “gap at 1”: it is NP-hard to decide full feasibility versus strong infeasibility, precluding any PTAS except for trivial cases (0712.1532).

Algebraically, this gap occurs as soon as the polymorphism algebra of $\Gamma$ admits only projections as term-operations (i.e., lacks any weak near-unanimity polymorphism). Examples include Max $k$-SAT, Max Cut, and Max DiCut, all having HSR=1 even when variable occurrences are bounded.

Problem Class	Constraint Type	HSR Phenomenon
Max $k$-SAT	Clause constraints	HSR=1 (no PTAS)
Max Cut	Inequality $(i\neq j)$	HSR=1 (no PTAS)
Max DiCut	Inequality $(i<j)$	HSR=1
Nonabelian group equations	Eqns over $G$	HSR=1

This formalizes the central threshold for inapproximability in discrete optimization: as soon as $\CSP(\Gamma)$ is NP-hard under the standard algebraic criterion, $\maxCSP(\Gamma)$ exhibits HSR=1, ruling out even bounded-approximation unless P=NP.

3. HSR as Empirical Metric in Decentralized Optimization

In privacy-preserving, decentralized combinatorial optimization for smart city applications, HSR is used as a main performance metric to capture the frequency of full satisfaction of hard (global) constraints across multiple trials:

• For $n$ agents, $m$ global constraint dimensions, $T$ independent trials, and admissible plan sets $P_i$ for each agent:

$$r = \frac{1}{T} \sum_{t=1}^T \mathbf{1}[\forall u: \mathcal{L}_u \leq g_u^{(t)} \leq \mathcal{U}_u]$$

• Optimization proceeds in two phases: (i) a cold-start phase using a feasibility-biased selection heuristic, and (ii) subsequent iterations with hard-constraint filtering and soft-objective minimization.
• Adaptations such as tree reorganization and rollback are triggered by constraint violations, stabilizing the system into a plan space where (by construction) HSR is maximized (Majumdar et al., 2023).

Empirical studies demonstrate that as hard-constraint "envelopes" tighten, the achievable HSR $r$ decreases, quantifying the trade-off between constraint strictness and solution diversity. Measured inefficiency costs rise as envelopes tighten. A behavioral shift in agent parameters is needed to maintain soft-cost optimality under hard-constraint satisfaction.

Scenario	Constraint Level	HSR ($r$)	Inefficiency Cost
Energy	L1 ($[1386,1388]$)	0.655	0.1006
Bike-sharing	L1 ($[-3,3]$)	0.560	0.7220
UAV Sensing	L1 ($[800,2200]$)	1.000	7.9950

Only hard-constrained runs achieve the constraint envelope; soft-constraint-only runs yield $r = 0$ (Majumdar et al., 2023).

4. HSR in Collective Decision Models and Influence Games

In opinion-leader–follower models (OLF) and their generalizations (gOLF, influence games), HSR becomes the count of actor-favorable configuration profiles:

• For any collective decision function $C_M:\{0,1\}^n \rightarrow \{0,1\}$, the satisfaction of actor $i$ is

$\HSR_M(i) = |\{x \in \{0,1\}^n : C_M(x) = x_i\}|$

• Computational analysis reveals computing $\HSR_M(i)$ is $\#P$-hard for gOLF and general influence models, but polynomial-time for strong hierarchical and star influence graphs (Molinero et al., 2016).

This measure coincides exactly with the Rae power index of actor $i$ in the associated simple game $\Gamma_M$, and up to an additive constant, with the Banzhaf value:

$\HSR_M(i) = \mathrm{Ra}_{\Gamma_M}(i) = 2^{n-1} + \mathrm{Bz}_{\Gamma_M}(i)$

Hence, HSR provides a direct technical link between social-choice satisfaction and computational game theory.

5. Algebraic Structure and Universality in HSR Hardness

The emergence of HSR=1 (hard gap at 1) in constraint satisfaction is tightly governed by the absence of weak near-unanimity polymorphisms in the constraint language's algebra (“projection-only factor” criterion). This algebraic characterization unifies the hardness of decision and optimization CSPs:

• If $\CSP(\Gamma)$ is NP-complete, then $\maxCSP(\Gamma)$ exhibits HSR=1. The underlying reductions use perfect implementation (primitive-positive definitions) and bounded-occurrence expander graphs, and are robust to variable degree constraints (0712.1532).
• The same algebraic framework connects inapproximability in constraint satisfaction to PCP theorem phenomena: the existence of any "hard" CSP template with HSR=1 implies the universality of hard gaps for all languages lacking rich polymorphisms.
• Open problems include classifying intermediate approximation thresholds (beyond the dichotomy of HSR=1 vs. presence of PTAS), tightening bounded occurrence bounds, and understanding the exact demarcation imposed by supermodularity.

6. Practical Impact and Significance

The notion of HSR underpins fundamental impossibility results in approximation complexity, delineates the true feasibility rates in decentralized and privacy-preserving optimization, and bridges algorithmic, algebraic, and game-theoretic perspectives in distributed decision making.

• In Max CSP, the HSR threshold precisely separates polynomial-time solvable cases from those that are not only NP-hard but inapproximable up to full constraint satisfaction (0712.1532).
• In decentralized algorithmics, HSR operationalizes the tension between distributed feasibility, agent privacy, and soft cost optimality in large-scale urban and infrastructure systems (Majumdar et al., 2023).
• In collective choice, HSR serves as a quantitative, actor-centric satisfaction measure, connected to classical power indices and revealing the intricate computational landscape of group influence and satisfaction (Molinero et al., 2016).

A plausible implication is that HSR provides a definitive yardstick for the realistic ambitions of algorithms in combinatorial domains, both in centralized and distributed settings, elucidating the price of strict feasibility and its algorithmic cost.

7. Connections to Related Work and Open Challenges

• The equivalence between the PCP theorem and the existence of CSPs with HSR=1 was known but the generality to all NP-hard core constraint languages is established in (0712.1532).
• Results extend and answer longstanding open questions on the hardness of satisfiable group equations and the impact of bounded variable occurrences.
• Ongoing challenges include classifying the entire landscape of approximation classes for Max CSP beyond HSR=1, pinpointing the supermodularity threshold, and systematically relating HSR to power indices across models.

HSR thus integrates hardness in logic, feasibility in optimization, and satisfaction in collective choice, serving as a potent analytic and empirical metric across computational domains.