VCSP: Valued Constraint Satisfaction

• VCSP is a framework that integrates constraint satisfaction with optimization by using cost functions to encode both hard and soft constraints.
• It relies on algebraic invariants such as polymorphisms, fractional polymorphisms, and multimorphisms to determine the tractability and complexity of combinatorial problems.
• Techniques like linear programming relaxations and binarization reduce VCSPs to simpler forms, enabling efficient algorithm design for tractable instances.

A Valued Constraint Satisfaction Problem (VCSP) is a general framework unifying classical constraint satisfaction with discrete optimization, where each constraint is evaluated via a cost function, possibly non-Boolean and possibly “soft” (i.e., non-hard) rather than crisp feasibility conditions. The VCSP formalism encompasses a wide class of combinatorial problems, including submodular function minimization, labeling problems in vision, metric labeling, Max-CSP, Min-Cost Homomorphism, and many others. Its mathematical and computational theory centers on “constraint languages” (sets of allowable cost functions), algebraic invariants (polymorphisms, fractional polymorphisms, multimorphisms, and their associated identities), and the role of linear programming relaxations in tractability classification. Over the last decade, deep and far-reaching dichotomy results have been established for finite domains, conservative languages, and more recently for certain classes of infinite structures, yielding a comprehensive algebraic understanding of tractability and hardness in the VCSP landscape.

1. Formal Structure and Definitions

A VCSP instance over a domain $D$ is specified by a valued constraint language $$\Gamma \subseteq \bigcup_{k \geq 1} \{f : D^k \to \mathbb{Q}_+ \cup \{\infty\}\}$$, a set of variables $X = \{x_1, \ldots, x_n\}$, and a multiset of valued constraints $\{(f_i, v_i)\}_{i=1}^m$, where $v_i = (x_{i1}, ..., x_{i,k_i}) \in X^{k_i}$ and $f_i \in \Gamma$. The objective is: $$\mathrm{Opt}(I) = \min_{x \in D^n} \sum_{i=1}^m f_i(x_{|v_i})$$ where $x_{|v_i}$ is the restriction of the assignment to the coordinates in $v_i$ (Cohen et al., 2016). The feasibility aspect is controlled by the cost $\infty$, which encodes “hard” constraints. A key special case is the classical CSP, where all $f$ are $\{0, \infty\}$-valued relations.

A constraint language $\Gamma$ (or simply “language”) is any set of cost functions over $D$; an instance of VCSP$(\Gamma)$ includes only cost functions from $\Gamma$ (Kolmogorov et al., 2015, Kolmogorov et al., 2011). $\Gamma$ is tractable if every instance can be solved in polynomial time; otherwise it is NP-hard.

The study of VCSPs seeks to classify, for each fixed $\Gamma$, the computational complexity of VCSP$(\Gamma)$ in terms of algebraic properties of $\Gamma$.

2. Algebraic Invariants: Polymorphisms and Multimorphisms

Central to VCSP theory is the concept of polymorphisms and their fractional generalizations. For $k \geq 1$, an operation $f : D^k \to D$ is a polymorphism of a cost function $\varphi$ if $f(x^1, ..., x^k) \in \text{Feas}(\varphi)$ for any $x^j \in \text{Feas}(\varphi)$. A collection of such operations is denoted Pol($\Gamma$) (Cohen et al., 2016).

A fractional polymorphism of $\varphi$ is a probability distribution $\omega$ over $k$-ary operations, satisfying: $$\sum_{g} \omega(g)\,\varphi(g(x^1, ..., x^k)) \leq \frac{1}{k}\sum_{i=1}^k \varphi(x^i)$$ for all feasible $x^1, ..., x^k$ (Cohen et al., 2016, Kolmogorov et al., 2015). Fractional polymorphisms capture the “algebraic symmetries” responsible for tractability.

For general (possibly infinite) languages and cost functions, weighted polymorphisms (where weights can be negative but only for projections) and “multimorphisms” (tuples of operations satisfying certain average inequalities) play a parallel role (Kozik et al., 2014, Kolmogorov et al., 2011).

Algebraic invariants preserved under binarizations (standard dual, extended dual) are crucial: there are explicit bijections between the sets of polymorphisms and fractional polymorphisms of the original and binary-encoded languages (Cohen et al., 2016).

Table: Core Algebraic Concepts

Concept	Definition	Role in Complexity Classification
Polymorphism	Operation preserving $\text{Feas}(\varphi)$	Underlies closure and invariant properties
Fractional polymorphism	Probability distribution over operations satisfying averaging inequality	Central to LP-exactness and tractability
Multimorphism	Tuple of operations obeying sum inequalities (e.g. submodularity)	Characterizes tractable classes (e.g., STP, MJN)
Weighted polymorphism	Weighting of operations, negative for projections only	Refines algebraic tractability analysis

3. Complexity Dichotomies and Tractability Criteria

The main dichotomy results for VCSPs on finite domains establish that tractability is controlled by explicit algebraic conditions in terms of fractional polymorphisms:

• Finite-valued Languages: VCSP$(\Gamma)$ is tractable if and only if $\Gamma$ admits a binary symmetric fractional polymorphism. Under this criterion, any instance can be solved exactly by the basic linear programming relaxation (BLP) (Thapper et al., 2012, Kolmogorov, 2012).
• General-valued Languages: The presence of a cyclic fractional polymorphism of arity at least two (or, equivalently, a Siggers polymorphism) is necessary (Kozik–Ochremiak) and, when the feasibility CSP is tractable, also sufficient for tractability (Kolmogorov et al., 2015).
• Conservative Languages: If $\Gamma$ contains all unary cost functions on $D$, then tractability is characterized by the existence of a symmetric tournament pair (STP) and a majority-minority-near-unanimity (MJN) multimorphism on a specific partition of pairs of $D$; otherwise, the problem is NP-hard (Kolmogorov et al., 2011, Kolmogorov, 2010, Kolmogorov et al., 2010).

In all cases, failure of the algebraic condition enables a reduction from a canonical hard problem: e.g., if no binary symmetric fractional polymorphism exists, then VCSP$(\Gamma)$ encodes Max-Cut (Thapper et al., 2012).

These results extend to broader settings, such as Boolean domains, four-element domains (with the emergence of 1-defect chain multimorphisms) (Jonsson et al., 2011), and various forms of “surjective” and “infinite-domain” VCSPs (Fulla et al., 2017, Bodirsky et al., 2018, Bodirsky et al., 2024).

4. Linear Programming Relaxations and Algorithms

For tractable languages, the “basic LP relaxation” (BLP) is provably exact:

• BLP for VCSP: The integral optimum of VCSP$(\Gamma)$ matches the solution to BLP if and only if $\Gamma$ admits symmetric fractional polymorphisms of all arities (Thapper et al., 2012, Kolmogorov, 2012).
• The BLP introduces marginal variables for variable-value and constraint-value pairs, with constraints enforcing local consistency.
• For submodular languages (i.e., those admitting the min, max polymorphisms), this specialization reduces VCSP to submodular function minimization, solvable in strongly polynomial time (Thapper et al., 2012).

When the algebraic condition fails and the LP relaxation is not tight, VCSP$(\Gamma)$ often becomes NP-hard, and sometimes even NP-hard to approximate with a constant factor (Dalmau et al., 2016).

5. Reduction and Binarization Techniques

A significant theoretical insight is that any VCSP over a finite language can be reduced, in polynomial time, to equivalent binary VCSPs, and further to minimum-cost homomorphism problems over fixed digraphs:

• Standard dual encoding: Transforms an arbitrary VCSP instance into a binary VCSP instance over “tuple-variables,” unary constraints for local costs, and binary matching constraints enforcing variable consistency across tuples. The dual retains exactly the same algebraic invariants as the original problem (Cohen et al., 2016).
• Extended dual encoding: Allows embedding a VCSP instance as a minimum-cost graph homomorphism (MinCostHom$(H)$) to a fixed digraph $H$ with appropriate unary costs (Cohen et al., 2016). This route unifies the analysis of VCSPs with the rich theory of digraph homomorphism problems.

These reductions preserve the set of polymorphisms and, critically, establish that the complexity of VCSP$(\Gamma)$ is polynomial-time equivalent to that of its binarized or MinCostHom representation.

6. Infinite-Domain and Special Structural Cases

VCSP theory extends to infinite domains in two principal directions:

• Piecewise Linear Homogeneous (PLH) Functions: For cost functions definable via first-order logic over the rationals with linear structure, submodularity again characterizes the tractable class. Submodular PLH functions form a maximally tractable class: adding any non-submodular PLH function renders the VCSP NP-hard (Bodirsky et al., 2018, Bodirsky et al., 2019).
• Temporal VCSPs over $(\mathbb{Q},<)$: When the template is preserved by all order-preserving automorphisms, tractability is determined by a finite list of canonical fractional polymorphisms (min, max, various lex/tropical operations), with complexity dichotomy P/NP-complete (Bodirsky et al., 2024).

Other infinite-domain dichotomies, e.g., for oligomorphic permutation groups, are proved via extensions of the Galois-theoretic framework and advanced model-theoretic tools (Bodirsky et al., 2023). The existence of a “pseudo-cyclic” fractional polymorphism characterizes tractability.

7. Applications, Algorithmic Implications, and Open Directions

The VCSP framework unifies and generalizes numerous classes of optimization and feasibility problems:

• Submodular and bisubmodular minimization: Captured via appropriate multimorphisms; BLP or combinatorial algorithms solve these instances efficiently (Thapper et al., 2012).
• Max-CSP and Min-CSP generalizations: The dichotomy for soft and hard constraints over small domains (Boolean, four elements, arbitrary size) is explained via algebraic framework (Thapper et al., 2012, Jonsson et al., 2011).
• Sparsification: Certain language classes admit highly efficient (near-linear) instance sparsification, generalizing graph cut sparsification to broader VCSPs (Filtser et al., 2015).

Current research focuses on:

• Characterization of constant-factor approximability and integrality gaps, using Lipschitz symmetric polymorphism conditions (Dalmau et al., 2016).
• Extension of dichotomy results to more general infinite-domain settings, guided by algebraic principles (Bodirsky et al., 2018, Bodirsky et al., 2024).
• The meta-complexity of deciding, for a given language, the presence of the relevant polymorphism (e.g., tractable in time $O(3^{|D|/3} \cdot \text{poly})$ with exponential-time hypothesis lower bounds) (Kolmogorov, 2018).

The algebraic approach, which combines universal algebra, polyhedral combinatorics, and linear programming duality, has led to a unified and “human-verifiable” theory that explains both tractability and hardness across the full spectrum of VCSPs and their encoding reductions.

• "Binarisation for Valued Constraint Satisfaction Problems" (Cohen et al., 2016)
• "The complexity of conservative valued CSPs" (Kolmogorov et al., 2011)
• "Sparsification of Two-Variable Valued CSPs" (Filtser et al., 2015)
• "The complexity of Boolean surjective general-valued CSPs" (Fulla et al., 2017)
• "Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains" (Bodirsky et al., 2018)
• "Testing the complexity of a valued CSP language" (Kolmogorov, 2018)
• "The Complexity of General-Valued CSPs" (Kolmogorov et al., 2015)
• "The complexity of finite-valued CSPs" (Thapper et al., 2012)
• "A dichotomy theorem for conservative general-valued CSPs" (Kolmogorov, 2010)
• "The complexity of conservative finite-valued CSPs" (Kolmogorov et al., 2010)
• "Algebraic Properties of Valued Constraint Satisfaction Problem" (Kozik et al., 2014)
• "Temporal Valued Constraint Satisfaction Problems" (Bodirsky et al., 2024)
• "Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation" (Bodirsky et al., 2019)
• "Min CSP on Four Elements: Moving Beyond Submodularity" (Jonsson et al., 2011)
• "The power of linear programming for valued CSPs: a constructive characterization" (Kolmogorov, 2012)
• "The Power of Linear Programming for Valued CSPs" (Thapper et al., 2012)
• "Towards a Characterization of Constant-Factor Approximable Finite-Valued CSPs" (Dalmau et al., 2016)
• "Greed is slow on sparse graphs of oriented valued constraints" (Kaznatcheev et al., 13 Jun 2025)
• "The Complexity of Resilience Problems via Valued Constraint Satisfaction" (Bodirsky et al., 2023)